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Topological generation of classical algebraic groups

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Topological generation of classical algebraic groups

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Content TOPOLOGICAL GENERATION OF CLASSICAL ALGEBRAIC GROUPS by Spencer Gerhardt A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MATHEMATICS) August 2017 Copyright 2017 Spencer Gerhardt Acknowledgments First and foremost I would like to thank my advisor Robert Guralnick for all of his support and guidance on this dissertation. I feel very fortunate to have been able to work on the problems considered below under his supervision, and benefited greatly from his wealth of knowledge and insight on these topics, as well as many others. Besides being unfailingly generous with thoughts and comments, he always seemed to instinctively know the right way of looking at things (even if it took me a while to realize it), and provided an inspiring approach towards the practice of mathematics. In addition, I would like to thank my wife Ellen for her patience in this undertaking, and for her sense of humor and perspective along the way. Certainly it made these past five years go by very quickly. I was truly lucky to find such an encouraging and thoughtful partner. I would also like to thank my family for their continued support. Lastly, I would like to thank my fellow graduate students and the faculty at USC, who created a friendly and productive environment for doing math. ii Table of Contents Acknowledgments ii Chapter 1: Introduction 1 1 Generation of algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Topological generation results . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 Random generation of finite groups of Lie type . . . . . . . . . . . . . . . . 6 4 Generic stabilizers of special linear algebraic groups . . . . . . . . . . . . . 9 Chapter 2: Preliminaries 11 Chapter 3: Topological generation of special linear algebraic groups 16 1 Remarks on topological generation . . . . . . . . . . . . . . . . . . . . . . . 24 2 Necessary conditions for topological generation . . . . . . . . . . . . . . . . 27 Chapter 4: Applications to random generation of finite groups of Lie type 30 1 Background results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2 Topological generation and random (r,s)-generation . . . . . . . . . . . . . 32 3 Random (r,s)-generation of special linear and unitary groups . . . . . . . . 35 Chapter 5: Applications to generic stabilizers of classical algebraic groups 38 Chapter 6: Topological generation of symplectic algebraic groups 41 1 Semisimple classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2 Unipotent classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Bibliography 67 iii Chapter 1 Introduction 1 Generation of algebraic groups This thesis considers a particular type of generation problem for algebraic groups, and uses the solution of this question to provide some interesting applications to the representation theory of algebraic groups, as well as random generation of finite groups of Lie type. Before stating the problem that will be discussed below, let us briefly pause to give some back- ground information and introduce the relevant notion of generation for algebraic groups. Specifyinggenerationpropertiesofagivengroup,algebra,module,etc. isafundamental algebraic problem. Depending on the structure under consideration, the appropriate notion of generation may vary. For instance in finite groups a given collection of elements will generate the group if there is no proper subgroup containing these elements. However when considering algebraic groups defined over algebraically closed fields, this finitary notion of generation is no longer applicable. Any finite collection of elements will generate a proper subgroup defined over a finite extension of the prime field. While one could attempt to specify infinite sets of elements which generate a given algebraic group, there is another, perhaps more natural, notion of generation arising from thegroupstopologicalstructure. Recallthatanalgebraicgroupisananealgebraicvariety whose group operations are morphisms of varieties, and a subset of an algebraic group is Zariski dense ifitsclosureintheZariskitopologyistheentiregroup. Thetopologicalnotion of density suggests a new finitary concept of generation, namely topological generation, or generating a Zariski dense subgroup of an algebraic group. Let G be an algebraic group and x 1 ,...,x e be elements of G. From now on, we say x 1 ,...,x e generates G (or generates G topologically)ifÈx 1 ,...,x e Í is Zariski dense inG. Topological generation of algebraic groups and related applications will be the main area of investigation in this thesis. In order for questions of topological generation to be meaningfully posed one small issue must first be addressed. In what follows, we assume algebraic groups are defined over algebraicallyclosedfields. Whenthefieldsunderconsiderationareofpositivecharacteristic, a slight restriction is necessary for the notion of density to be significant. In particular if G is an algebraic group defined over the algebraic closure of a finite field, then the closure 1 of any finitely generated subgroup will be contained in some proper subfield subgroup, and hence will not be Zariski dense. To avoid this problem when working with algebraic groups defined over fields of positive characteristic,weassumethefieldshavetranscendencedegreeatleastoneoverthealgebraic closure of a finite field. In this context, topological generation behaves exactly in the same manner as it does in characteristic zero. In fact for the proofs of theorems it will be convenient to assume something stronger: that the algebraically closed fields under consideration are uncountable. Once the generation results have been proved over these fields, analogous results for fields of small transcendence degree are easily recovered. Hence (unless otherwise stated) an algebraic group G will be taken to be the set of k-rational points of G,where k is an uncountable algebraically closed field of characteristic pØ 0. Similar to complex Lie groups, semisimple algebraic groups can be classified in terms of root systems, with the simple algebraic groups (those having no closed, connected, normal subgroups) dividing into two cases: the classical groups (such asSL n (k),Sp 2n (k),SO n (k)), and the exceptional groups (F 4 (k), E 6 (k), E 7 (k), E 8 (k), and G 2 (k)). We will examine topological generation properties of simple algebraic groups, and more specifically of the classical groups SL n (k) and Sp 2n (k). The tools we develop will be applied to establish topological generation and related applications for other classical and exceptional groups in future work, so in many cases the necessary background results and lemmas are proved in greater generality than is strictly required for the linear and symplectic cases. The main question to be considered is as follows. Let G be a symplectic or special linear algebraic group and let C 1 ,...,C e be some specified conjugacy classes of G. What are necessary and sucient conditions for the existence of elements x 1 ,...,x e œ C 1 ,...,C e such thatÈx 1 ,...,x e Í is Zariski dense in G? Thisquestionisquitenaturalfromagroup-theoreticstandpoint,wherestrongproperties often reduce to the problem of picking elements from some fixed set of conjugacy classes which together have some specific attribute. For instance the Deligne-Simpson problem, which geometrically specifies information about the monodromy of regular linear systems, reduces to the question of finding necessary and sucient conditions for the existence of elements x 1 ,...,x e in conjugacy classes C 1 ,...,C e ofSL n (C) such thatÈx 1 ,...,x e Í is an irre- ducible subgroup of SL n (C), and x 1 x 2 ...x e = 1. While partial solutions to this problem have been found by Kostov [20] [21] [22], Simpson [34], and others, some questions still remain. Likewise, in Guralnick-Malle [15] the existence of elements x 1 ,x 2 ,x 3 in three par- ticular conjugacy classes of E 8 (k) such that x 1 x 2 x 3 = 1 is used to prove that the simple group E 8 (p) occurs as a Galois group overQ (for all good primes p). 2 Besides the natural quality of the question under consideration, we also have some specific applications in mind. The study of Zariski dense subgroups of semisimple algebraic groups provides connections between many dierent areas of mathematics, such as group theory, arithmetic geometry, and combinatorics. In particular, the appearance of Zariski densesubgroupswithsomespecialpropertyoftenleadsinsurprisingdirections. Forinstance inarecentpaperofBreuillard-Green-Guralnick-Tao[6]theexistenceofnon-abelianstrongly dense free subgroups of semisimple algebraic groups (free subgroups such that every pair of non-commutingelementsgeneratesaZariskidensesubgroup)yieldsnewresultsonexpander graphs, the random generation of finite simple groups, and leads to a strengthening of a theorem of Borel and Deligne-Sullivan about the Banach-Tarski paradox. Reflecting this philosophy (and adopting some aspects of their general approach), the solution to our topological generation question is used to establish two interesting and quite distinct applications. One concerns the representation theory of simple algebraic groups, and the other random generation of finite groups of Lie type. These applications and their relevant backgrounds will be introduced following the statement of the main theorems. 2 Topological generation results In this section let G be a classical algebraic group, C 1 ,...,C e be conjugacy classes of G, and = C 1 ◊ ...◊ C e .Wesayatuple Ê := (x 1 ,...,x e ) œ topologically generates G if Èx 1 ,...,x e Í is Zariski dense in G. Let “ i be the dimension of the largest eigenspace of C i , and recallC i is called quadratic if it has a degree two minimal polynomial. The main result for linear groups is as follows. Theorem 1. Let C 1 ,...,C e be non-central conjugacy classes of G = SL n (k),with n Ø 3. Then there exists a tuple Ê œ topologically generating G if and only if the following conditions hold: (i) q e i=1 “ i Æ n(e≠ 1); (ii) it is not the case that e=2 and C 1 ,C 2 are quadratic. Whenconsideringsymplecticgroups,thereareafewexceptionstotopologicalgeneration involving semisimple classes of small rank groups. In order to describe these exceptional cases let us fix some notation. Pick a semisimple conjugacy class C in Sp 2n (k). Assume elements xœ C have eigenspace decomposition V =ü k i=i E n i – i on the natural module, where E n i – i is an eigenspace of dimension n i , with corresponding eigenvalue – i .Wewillwrite C =ü k i=i E n i – i . Furthermore, assume E ⁄ is an eigenspace such that ⁄ ”= ±1. 3 Theorem 2. Let C 1 ,...,C e be non-central conjugacy classes of G = Sp 2n (k) such that C 1 ,...,C e are either all unipotent, or all semisimple. Furthermore, assume the field k does not have even characteristic. Then there exists a tuple Ê œ topologically generating G if and only if the following conditions hold: (i) q e i=1 “ i Æ 2n(e≠ 1); (ii) it is not the case that e=2 and C 1 ,C 2 are quadratic; (iii) dim Ø dim G+rank (G); (iv) the classes C 1 ,...C e and group G do not appear in Table 1 below. Table 1: Exceptions to topological generation in small rank symplectic groups G Number of classes C 1 ,...,C e Description of classes Sp 4 (k) 3Æ eÆ 4 C 1 ,...,C e =E 2 1 ü E 2 ≠ 1 Sp 4 (k) 3 C 1 ,C 2 =E 2 1 ü E 2 ≠ 1 ; C 3 =E 2 ⁄ ü E 2 ⁄ ≠ 1 Sp 6 (k) 3 C 1 ,C 2 ,C 3 =E 4 ±1 ü E 2 û 1 Note for linear groups we chose arbitrary conjugacy classes C 1 ,...,C e , while for sym- plectic groups we restrict to classesC 1 ,...,C e that are either all semisimple or all unipotent. In addition, for symplectic groups the theorem is stated over fields of odd characteristic (or characteristic zero). The proofs make crucial use of the maximal subgroup structures of classical algebraic groups, and symplectic groups in even characteristic have diering subgroup structures, slightly altering the results. In most cases, conditions (i) and (ii) from Theorem 2 imply that condition (iii) also holds (this is always true for linear groups, and hence this condition is not required for Theorem 1). For future use, it will be helpful to record the few instances where this statement fails (see Lemma 28 below). Table 2: Classes satisfying conditions (i),(ii) of Theorem 2, but not condition (iii) G n even/odd Description of classes C 1 and C 2 Sp 2n (k) n even C 1 =E n ±1 ü E n û 1 ; C 2 =E n ±1 ü E n≠ 2 û 1 ü E 1 ⁄ ü E 1 ⁄ ≠ 1 , Sp 2n (k) n odd C 1 =E n+1 ±1 ü E n≠ 1 û 1 ; C 2 =E n≠ 1 ±1 ü E n≠ 1 û 1 ü E 1 ⁄ ü E 1 ⁄ ≠ 1 Modulo the few small rank exceptions appearing in Table 1, the topological generation results given above are as nice as one could possibly hope for, in the sense that a generating 4 tuplealwaysexistsexceptwhenthereisaclearreasonwhyoneshouldn’t. Toexplainthis,we fix some representation-theoretic terminology. If V is a vector space and Ï :Gæ GL n (V) is a morphism which respects the group operation onG,thenV is a rationalkG-module. V isreducible ifthereexistsapropernontrivialsubspaceW µ V suchthatW isarationalkG- module,andV isirreducibleotherwise. AsubgroupH ofaclassicalgroupGisreducibleifH acts reducibly on the natural module. IfÈx 1 ,...,x e Í is a reducible subgroup ofG,thenevery Ê =(x 1 ,...,x e ) œ will topologically generate a group living inside a maximal parabolic subgroup, or the stabilizer of a non-degenerate space, and hence will not be Zariski dense. Now if condition (i) in the statements of Theorems 1 or 2 are violated, all tuples Ê œ will generate a reducible subgroup of G, and hence not generate G topologically. Similarly it is a nice linear algebra exercise to check that two quadratic elements in GL n (k) generate a reducible subgroup when dimV Ø 2, so there will be no generating tuples when condition (ii) is violated. Next, when dim < dim G + rank(G) tuples in generate groups having fixed points on the adjoint module, and again will not generateG topologically (see Lemma 17 below). So the families of exceptions to topological generation listed in Theorems 1 and 2 all appear for quite natural and easily understood reasons. Finally note that while the appearance of a single generating tuple Ê œ mayseemlike a rather specific property to search for, using the topological structure of algebraic groups it in fact reveals a general phenomenon. The existence of a tuple Ê œ topologically generating G can be used to show that generically tuples in share this property, with generic taken in the appropriate sense. In characteristic p = 0, the property of generating a Zariski dense subgroup of G is equivalent to acting irreducibly on some finite collection of modules, which is an open condition. Hence if the set of tuples in generating G is non-empty, then an open dense subset of tuples in will share this property. In positive characteristic this is not quite true, as the property of being a generating tuple is no longer an open condition. However the appearance of a generating tuple in ensures the set of tuples that fail to generate G topologically is small. To make this precise, say a property P holds generically on X if P holds on a set W µ X which contains the complement of a countable union of proper closed subvarieties. The complement of a generic set is a meagre set. Since the fields under consideration are uncountable, meagre sets are not large. Adopting this language, the existence of a single generating tuple in ensures that generically tuples in will generate G topologically, where G is an algebraic group of any characteristic. Now let us introduce the intended applications of our generation results. 5 3 Random generation of finite groups of Lie type Algebraic groups provide natural algebraic analogues to the notions of topological groups in topology, or Lie groups in geometry or analysis. In addition, they provide a natural context for understanding actions on algebraic varieties. Perhaps less obviously, algebraic groups also provide a framework for understanding the structure theory of finite groups. In particular, the classification of finite simple groups states that every finite simple group appears in one of the following four families: (i) Cyclic groups of prime order (ii) Alternating groups (iii) Groups of Lie type (iv) Twenty six sporadic groups Furthermore, ‘most’finitesimplegroupsareoftype(iii)listedabove. Groupsappearing in this class are finite analogues of the simple algebraic groups introduced earlier. More precisely by pioneering work of Steinberg, Chevalley and others from the 1950s and 60s, all groups of type (iii) can be recovered from a simple algebraic group G by taking the fixed points G F of a Steinberg endomorphism F : G æ G. For instance if G = SL n (k) and F : G æ G is the Frobenius endomorphism which sends x ‘æ x q , we can recover the finite group G F = SL n (q) by taking fixed points G F = {x œ G | F(x)= x}. The group G F is simple modulo the center. Similarly, if G = SL n (k) and ‡ : G æ G is the graph automorphism corresponding to the transpose inverse map (sending x ‘æ x ≠ tr ), then the fixed points of ‡ ¶ F : G æ G yields the unitary group SU n (q), which again is simple modulo the center. With some additional complications (especially in the case of certain exceptional groups), it can be shown that the remaining finite simple groups of Lie type can be recovered by taking fixed points of an appropriate Steinberg endomorphism. Since algebraic groups are defined over algebraically closed fields, they have many nice properties not shared by the corresponding finite groups. For instance, classical groups come equipped with an appropriate notion of Jordan form. Furthermore algebraic groups and their subgroups possess a natural notion of dimension, which can make them much easier to work with than the corresponding finite groups. In many cases, the simplified structure of algebraic groups oers a method of proof for the corresponding finite groups of Lie type. This involves working in relevant algebraic groups to establish some property, taking fixed points, and proving some variant of the 6 desiredpropertyholdsinthecorrespondingfinitegroups. Ourfirstapplicationoftopological generationproceedsexactlyinthismanner. Weconnectthenotionoftopologicalgeneration in algebraic groups to the notion of random generation in the corresponding finite groups of Lie type. This has the appealing feature of linking two fundamental probabilistic and topological, or discrete and continuous, phenomena. Furthermore, the latter question is a classicalgroup-theoreticproblem. Itwillbeshownthattheappearanceofaparticulartuple Ê œ topologically generating G can be used to prove that randomly elements of orders r,s will generate the corresponding finite groups of Lie type G F . For the present moment, let H be a finite group, J,K be subsets of H, and P(J,K)be the probability that a random pair of elements in J◊ K generate H. In other words, P(J,K)= |{(x,y)œ J◊ K :Èx,yÍ =H}| |J◊ K| Then P(H):= P(H,H) is the probability a random pair of elements in H generate the group H. The question of whetherP(H)æ 1 as |H|æŒ forH a symmetric or alternating group dates as far back as the nineteenth century. This fact was eventually proved by Dixon [8] in the 1960s using new methods of statistical group theory introduced by Erd˝ os and Turan. Dixon then conjectured the same property holds for all finite simple groups. After much work, this result was established by Kantor-Lubotsky [19] and Liebeck-Shalev [24] in the 1990s using probabilistic methods. The solution of Dixon’s conjecture led to many more fine-grained questions concerning random generation of finite simple groups. A natural refinement of the above question concerns whether two random elements of orders r,s generate G. In other words, if I t (H) are the elements of order t in H, and P r,s (H)= |{(x,y)œ I r (H)◊ I s (H):Èx,yÍ =G}| |I r (H)◊ I s (H) | when does P r,s (H) æ 1 as |H|æŒ ? This property is called random (r,s)- generation. Perhaps surprisingly given the strength of the statement, this property has also been found to hold in many cases [25], although counterexamples do exist [26], and some questions remain. In particular, by work of Liebeck-Shalev [25] it is known that finite simple classical groupshaverandom(r,s)-generationwhentherankofthegroupislargeenough(depending on the orders of r,s). However by looking at the case of random (2,3)-generation, which is 7 now completely understood, a slightly better result seems possible. In Liebeck-Shalev [26], it is shown that all finite simple classical groups have random (2,3)-generation except rank two symplectic groups. Given this fact one might expect the dependency of the rank on the orders of r,s in the above stated result to be an artifact of the method of proof, rather than an essential ingredient. In Chapter 4 we use topological generation results to prove this for linear and unitary groups, and also prove a theorem that will allow us to carry out this procedure for the remaining simple groups. For now, let us state the results and suggest how topological generation may be linked to random generation of finite groups. Inwhatfollows, letG(q):=G F forsomeSteinbergendomorphismF, andforanysubset E ofG,letE(q):=Efl G(q). Given a Steinberg endomorphismF :Gæ G, say that classes C Õ 1 ,C Õ 2 of elements of orders r,s in G are bad if (i) Õ (q):=C Õ 1 (q)◊ C Õ (q)”=ÿ (ii)thereexistsnotuple Ê Õ œ Õ topologically generating G Similarly, call classes C 1 ,C 2 of elements of orders r,s in G good if (i)( q)”=ÿ (ii)thereisatuple Ê œ topologically generating G (iii) for any bad classes C Õ 1 ,C Õ 2 of G,dim Õ < dim The next theorem uses tools of algebraic geometry to connect topological generation of classical algebraic groups to random (r,s)-generation of finite groups of Lie type. Theorem3. LetGbeaclassicalalgebraicgroup, and(r,s)beprimepowers. LetF :Gæ G be a sequence of Steinberg endomorphisms such that for each F: (i) G F contains elements of orders r and s,and (ii) there exist good classes C 1 ,C 2 in G. Then P r,s (G(q)) æ 1,as |G(q)|æŒ . Proving the existence of such good classes in SL n (k)when(r,s)”=(2,2), we establish random (r,s)-generation for special linear and unitary groups. This strengthens part of Theorem 1 in [25], and provides a new method for establishing random (r,s)-generation of simple groups of Lie type in the fixed rank case. (Proving random (r,s)-generation for prime powers r,s in G F is sucient to prove random ( r,s)-generation for primes r,s in the corresponding simple group S). 8 Theorem 4. Let G = SL n (k),and (r,s) ”=(2,2) be prime powers. Assume F : G æ G is a sequence of Steinberg endomorphims such that each G F contains elements of orders r and s. Then P r,s (G(q))æŒ ,as |G(q)|æŒ . Infact,amoregeneralversionofTheorem3canbeprovenforallsimplealgebraicgroups. Hence a characterization of the topological generation properties of conjugacy classes of prime power elements in simple algebraic groups can be used to classify random (r,s)- generation for finite simple groups of Lie type (with the fixed rank independent of the order r,s). This is a project we are currently undertaking. To conclude this introductory chapter, let us turn to the final application of topological generation involving the representation theory of simple algebraic groups. 4 Generic stabilizers of special linear algebraic groups In this section, unless otherwise stated, letG be a simple algebraic group andV be a linear G-variety such that V G =0(i.e. G has no fixed points on V). Recall the stabilizer of a point xœ V is defined G x := {gœ G |gx =x}.Wesaya generic stabilizer has a property P if there exists a non-empty open subvariety V 0 µ V such that G x shares this property for each x œ V 0 . In particular, a generic stabilizer has some given cardinality if an open subvariety of point stabilizers share this size. In characteristic p = 0, the size of a generic stabilizer for a given irreducible rational kG-module has been carefully investigated. The question of when a generic stabilizer of a simple algebraic group is finite (or trivial) is studied in works of Popov and ´ Elaˇ svili (see [30], [31], [32], [9], and[10]). Morerecently, thequestionofwhenagenericstabilizerisfinite but nontrivial (again in characteristic zero) appears in works of Bhargava, B.H. Gross, X. Wang and W.Ho (see [2], [3], [4], [5]). Furthermore the cardinality of a generic stabilizer is closely related to base sizes for algebraic groups, an area of investigation recently initiated by Burness, Guralnick and Saxl (see [7]). In what follows, we use our topological generation results to establish a bound – on the dimension of a linear G-variety V such that a generic stabilizer is trivial whenever dim V>– . This bound holds in arbitrary characteristic. Such bounds relate to recent work of Guralnick-Lawther-Liebeck-Testerman [17] on irreducible modules of simple algebraic groups. Again, we first focus our attention on the case of G =SL n (k), and in future work extend these bounds to include other simple algebraic groups. Now let V(C)= {v œ V| gv = v for some g œ C}, and let C be the collection of all semisimple conjugacy classes of prime power orders, together with all unipotent classes 9 in G. Using results from the literature [7], it is possible to show if dim V(C) < dim V for all CœC , then a generic stabilizer is trivial. Assume d elements from C generate G topologically. Applying our generation results, it is possible to show: dim V(C)Æ d d≠ 1 ·dim V +dimC< dim V whenever dimV>d·dim C. Now let C d be the following collection of conjugacy classes in C: C d := {CœC| d elements of C are required to generate G topologically} Furthermore, let – d := max {dim C | CœC d }; and – := max {– d ·d | 2Æ dÆ n}. Again making use of topological generation, it is possible to compute an upper bound for – .In this case, dim V(C)Æ –< dim V for all CœC , proving a generic stabilizer is trivial when dimV>– . For SL n (k), this leads to the following result. Theorem 5. Let G = SL n (k) and V be a linear G-variety such that V G =0.Ifdim V> 9 4 n 2 , then a generic stabilizer is trivial. Before proving topological generation results and related applications, it will be helpful to collect a few general facts about algebraic groups required for the arguments. 10 Chapter 2 Preliminaries In this chapter, let G be a simple algebraic group, C 1 ,..,C e be conjugacy classes of G, and = C 1 ◊ ...◊ C e . Noteuptoconjugacy,Ghasonlyfinitelymanyclosedmaximalsubgroups M 1 ,...,M k . Now for each M i ,let: - i := r e j=1 (C j fl M i ), - X i := {(x 1 ,...,x e )œ G◊ ...◊ G|Èx 1 ,...,x e Íœ M g i , for some gœ G} - Ï i :G◊ i æ X i be the map (g,(x 1 ,...,x e ))‘æ (x g 1 ,...,x g e ) - Y i =im(Ï i ) The following elementary observation shows if contains no tuple which generates G topologically, then µ Y i for some i. Lemma 1. Assume G is a simple algebraic group, and up to conjugacy the maximal closed subgroups of G are M 1 ,...,M k .If ”µ Y i for 1 Æ i Æ k, then there exists a tuple Ê œ topologically generating G. Proof. To begin, assume µ t gœ G M g i . Pick some tuple Ê =(x 1 ,...,x e ) œ . Then Ê g œ i for some g œ G, and (x g 1 ,...,x g e ) œ Y i . It follows that ”µ Y i ∆ ”µ t gœ G M g i . In particular, if ”µ Y 1 fi ...fi Y k ,thereexistsatuple Ê œ such that Ê/œ t gœ G M g i for 1Æ iÆ k, and hence Ê œ generates G topologically. In other words, if no tuple Ê œ generates G topologically, then µ Y 1 fi ...fi Y k . Note is an irreducible subset of G e , and each Y i is a closed subset of G e . An irreducible set contained in a finite union of closed sets must be contained in one of these sets. So if no tuple in generates G topologically, then µ Y i , for some i. Although not dicult to prove, the following lemmas provide valuable bounds on the dimensions of conjugacy classes, and certain fixed point spaces. These facts will be used on several occasions in order to prove the existence of a generating tuple in. Lemma 2. Let M i be a maximal closed subgroup of a simple algebraic group G, and let C 1 ,...,C e be conjugacy classes of G.If µ Y i , then dim Æ dim i +dim G/M i . 11 Proof. Assume µ im(Ï i )= Y i . We compute an upper bound on dim Y i . Any (x 1 ,...,x e ) g œ im(Ï i ) will be in the same orbit as (x 1 ,...,x e ) œ i , and hence their fibers will have the same dimension. So without loss of generality assume (x 1 ,...,x e ) œ i , and restrict to the map Ï Õ i :M i ◊ i æ i ,(g,(x 1 ,...,x e ))‘æ (x g 1 ,...,x g e ). Clearly im(Ï Õ i )= i , and every fiber Ï Õ≠ 1 i ((c g 1 ,...,c g e )) has dimension at least dimM i . Hence by the fiber theorem (see Corollary 4 in [12]), dim Y i Æ dim+ dim G/M i . The following is a version of Burnside’s lemma for algebraic groups. It will allow us to recasttheabovelemmainaparticularlygeometriclight,andyieldinsightintoourcondition on the largest eigenspaces. Lemma 3. Let M be a closed subgroup of an algebraic group G,with G acting on G/M. Then for xœ M, dim G/M≠ dim (G/M) x = dim C≠ dim (Cfl M) where (G/M) x is the set of fixed points of x,and C is the conjugacy class of x in G. Proof. This is Proposition 1.14 in [23]. Given its importance in what follows and relative ease, we provide the proof. Define V = {(g,Ê )œ G◊ G/M :gÊ =Ê } If fi,Ï : G◊ G/M æ G/M are the morphisms defined by fi (g,Ê )= Ê , and Ï (g,Ê )= gÊ , then V = {(g,Ê ) œ G◊ G/M : fi (g,Ê )= Ï (g,Ê )}. Hence V is a closed subvariety of G◊ G/M. For xœ M,define V x = {(x g ,Ê ):gœ G,Ê œ G/M,x g Ê =Ê }. ThenV x is a variety, and the elements in the image of the mapV x æ x G given by (x g ,Ê )æ x g have fibers of dimension dim (G/M) x ,so dim V x =dim x G +dim(G/M) x 12 On the other hand, elements in the image of the map V x æ G/M given by (x g ,Ê ) æ Ê have fibers of dimension dim (x G fl M), so dim V x =dim G/M+dim(x G fl H) The conclusion follows. Combined with Lemma 2, the preceding lemma yields the following useful bounds on fixed point spaces. Lemma 4. Let M i be a maximal closed subgroup of G,and x 1 ,...,x e be elements of the conjugacy classes C 1 ,...,C e .If q e j=1 dim (G/M i ) x j < (e≠ 1)·dim G/M i , then ”µ Y i ,. Proof. Assume µ Y i . Lemma 2 says: e ÿ j=1 dim C j Æ e ÿ j=1 dim (C j fl M i )+dim G/M i Then summing from 1 to e, Lemma 3 yields: e ÿ j=1 dim C j = e ÿ j=1 dim G/M i ≠ e ÿ j=1 dim (G/M i ) x j + e ÿ j=1 dim (C j fl M) Combining these facts: e ÿ j=1 dim (G/M) x j Ø ( e ÿ j=1 dim G/M i )≠ dim G/M i Note in particular if M i is the stabilizer of a 1-dimensional subspace of SL n (k) or Sp 2n (k), then M i is a parabolic subgroup of G, G/M i is isomorphic to projective space P 1 (V), and (G/M i ) x j is isomorphic to projective space inside the largest eigenspace of x j œ C j (with x j acting on the natural module). In the next chapter we show this implies: e ÿ j=1 dim (G/M i ) x j < (e≠ 1)·dim G/M i … e ÿ j=1 “ j Æ n(e≠ 1) Furthermore, in the special case of parabolic subgroups, we have: X i = € gœ G (M g i ◊ ...◊ M g i ) 13 (see Lemma 3.2 in [6]) and hence µ Y i if and only if µ X i . In other words, the assumption on the largest eigenspaces given as condition (i)inthe statements of Theorems 1 and 2 is equivalent to assumption that there exists a tuple Ê œ generating a subgroup which does not fix any common 1-dimensional subspace W of V. While this seems like a rather weak condition, it is equivalent to something seemingly much stronger: that generically tuples in share this property. To see this, pick some rational kG-module V, and let R d,e (V) be the set of tuples (g 1 ,...,g e )œ G e suchthatÈg 1 ,...,g e Ífixesacommond-dimensionalsubspaceofV. Similarly, let I d,e (V) be the set of tuples (g 1 ,...,g e ) œ G e such that Èg 1 ,...,g e Í fix no d-dimensional subspace of V, and I e (V):= {(g 1 ,...,g e )œ G e |Èg 1 ,...,g e Í acts irreducibly on V} The following well-known lemma shows the set of tuples fixing a commond-dimensional space is closed. This implies the set of tuples which fix no d-dimensional space is open, and that acting irreducibly on a single (and hence any finite collection) of modules is an open condition. Lemma 5. Let G be a simple algebraic group over an an algebraically closed field k,and let V be an n-dimensional rational kG-module. Then (i) R d,e (V) is a closed subvariety of G e . (ii) I d,e (V) is an open subvariety of G e (iii) I e (V) is an open subvariety of G e . Proof. Part (i) follows immediately from Lemma 11.1 in [15]. However several versions of this argument will appear in the proceeding, so we provide a proof. Note the set P d (V) of d-dimensional subspaces of V (the d-th Grassmanian of V)isa completevariety. Furthermore,GactsonP d (V)byx 1 ·x 2 (y)=x 1 (x 2 (y))andx ≠ 1 1 ·x 1 (y)=y, so the elementsx 1 ,...,x e œ G fix a pointyœ P d (V) if and only ifÈx 1 ,...,x e Í fixesyœ P d (V). Hence it suces to show the set of tuples in G e fixing a point inP d (V) is a closed subvariety of G e . Let Ï :G e ◊ P d (V)æ G e be the natural projection map. Since P d (V) is complete, Ï is a closed map. Clearly Z = {(x 1 ,...,x e ,y)œ G e ◊ P d (V) |x 1 y =x 2 y =... =x e y =y} 14 is a closed subvariety of G e ◊ P d (V), and hence Ï (Z) is closed. But this is precisely the set of tuples in G e which fix a common d-dimensional subspace of V.So R d,e (V) is a closed subvariety of G e . For part (ii), note that R d,e (V) c = I d,e (V) is the set of tuples (x 1 ,...,x e ) œ G e such that for every y œ P d (V) there exists some x i ,1Æ iÆ e, such that x i y ”= y. Finally part (iii)isimpliedby(ii), asfi 1Æ dÆ n I d,e (V)= I e (V). Lemma 5 implies I 1,e (G)fl (the set of tuples in that fix no 1-dimensional subspace) is open in, and hence the assumption on the largest eigenspaces given in the statements of Theorem 1 and 2 is equivalent to the statement that an open dense subset of tuples in fix no 1-dimensional subspace of V. This will be a crucial ingredient for establishing topological generation below. Similarreasoningallowsustoestablishanothervaluableproperty. Namely,theexistence of a single generating tuple in ensures that an open dense subset of tuples in will generate at least a subfield subgroup ofG. Note in characteristicp = 0 such proper subfield subgroups do not occur, and in this case it is possible to conclude that the appearance of a generating tuple in ensures that an open dense subset of tuples in generate G topologically. In positive characteristic topological generation is longer an open condition, but it is a generic one. Let S e (G) be the set of tuples in G e that topologically generate a group containing some subfield subgroup G(q 0 ) of G (for q 0 appropriately large), and let U be some finite collection of rational kG-modules. Finally, let fl WœU I e (W) be the set of tuples in G e that act irreducibly on every module WœU . By Lemma 5, fl WœU I r (W) is an open subset of G e . The following lemma shows generating a group containing a subfield subgroup is an open condition. Lemma 6. Let G be a simply connected classical algebraic group of rank > 2. Then there exists a finite collection of rational kG-modules U such that S e (G)=fl WœU I e (W). Proof. This follows from Theorem 11.5 and Corollary 11.4 in [15] Inparticular,S e (G)fl (thesetoftuplesintopologicallygeneratingagroupcontaining some subfield subgroup) is an open dense subset of. This will be a crucial ingredient for establishing the applications to random generation of finite groups of Lie type. We now have the necessary tools to prove the main topological generation results. 15 Chapter 3 Topological generation of special linear algebraic groups In this section, let C 1 ,...,C e be non-central conjugacy classes of the simple algebraic group G =SL n (k). Furthermore let = C 1 ◊ ...◊ C e , V be the natural module for G, and “ i be the dimension of the largest eigenspace of an element of C i . Using facts from the previous section, we prove the main topological generation result for special linear algebraic groups. Theorem 1. Let C 1 ,...,C e be non-central conjugacy classes of SL n (k),with nØ 3. Then there exists a tuple Ê œ topologically generating G if and only if the following conditions hold: (i) q e i=1 “ i Æ n(e≠ 1); (ii) it is not the case that e=2 and C 1 ,C 2 are quadratic. The “only if” direction follows without much diculty, and will be proven at the end of the chapter. The basic idea is that if either q e i=1 “ i >n(e≠ 1), or e = 2 and C 1 ,C 2 are quadratic, then tuples in will generate a reducible subgroup on the natural module. The converse direction requires considerable eort, and will take up the remainder of the following few sections. The argument will proceed by induction on the dimension of the natural module. For both the base case and the induction, it will be helpful to record some information about the maximal subgroup structure of SL n (k). This follows from more general work on the maximal closed subgroup structures of classical groups first carried out by Aschbacher [1], Liebeck-Seitz and others. In outline, the results say that the positive dimensional maximal closed subgroups of classical algebraic groups occur in four natural geometric families, and onefamilyofgroupsthatactirreduciblyandtensorindecomposablyonthenaturalmodule. The geometric families can be described in a uniform fashion, and are often labelled C 1 ≠C 4 . The members of the latter family are less easily described in a systematic fashion, but have thenicepropertythattheirconnectedcomponentisanalmostsimplegroupmoduloscalars. L¨ ubeck [28] provides useful information about these groups (in particular concerning the 16 degrees of their representations) that will be quite helpful in what follows. The classical groups in this family are sometimes labelled C 6 . In the following table, the listed structures should be read as intersecting with SL n (k) in order to obtain the appropriate maximal subgroups of SL n (k). Theorem 6. The maximal positive-dimensional closed subgroups of SL n (k) contained in C 1 fi ...fiC 4 fiC 6 have one of the following forms. Maximal subgroups in C 1 fi ...fiC 4 fiC 6 in SL n (k) Class structure conditions rank C 1 P m 1Æ mÆ n≠ 1 n≠ 2 C 2 GL m ÓS t n =mt,tØ 2 t(m≠ 1) C 4 GL n 1 ¢ GL n≠ 2 n =n 1 n 2 ,2Æ n 1 <n 2 n 1 +n 2 ≠ 2 (¢ t i=1 GL m ).S t n =m t , mØ 3,tØ 2 t(m≠ 1) C 6 Sp n n even n 2 SO n p”=2 Â n 2 Ê Proof. See Proposition 18.13 in [29]. The groups in C 1 are maximal parabolic subgroups of SL n (k). They stabilize an m - dimensional subspace of V,with1 Æ m Æ n≠ 1. The groups in C 2 are irreducible and imprimitive: they fix no nontrivial proper subspace, but permute t subspaces of dimension m,where n =mt. The groups in class C 4 preserve or permute a tensor decomposition. The groups in C 6 are discussed above. For the base case, it will be convenient to record the maximal closed subgroup structure of G =SL 3 (k) and dimensions of non-central conjugacy classes in G. Corollary 1. Assume M i is a maximal closed subgroup of SL 3 (k). Up to conjugacy M i has one of three possible forms: (i) M i is irreducible and primitive. In this case, M i is a finite group, or M i ≥ =SO 3 (k). (ii) M i is irreducible and imprimitive. In this case, M i is the normalizer of a torus GL 1 (k)ÓS 3 . (iii) M i is reducible. In this case M i stabilizes a line or a hyperplane. Lemma 7. Let C be a non-central conjugacy class of SL 3 (k). Then dim C =4 if C is quadratic, and dim C=6 otherwise. 17 Proof. Fix a non-central conjugacy class C inSL 3 (k). Pick xœ C, and consider its Jordan decomposition x =x s x u ,where x s , x u are commuting semisimple and unipotent elements. Then up to conjugacy x has one of the following forms: (i) x u =J 3 and x s =⁄I,⁄ 3 = 1. In this case x is not quadratic, and will centralize elements of the form y = S W W W U ab c 0 ab 00 a T X X X V with a 3 = 1. Hence dim C=8≠ 2 = 6. (ii) x u =J 2 ü J 1 and x s =I. In this situation, x is quadratic and centralizes determinant one matrices of the form y = S W W W U ab c 0 a 0 d 0 e T X X X V ,sodim C=8≠ 4 = 4. (iii) x u =J 2 ü J 1 and x s =diag[⁄,⁄,⁄ ≠ 2 ], ⁄ ”= 1. Elements commuting withx have the formy = S W W W U ab 0 0 a 0 00 a ≠ 1 T X X X V ,sodimC=8≠ 2 = 6. Since ⁄ ”= 1, x is not quadratic. (iv) x u =I and x s =diag[⁄ 1 ,⁄ 2 ,(⁄ 1 ⁄ 2 ) ≠ 1 ], with ⁄ 1 ,⁄ 2 ,(⁄ 1 ⁄ 2 ) ≠ 1 distinct. In this case x is semisimple, non-quadratic, and C G (x) is a maximal torus. So dim C=8≠ 2 = 6. (v) x u =I and x s =diag[⁄,⁄,⁄ ≠ 2 ]. Finally, in this situation x is quadratic and its centralizer is GL 2 (k). So the dimension of its conjugacy class C is 8≠ 4 = 4. 18 Now let M i =P 1 be the stabilizer of a 1-dimensional subspace of V. G acts transitively on 1-spaces, so G/M i is isomorphic to projective space P 1 (V). Furthermore for parabolic subgroups, X i = t gœ G (M g i ◊ ...◊ M g i ) (see Lemma 3.2 in [6]). Hence if there is a tuple in fl X c i , then some tuple in fixes no 1-dimensional subspace of V, and by Lemma 5 generically tuples in will share this property. This will be a crucial fact for establishing both the base and inductive cases below. Lemma 8. Let C 1 ,...,C e be conjugacy classes of SL n (k),with “ i the dimension of the largest eigenspace of C i .Assumethat q e j=1 “ j Æ n(e≠ 1).If M i is a stabilizer of a 1-space or a hyperplane, then ”µ X i . Proof. AssumeM i =P 1 ,and µ X i .WhenM i isaparabolicsubgroup, µ X i … µ Y i . Applying Lemma 4, e ÿ j=1 dim (G/M i ) x j Ø (e≠ 1)·dim G/M i In P 1 (V), we have v≥ –v for all – œ k ◊ , and hence P(V) x j = {v |x j ·v = –v } =fi r i=1 E – i , where E – i are the eigenspaces of x j viewed inside projective space. In other words, dim (G/M i ) x j is equal to the dimension of projective space inside the largest eigenspace of x j . Hence dim (G/M i ) x j =“ j ≠ 1, and dim G/M i =n≠ 1. Applying this information to the above inequality yields ( q e j=1 “ j )≠ eØ (e≠ 1)(n≠ 1). This in turn implies q e j=1 “ j Ø n(e≠ 1)+1, contradicting our hypothesis on the largest eigenspaces. NextnotethetransposeinversemapÏ :Gæ Gwhichsendsg‘æ g ≠ tr isanisomorphism between P 1 and P n≠ 1 . Since this map is an outer automorphism of SL n (k), we have SL n (k)/P n≠ 1 ≥ = SL n (k)/P 1 ≥ = P 1 (V), and the above argument can be repeated to show M i is not the stabilizer of a hyperplane. NoteintheprecedinglemmanorestrictionwasplacedontheconjugacyclassesC 1 ,...,C e of SL n (k). This was due to the fact that only 1-dimensional subspaces of V were being considered. When moving to the stabilizers of 2-dimensional or higher subspaces, dierent arguments will be required to compute (G/M i ) x j depending on whether xœ C j is semisim- ple, unipotent, or mixed. This issue will arise when considering symplectic groups below. We are now in a position to prove the base case of Theorem 1. Lemma 9. Let C 1 ,..,C e be non-central conjugacy classes of SL 3 (k).Assume q “ i Æ n(e≠ 1),anditisnotthecasethat e=2 and C 1 ,C 2 are quadratic. Then there is a tuple Ê œ topologically generating SL 3 (k). 19 Proof. Up to conjugacy, let M 1 ,...,M k be the maximal closed subgroups of SL 3 (k). Assume it is not the case that q e j=1 “ j >n(e≠ 1), or e = 2 and C 1 , C 2 are quadratic. Applying Lemma 1, it suces to show ”µ Y i for 1Æ iÆ k. By Corollary 1, it suces to consider the following cases: (i) M i is finite. Pick some class C j from C 1 ,...,C e . By Lemma 7, dim C j =4if C j is quadratic, and dim C j = 6 otherwise. Under the assumptions placed on C 1 ,...,C e in Lemma 9 it follows that dim > 8, and hence by Lemma 2 that ”µ Y i . M i = SO 3 (k). This maximal subgroup comes from the natural action of SL 2 (k) on the second symmetric power. However to understand how the elements of M i appear in SL 3 (k), it is convenient to view M i as SO 3 (k) on the natural module. Again pick a class C j ,where1Æ jÆ e, and assume xœ C j occurs in some conjugate of M i . Consider the Jordan form of x on the natural module for SO 3 (k). Note even sized Jordan blocks come in pairs, and any eigenvalue – ”= ±1 of x comes paired with another eigenvalue – ≠ 1 . It follows that dim (C j fl SO 3 (k))”=0implies either C j is a unipotent class with Jordan form J 3 , or C j is a semisimple class. In either of these situations an element x œ C j fl SO 3 (k) centralizes a 1-dimensional subgroup in SO 3 (k), so dim (C j fl SO 3 (k))Æ 2. As dim G/M i = 5, it follows that dim > dim i +dim G/M i , given the assumptions placed on C 1 ,...,C e in the statement of Lemma 9. (ii) M i is the normalizer of a torus. A maximal torus T in SL 3 (k) is two dimensional. Since N G (T)/C G (T) is finite and C G (T)=T, we have that dim (M i fl C j )Æ 2. If µ Y i , it follows that dim Æ 2e+6. By Lemma 7, this inequality holds only when (a) e = 2 and at most one class is non-quadratic, or (b) e = 3 and all classes are quadratic. However, when C j is quadratic, N G (T)fl C j is finite except in the case when C j is a class of involutions. In this situation, N G (T)fl C j is a union of outer involutions in N G (T) (there are two such classes if the characteristic p”= 2, and one if p = 2). These outer involutions have a one dimensional centralizer in N G (T), and hence if C j is quadratic dim N G (T)fl C j Æ 1. So dim > dim i +dim G/M i under the assumptions placed on C 1 ,...,C e . 20 (iii) Applying Lemma 8, M i can not be the stabilizer of a 1-space or a hyperplane. To conclude the proof of Theorem 1, we argue by induction that there is a tuple Ê œ topologically generating a large rank subgroup with a few special properties. To argue in this fashion, it is necessary to restrict the classes C 1 ,...,C e of SL n (k) to classes C Õ 1 ,...,C Õ e of SL n≠ 1 (k) and check the inequality q e j=1 “ Õ j Æ (n≠ 1)(e≠ 1) still holds (where “ Õ j is the dimension of the largest eigenspace in the conjugacy class C Õ j ). To begin, pick a class C j and some element x = x s x u œ C j ,where x s and x u are commuting semisimple and unipotent elements. Assume x u has Jordan form ü l i=1 ⁄ i J r i i on the natural module. Let ⁄ be an eigenvalue corresponding to a maximal dimensional eigenspace E ⁄ of C j , and let J k be a Jordan block of minimal dimension such that ⁄J k occurs in the Jordan decomposition of x. Replace the block ⁄J k in the Jordan form of x with ⁄J k≠ 1 (if k = 1, remove the block ⁄J 1 ). Note an element x Õ with this new Jordan form (say viewed in GL n≠ 1 (k)) has determinant ⁄ ≠ 1 . However, by scaling the eigenvalues of x Õ by ⁄ 1/(n≠ 1) and restricting conjugation to elements of determinant one, we get a new conjugacy class C Õ i of SL n≠ 1 (k). Note every class C Õ i can be extended to C i in the obvious way, by reintroducing the Jordan block ⁄J k , and performing the appropriate rescaling. The following lemma shows that the condition on the largest eigenspaces is met when restricting classes C 1 ,...,C e of SL n (k) in this manner. Lemma 10. Assume C 1 ,...,C e are non-central conjugacy classes of SL n (k) such that (i) q e i=1 “ i Æ n(e≠ 1) (ii) it is not the case that e=2,and C 1 and C 2 are quadratic Furthermore, assumeC Õ 1 ,...,C Õ e are classes ofSL n≠ 1 (k) as described above. Then q e i=1 “ Õ i Æ (n≠ 1)(e≠ 1). Proof. Pick a class C i of SL n (k), and consider the corresponding class C Õ i of SL n≠ 1 (k). Assume x œ C i , and ⁄ k J k is the Jordan block removed from ü l i=1 ⁄ i J r i i in order to get x Õ œ C Õ i .If k = 1, then “ Õ i = “ i ≠ 1. Ifk> 1, then “ Õ i = “ i . Furthermore in the second case, all Jordan blocks of C i have have size greater than one, so we have that “ Õ i Æ n 2 ,with equality if and only if C i is quadratic. We argue by induction on the number of classes e. For the base case, assume e = 2. Note by the assumption on the largest eigenspaces “ 1 +“ 2 Æ n.IfeitherC 1 orC 2 contain a 21 Jordan block of size one, then “ Õ 1 +“ Õ 2 Æ “ 1 +“ 2 ≠ 1Æ n≠ 1. So assume C 1 and C 2 contain no Jordan blocks of size one. Then, as it is not the case that bothC 1 andC 2 are quadratic, “ Õ 1 +“ Õ 2 Æ n≠ 1. Now assume q e≠ 1 i=1 “ Õ i Æ (e≠ 2)(n≠ 1). Since C e is a non-central class, “ Õ e Æ n≠ 1. In particular, q e i=1 “ Õ i Æ (e≠ 1)(n≠ 1). Using the inductive hypothesis we may assume there is a tuple Ê Õ œ Õ topologically generating SL n≠ 1 (k). By rescaling and extending Ê Õ to a tuple Ê œ , it follows there is a tuple Ê œ topologically generating a group containing SL n≠ 1 (k). This fact will be used several times in order to ensure the existence of a tuple Ê œ generating a group with special properties. To begin, note the existence of a tuple topologically generating a group containing SL n≠ 1 (k) implies that there is a tuple inÊ œ generating a group which has a composition factor of dimension at least n≠ 1 on the natural module. By Lemma 5 this is an open condition, so generically tuples in share this property. By Lemma 8 generically tuples do not fix a 1-space or a hyperplane (by considering the dual space), and so generically act irreducibly on the natural module. To establish the remaining properties, it is helpful to draw a connection between a prop- erty holding generically on a simple algebraic group G and a property holding generically for tuples in. In what follows let W be the free group on e letters. For any w œ W,let Ï w :G◊ ...◊ Gæ G be the word map (x 1 ,...,x e )‘æ w(x 1 ,...,x e ). Proposition 1. Let G be a semisimple algebraic group over an algebraically closed field. For any nontrivial wœ W,thewordmap Ï w is dominant. In particular, µ G is a generic subset of G if any only if {(x 1 ,...,x e )œ G e | w(x 1 ,...,x e )œ } is a generic subset of G e . Proof. This is a result of Borel, and appears as Proposition 2.5 in [6]. Let us state the following two lemmas in slightly greater generality than necessary, so that they may be applied in the case of symplectic groups below. Lemma 11. Assume G is a semisimple algebraic group containing elements of infinite order, and C 1 ,...,C e are conjugacy classes of G. Then generic tuples in will generate a group containing elements of infinite order. Proof. The set n = {x œ G | x n =1} of elements of order at most n is closed in G. Hence t nØ 1 n , the set of elements of finite order in G, is a countable union of proper closed subvarieties. Over an uncountable field, = ( t nØ 1 n ) c is a generic subset of G. 22 Picking some word w(x 1 ,...,x e ) of infinite order and applying Proposition 1 generic tuples in generate groups containing infinite order elements. Lemma 12. Let G be a classical algebraic group whose natural module V has dimension d, and let C 1 ,...,C e be conjugacy classes of G. Assume there is a tuple Ê œ topologically generating a subgroup H such that H ¶ has an irreducible composition factor of dimension greater than d 2 on V. Then generically tuples do not generate an imprimitive subgroup of G. Proof. Note any imprimitive group H has a normal subgroup H 0 of index at most d!, and that all composition factors of H 0 have dimension at most d/2. Furthermore, there is a subgroup W Õ of W such that W Õ is contained in any subgroup of index less than d!in W. Let w 1 ,...,w m be words generating W Õ . For some fixed Ê := (x 1 ,...,x e ) œ , let U Ê :=Èw i (x 1 ,....x e ) | 1Æ iÆ mÍ, and Z := {Ê œ | U Ê has composition factors of dimension at most d/2} ByLemma5,Z isclosedin. Howeverthesetoftuplesinwhichgenerateanimprimitive subgroup is contained in the previous set, and so is contained in a proper closed subset of . Hence if any tuple Ê œ topologically generates a subgroup H such that the connected component of H has a composition factor greater than d/2, then generically tuples in will not generate an imprimitive subgroup. Finally, there is one additional valuable property that can be recovered from the exis- tence of a tuple Ê œ which topologically generates a group containing SL n≠ 1 (k). This property comes from examining the Lie algebra of G. Fix a maximal torus T of G, and let Ad :Gæ GL n (Lie(G)) be the adjoint representation. Note Lie(G) ≥ =V ¢ V ú ,where V is the natural module for G. Recall an element x œ G is regular if the dimension of its centralizer is as small as possible, and regular semisimple if all its eigenvalues are distinct. Note if an element tœ T has eigenvalues “ 1 ,...,“ n onV,thent has eigenvalues “ i ·“ ≠ 1 j onV ¢ V ú , for 1Æ iÆ n,1Æ jÆ n. Call a regular semisimple element of G strongly regular if for every 1Æ i,j,k,lÆ n, we have that “ i · “ ≠ 1 j = “ k · “ ≠ 1 l implies i = j and k = l. In other words, t is strongly regular if for any two distinct roots – i ,– j of G, – i (t)”= – j (t). It is a well known fact (see Theorem 2.5, [18]) that regular semisimple elements are open and dense in G. It follows 23 that generically elements inG are strongly regular: if“ 1 ,...,“ n are generically distinct, then so are “ i ·“ ≠ 1 j and “ k ·“ ≠ 1 l for i”=j,k”=l. NowtheexistenceoftupletopologicallygeneratingagroupcontainingSL n≠ 1 (k)ensures that there is a tuple (x 1 ,...,x e ) œ and word w œ W such that w(x 1 ,..,x e ) is strongly regular. Since generically elements of G are strongly regular, Proposition 1 implies that generic tuples in will generate a group containing strongly regular elements. Combining the above facts, the inductive hypothesis ensures that there is a tuple Ê œ topologically generating an irreducible, primitive subgroup, containing strongly regular elements of infinite order. This is sucient to prove topological generation for SL n (k). Lemma 13. LetG =SL n (k),andC 1 ,...,C e be non-central conjugacy classes ofG.Assume there is a tuple Ê œ topologically generating a group that acts irreducibly and primitively on the natural module, and contains a strongly regular element of infinite order. Then there is a tuple Ê œ topologically generating G. Proof. Let Ê œ be a tuple topologically generating a group H with the above listed properties, and let t œ H be a strongly regular element of infinite order. Since the non- trivial eigenvalues of t on V ¢ V ú are distinct, it follows that Lie(G)is H-invariant, and in particular that Lie(H) is generated by root subspaces of Lie(G). Furthermore, since t has infinite order, Lie(H) is non-trivial. In other words, Lie(H)=T 0 ü N – 1 ...ü N – j where the – i are roots of G, T 0 µ T is a torus of G, and the N – i are one-dimensional root subspaces of Lie(G). If Lie(H)=Lie(G) we are done, so assume that Lie(H) (Lie(G). Then there is a one dimensional root subspace N – ™ Lie(G) such that N – ”™ Lie(H). In particular, we have Lie(H)µ T ü N – 1 ...ü N – j (Lie(G) Since T is a maximal torus of G, this implies that H is contained in a maximal rank subgroup of G. From Section 6 of [28] and Table 18.2 of [29], we see there are no proper maximal rank irreducible primitive subgroups for nØ 4. 1 Remarks on topological generation NotetheassumptioninTheorem1thatthefieldkisanuncountablealgebraicallyclosedfield was used in the proof of the theorem to ensure Lie(H) was positive dimensional. While several variants of the proof method introduced above can be given in order to ensure 24 topological generation (one will be considered later in the case of symplectic groups), in each case some explicit use is made of the fact that the underlying field is uncountable. In essence this allows us to exclude the possibility that we are generating subfield subgroups G(q) of G for each q. Certainly some hypothesis is necessary, as Theorem 1 does not hold when k =K is the algebraic closure of a finite field. However, it seems likely there should be a proof that works over fields of smaller transcendence degree. For instance, if one could show working over a field of small transcendence degree that generically tuples Ê œ generate a group containing strongly regular element of infinite order, then the same proof given above would yield topological generation over this smaller field. Unfortunately, it is not immediately clear how to do this. On the other hand, in terms of the statement of results, once topological generation has been proved over an uncountable algebraically closed field, it is easy to recover the same result over fields of much smaller transcendence degree. This is intuitively clear, as we are looking a finite tuple defined over a field of some finite transcendence degree over K. Corollary 2. Let G =SL n (k) where k is an uncountable algebraically closed field of char- acteristicp> 0, K =F p is the algebraic closure of a finite field, and K Õ is a subfield of k such that trdeg(K Õ /K) Ø n· dim G.Assume C Õ 1 ,...,C Õ e are classes of SL n (K Õ ),andthat C 1 ,...,C e are the corresponding classes defined overG. If there is a tuple Ê œ topologically generating G, then there there is a tuple Ê Õ œ Õ topologically generating SL n (K Õ ). Proof. Let(x 1 ,...,x e )œ topologicallygenerate G. ConsiderthesubfieldK 1 ofk generated by K and the coecients of x 1 ,...,x e .Then trdeg(K 1 /K)Æ e·dim G.Sothereisatuple defined over G(K 1 ) that topologically generates SL n (K 1 ). Now note q n i=1 “ i Æ n(n≠ 1) for any non-central conjugacy classes C 1 ,...,C n of G. Hence for any C 1 ,...C e that topologically generate G, there will always be a field K 1 of transcendence degree of at most n·dim G such that there is a tuple defined over G(K 1 ) topologically generatingSL n (K 1 ). Futhermore K 1 can be embedded in any field extension K Õ where trdeg(K Õ /K)Ø n·dim G. It seems probable that topological generation will also hold over fields of even smaller transcendence degree, likely transcendence degree one. However again it is not immediately clear how to prove this. On the other hand, if one were to slightly reduce their aims, a closely related result could be established over the algebraic closure of a finite field. Corollary 3. Let G =SL n (K) where K =F p is the algebraic closure of a finite field, and n Ø 3. Let C 1 ,...,C e be non-central conjugacy classes of G. There exists a tuple Ê œ 25 generating a group containing SL n (q) for some arbitrarily large q = p n if and only if the following conditions hold: (i) q e i=1 “ i Æ n(e≠ 1); (ii) it is not the case that e=2 and C 1 ,C 2 are quadratic. Proof. The base case proceeds exactly as Lemma 9. For induction, note in the proof of Theorem 1 that the existence of a tuple Ê œ topologically generating a group acting irreducibly on the natural module and containing a strongly regular element did not depend ontheuncountabilityofthefield. HencewemayassumethereisatupleÊ œ topologically generating a group that contains at least a subfield subgroup G(q 0 ) of G. By Lemma 6 (which holds for classical algebraic groups defined over any algebraically closed field), generating a group containing G(q 0 ) is an open condition, and the only subgroups properly containing G(q 0 ) are other subfield subgroups of SL n (k). Now pick some arbitrarily large subfield subgroup G(q) of G. As shown in Lemma 11, the set up tuples generating a group of cardinality greater than |G(q)| is an open condition. Hence there is a tuple in generating a subfield subgroup properly containing G(q). The only if condition follows from Lemmas 15 and 16 below. Finally, the choice of starting fromSL 3 (k) in the statement of Theorem 1 was made for purposes of uniformity. Using the above arguments it is easy to prove the analogous result forSL 2 (k), although the statement is slightly dierent. Since this result will be used when considering Sp 2 (k) ≥ = SL 2 (k) below, we provide a short proof of this fact. Note since all non-central classes in SL 2 (k) have a largest eigenspace of dimension one, the condition on the largest eigenspaces may be dropped in this setting. Lemma 14. Let G = SL 2 (k) and C 1 ,...,C e be non-central conjugacy classes of G. Then there exists a tuple Ê œ topologically generating G if and only if e=2 and C 1 ,C 2 are classes of involutions modulo the center. Proof. According to Theorem 6 the conjugacy classes of maximal closed subgroups M 1 ,...,M k of SL 2 (k) are finite groups, a Borel subgroup, and the normalizer of a torus. Again it suces show ”µ Y i , for 1Æ iÆ k. First assume M i is a finite group. Note any non-central conjugacy class C j ofSL 2 (k)is twodimensional. Asdim fl M i = 0anddimG/M i = 3, byLemma2wehave ”µ Y i . Next assume M i is a Borel subgroup. First note if C 1 ,C 2 are semisimple classes, then elements in C j fl M i will have a one dimensional centralizer. Hence dim > dim i +dim G/M i , 26 and ”µ Y i . Similarly assume at least one of C 1 ,C 2 are unipotent (up to scalars). The unipotent class will be regular unipotent, and up to conjugacy regular unipotent elements live in a unique Borel subgroup. So we may pick an element in the other class which does not live in this Borel subgroup, and ”µ Y i . Finally, assume M i is the normalizer of a torus. Note dim C j fl M i = 0, unless C j is a class of outer elements of order two. There are two classes in SL 2 (k) corresponding to outer elements of order two in M i .Either C j is a class of involutions, or C j is a class of involutions modulo the center. Excluding these cases, dimC j fl M i = 0 and dimG/M i = 3, andhence ”µ Y i .WhenC 1 ,C 2 are involutions modulo the center, alltuples will generate a group contained in the normalizer of torus, and hence will not generateG topologically. 2 Necessary conditions for topological generation To complete the proof of Theorem 1, we must check the conditions listed in the statement of the theorem are in fact necessary. In other words if (i) q “ i >n(e≠ 1) or (ii) e = 2 and C 1 ,C 2 are quadratic, there are no tuples Ê œ that generate G topologically. Lemma 15. Let C 1 ,...,C e be conjugacy classes of a classical algebraic group G whose natural module V is n-dimensional. If q e i=1 “ i >n(e≠ 1), then there is no tuple Ê œ topologically generating G. Proof. Let C 1 ,...,C e be conjugacy classes of a classical algebraic group G whose natural moduleV isn-dimensional. For any elementxœ C i ,letE – i be a “ i -dimensional eigenspace of x on V. We would like to show that q e i=1 “ i >n(e≠ 1) implies dim ( e ‹ i=1 E – i )= e ÿ i=1 “ i ≠ n(e≠ 1) This is proved by induction on the number of classes. If e = 2 and “ 1 +“ 2 >n,then clearly dim (E – 1 fl E – 2 )=“ 1 +“ 2 ≠ n. So assume q k i=1 “ i >n(k≠ 1), and dim ( k ‹ i=1 E – i )= k ÿ i=1 “ i ≠ n(k≠ 1) Then dim ( k ‹ i=1 E – i fl E – k+1 )=( k ÿ i=1 “ i ≠ n(k≠ 1))+“ k+1 ≠ n Hence if q e i=1 “ i >n(e≠ 1), then every tuple Ê œ generates a reducible group on the natural module. 27 The second statement is a nice linear algebra exercise. Lemma 16. LetC 1 ,C 2 be quadratic conjugacy classes ofG =SL n (k),andV be the natural module for G.IfdimV> 2, then every tuple Ê œ generates a reducible subgroup of V. Proof. Let x 1 ,x 2 be quadratic elements in SL n (k), and H = Èx 1 ,x 2 Í. We claim if V is irreducible as an H-module, then dim V Æ 2. To begin note since x i is quadratic, either (i) x i is semisimple with two distinct eigen- values, or (ii) x i has a single eigenvalue, and contains all Jordan blocks of size at most two (with at least one block of size two). Foreaseofnotationinwhatfollowsassumethatx 1 haseigenvalues⁄ 1 ,⁄ 2 (wherepossibly ⁄ 1 = ⁄ 2 ), and x 2 has eigenvalues — 1 ,— 2 (where again possibly — 1 = — 2 ). Let V ⁄ i =(x 1 ≠ ⁄ i I)V, and F ⁄ i := {vœ V | (x 1 ≠ ⁄ i I)v”=0but(x 1 ≠ ⁄ i I) 2 v=0}.Then V ⁄ 1 =E ⁄ 2 if x 1 is semisimple, and V ⁄ 1 =F ⁄ 1 otherwise. Similarly define V — i to be (x 2 ≠ — i I)V. Note if V ⁄ i fl V — j is nontrivial for some 1Æ i,jÆ 2, then V is a reducible H-module. So we may assume V ⁄ i fl V — j is trivial, and hence dim V ⁄ i =dim V — j = n 2 , and V =V ⁄ i ü V — j . Now let Ï (⁄ i ,— j ) be the vector space isomorphism Ï (⁄ i ,— j ) : V ⁄ i æ V — j which sends v‘æ (x 2 ≠ — j I)v. Pick a basis {v 1 ,...v k } forV ⁄ 1 , and let Ï (⁄ 1 ,— 1 ) {v 1 ,...v k } be a basis forV — 1 . By the preceding {v 1 ,...v k }fi Ï (⁄ 1 ,— 1 ) {v 1 ,...v k } provides a basis for V.Withrespecttothis basis, x 1 and x 2 have the following forms: y 1 = Q a ⁄ 2 IM 0 ⁄ 1 I R b y 2 = Q a — 1 I 0 I — 2 I R b (2.1) where M is an invertible matrix. Let P be a matrix such that PMP ≠ 1 = Q is upper triangular. Then z 1 = Q a P 0 0 P R b Q a ⁄ 2 IM 0 ⁄ 1 I R b Q a P 0 0 P R b ≠ 1 = Q a ⁄ 2 IQ 0 ⁄ 1 I R b (2.2) z 2 = Q a P 0 0 P R b Q a — 1 I 0 I — 2 I R b Q a P 0 0 P R b ≠ 1 = Q a — 1 I 0 I — 2 I R b (2.3) It follows that W = È(1,0,...,0),(0,...,1,0,...,0)Í is H-invariant, and hence V reducible if dimV> 2. 28 This concludes the proof of Theorem 1. We now turn our attention to some applications of topological generation. 29 Chapter 4 Applications to random generation of finite groups of Lie type 1 Background results In this chapter, we use topological generation of algebraic groups to answer questions con- cerning random generation of finite groups of Lie type. Before stating the results, let us briefly review some relevant facts from the literature. In what follows, let H be a finite group. Recall if J,K are subsets of H,then P(J,K) is the proportion of pairs in J ◊ K that generate H. From the early days of group theory it was known that many pairs of elements in H = Alt n or Sym n generate H. Making this precise, in the late nineteenth century Netto conjectured that P(H):=P(H,H)æ 1 as |H|æŒ ,when H = Alt n or H=Sym n .This was first proved by Dixon in [8], where he conjectured that the same property holds for all finite simple groups. After considerable eort, Dixon’s Conjecture was established by Liebek-Shalev [24] and Kantor-Lubotsky [19] in the 1990s using probabilisitc methods. The solution of Dixon’s Conjecture led to many more fine-grained questions concerning random generation of finite groups, such as when do two random elements of prime orders r and s generate a finite simple group? This property is called random (r,s)-generation. While this is a natural refinement of random generation for finite simple groups, it is also a generalization of another classical group-theoretic question, the (2,3)-generation problem. The (2,3)-generation problem asks to classify the finite simple groups which are gen- erated by elements of orders two and three, and was originally motivated by geometrical considerations. It had been known since the early twentieth century that the modular groupPSL 2 (Z) was isomorphic to the free product C 2 ú C 3 . Hence in its original context the (2,3)-generation problem asked to classify the finite simple groups appearing as quo- tients ofPSL 2 (Z). The solution to the question yields information about fractional linear transformations. In addition, (2,3)-generation gives information concerning the existence of Hurwitz groups. These are groups acting on compact Riemann surfaces of genus g Ø 2 such that the order of the group attains the Hurwitz upper bound 84(g≠ 1). Groups of 30 this type have generators x,y such that x 2 = y 3 =(xy) 7 , and so are (2,3)-generated. Throughout the years, these questions have attracted attention in the literature. See [25] for background on these problems, and recent results using random generation properties. Studying (r,s)-generation is a natural generalization of these classical problems. The following result shows random (r,s)-generation holds for classical groups when the rank of the group is large enough (depending on the orders of r,s). Theorem 7. Let (r,s) be primes with (r,s)”=(2,2). There exists a positive integer f(r,s) such that if H is a finite simple classical group of rank at least f(r,s), then P r,s (H)æ 1 as |H|æŒ Proof. This is Theorem 1 of Liebeck-Shalev [25]. By other work of Liebeck-Shalev [26], it is known that all finite simple classical groups have random (2,3)-generation except PSp 4 (in particular there are at most finitely many exceptions to (2,3)- generation aside from PSp 4 ). Using random (2,3)-generation as a guide, one might expect random (r,s)-generation to hold with orders r,s independent of the rank in all but a few small rank cases. We use topological generation to prove this stronger result below for linear and unitary groups in the fixed rank case. From now on, let G be a classical algebraic group of simply-connected type, F :Gæ G be a Steinberg endomorphism, andG F orG(q) be the fixed points ofF onG. Furthermore, let E(q):=Jfl G(q) for any subset E of G(q). A brief sketch of the proof idea is provided below. To start, choose a sequence of Steinberg endomorphisms F :Gæ G, and groups of Lie typeG F suchthateachG F containselementsoforderr ands. ForeachG F inthissequence, we show there exist classesC,D inG meeting the conditions for topological generation such that( q)”=ÿ , and for any bad classesC Õ ,D Õ that fail to meet the conditions for topological generation, dim Õ < dim. By Lemma 6, the set W = {(x,y)œ G◊ G|Èx,yÍ contains G(q 0 ), for q 0 large enough } is open in G◊ G, and hence the existence of a generating tuple in implies that W fl is open and dense in. Using some algebraic geometry, it is shown that random tuples in ( q) are contained in W, and so will generate at least a subfield subgroup of G(q). By a 31 probabilistic argument, it is then shown that the proportion of tuples generating a proper subfield subgroup of G(q) goes to zero as |G(q)|æŒ . HenceitispossibletochooseasequenceofgoodclassesC,D suchthatP(C(q),D(q))æ 1 as |G(q)|æŒ . In general there are only finitely many conjugacy classes of elements of orders r,s in a simple algebraic groupG, and for any bad classesC Õ ,D Õ of orders r,s (those which do not meet the hypothesis for topological generation) we have dim Õ < dim. Hence these classes will not contribute asymptotically to the proportion of generating pairs. So P r,s (G(q))æ 1 as |G(q)|æŒ . Finally, notethatwhiletheresultsofLiebeck-Shalevarestatedforfinitesimpleclassical groups, G F itself is not typically simple. This small discrepancy is easily resolved. If G is a simply connected simple algebraic group, then S = G F /Z(G) F is (almost always) a finite simple group. More generally, [G F ,G F ] modulo its center is simple except for a very small number of cases. Let Ï : G æ S be the natural projection map. The center of G F is contained in the Frattini subgroup of G, and hence the elements x 1 ,...,x e œ G F will generate G F if and only the elements x 1 ,...,x e œ S generate S (of course, the orders of the elements may be dierent in S). Hence if we are only concerned with proving random (r,s)- generation for elements of prime order the simple group S, it suces to prove the relevant random (r,s)-generation results for all prime power order elements r,s in the groups of Lie type G F . 2 Topological generation and random (r,s)-generation In this section we connect topological generation in algebraic groups to random (r,s)- generation in finite groups of Lie type. We show under the appropriate hypothesis that topological generation in a classical algebraic group G can be used to ensure random (r,s)- generation in the corresponding finite group of Lie type G F . To begin, let F :Gæ G be a Steinberg endomorphism. Recall conjugacy classes C Õ ,D Õ of elements of orders r,s in G are bad if (i) Õ (q) ”= ÿ , and (ii) C Õ ,D Õ do not meet the conditions necessary for topological generation. Similarly, classes C,D of elements of order r,s in G are called good if (i)( q)”=ÿ ; (ii)Thereisatuple Ê œ topologically generating G; (iii) For any bad classes C Õ ,D Õ in G,dim Õ < dim. 32 Note our topological generation results deal with particular conjugacy classes C 1 ,...,C e of an algebraic groupG, while random (r,s)-generation deals with all elements of ordersr,s in G F . To connect these two occurrences, it will be shown that a sequence of good classes in G is sucient to ensure random ( r,s)-generation. It is first helpful to prove a lower bounds on the dimension of = C 1 ◊ C 2 when contains a tuple topologically generating G. This lemma also proves the necessity of condition (iii) given the statement of Theorem 2. Lemma 17. Let C 1 , C 2 be conjugacy classes of the classical algebraic group G.Assume fl W ”= ÿ.If G is a symplectic group defined over a field of odd characteristic, then dim Ø dim G+rank(G). In general, dim Ø dim G+rank(G)≠ ” , where ” Æ 1 unless G =D m , mØ 4,and p=2. In the latter case, ” =2. In every situation, dim > dim G. Proof. The following argument is a slight modification of Lemma 2.4 in [14]. Since fl W ”= ÿ ,thereisatuple(x,y)œ generating a subfield subgroup H of G. It follows that Èx,yÍ has the same set of invariant subspaces on A = Lie(G) as G does. Set z = xy, and let E be the conjugacy class of z in G. Note that dim [x,A]=dim A≠ dim A x Æ dim G ≠ dim C G (x), so that dim [x,A] Æ dim C. Similarly, dim [y,A] Æ dim D, and dim [z,A] Æ dim E. Furthermore, dim E Æ dim G≠ r as regular classes in G have dimension G≠ r. Applying Scott’s Lemma [35], we have dim C+dim D+dim G≠ rØ dim A+dim[G,A]≠ dim A G Since A=[G,A] and dim A=dim G, it follows that dim C+dim DØ dim G≠ dim A G +r If G is a symplectic group in odd characteristic, then A is irreducible and dim A G = 0. More generally, dim [G,A]≠ dim A G =dim G≠ ” ,where ” is at most the number of trivial composition factors of G on A.Thus ” Æ 1unless G =D m , mØ 4 and p = 2, as desired. The last statement follows immediately from the preceding, except possibly whenr = 1. However, every nontrivial conjugacy class has dimension at least 2, so dim > dim G also holds in this setting. 33 Now call a set I µ G normal if I is a union of conjugacy classes in G. Let C t be the set of conjugacy classes of elements of order t in G, and let J,K be the following normal sets of elements of order r,s in G J = € {CœC r | C(q)”=ÿ} ; K = € {CœC s | C(q)”=ÿ} The following lemma shows the probability that a random pair from J(q)◊ K(q) generates a proper subfield subgroup of G(q) is small. Lemma 18. Let G be a classical algebraic group, and J,K be normal subsets of elements of orders r and s in G such that dim J◊ K = G+e> dim G,and J(q)◊ K(q)”=ÿ .The probability that a random pair from J(q)◊ K(q) is contained in H◊ H for some conjugate H of a proper subfield subgroup of G(q) is at most O(q ≠ 1/2 ). Proof. This is argument is modified from Theorem 2.5 in [14]. SinceJ,K are normal sets of G, they contain conjugacy classes C,D in G such that dim C=dim J,dim D=dim K, and dim( q)=C(q)◊ D(q)”=ÿ . Hence it suces to show the probability that a random pair from C(q)◊ D(q) is contained in a conjugate of a proper subfield subgroup H◊ H of G(q) is at most O(q ≠ 1/2 ). First note the order of( q) is approximately q dim C+dim D =q dim G+e , and the order of ( q 1/b ) is approximately q (dim G+e)/b . Furthermore, the number of conjugates of G(q 1/b )is at most |G(q):G(q 1/b )|≥q (dimG)(1≠ 1/b) . SummingoverallsubfieldsubgroupsofG(q)showsthenumberofpairsin( q)contained in a proper subfield subgroup is at most c ÿ b q (dimG+e)/b |G(q):G(q 1/b )|Æ c Õ ÿ b q (dimG+e)/b+dimG(1≠ 1/b) where c,c Õ are constants depending on G, and the summation is over all primes b dividing a where q = p a . Thus the probability that a random pair in C(q)◊ D(q) is contained in a conjugate of a subfield subgroup H◊ H is on the order of q b q e/b /q e , which is bounded above by O(q ≠ 1/2 ). Finally, assume F : G æ G is a Steinberg endomorphism, and C,D are good classes of G. Let C Õ ,D Õ be classes of G such that dim C Õ Ø dim C,dim D Õ Ø dim D, and C Õ (q)◊ D Õ (q)”=ÿ . Let = J◊ K.Then “ Õ 1 Ø “ 1 ,“ Õ 2 Ø “ 2 , and in particular there is a tuple Ê œ C Õ ◊ D Õ topologically generating G. So we may assume without loss of generality that dim = dim C◊ D. 34 Theorem3. LetGbeaclassicalalgebraicgroup, and(r,s)beprimepowers. LetF :Gæ G be a sequence of Steinberg endomorphisms such that for each F, (i) G F contains elements of orders r and s,and (ii) G contains good classes C and D. Then P r,s (G(q)) æ 1,as |G(q)|æŒ . Proof. Pick a Steinberg endomorphism F, and good classes C,D in G such that dim = dim. Assume Ê œ topologically generates G, and pick a q 0 such that C,D are defined over F q 0 . Let F q be the field of q =q m 0 elements, and W = {(x,y)œ G◊ G|Èx,yÍ´ G(q 0 )} By Lemma 6, W fl is open and dense in . Note |( q)| is on the order of q s ,with s=dim C◊ D.SinceW is open, the points of not in W are contained in a proper closed subvariety. In particular, applying the alternate version of Lang-Weil given as Corollary 4 in [36] (which does not depend on the variety being irreducible), |( q)\W| has order at most O(q s≠ 1 ). Hence for q large enough, almost all points of( q) are contained in W. Next, tuples (x,y)œ W fl ( q) generate at least a subfield subgroup G(q 1 ) of G(q) for some q 1 |q, and q 0 |q 1 . However by Corollary 6, the proportion of pairs that are contained in a conjugate of G(q 1 ) for some q 1 <q is at most O(q ≠ 1/2 ). Hence the proportion of pairs in( q) that generate G(q) is at least 1≠ O(q ≠ 1/2 ), and goes to 1 as qæŒ . 3 Random (r,s)-generation of special linear and unitary groups Theorem 3 reduces random (r,s)-generation for finite classical groups to proving the exis- tence of a sequence of good classes in G. The following lemma establishes that such a sequence of classes exist for G =SL n (k). Lemma 19. Let G = SL n (k) and (r,s) ”=(2,2) be prime powers. Let F : G æ G be a sequence of Steinberg endomorphisms such that G F contains elements of orders r and s. Then for each F :Gæ G, there exist good classes C 1 ,C 2 in G. Proof. To begin, pick a Steinberg endomorphism F :Gæ G, such that G F =SL n (q). We want to find classes C 1 ,C 2 in G where (i)( q) ”= ÿ ,(ii) “ 1 +“ 2 Æ n, and (iii) for any classes C Õ 1 ,C Õ 2 of orders r, s in G such that Õ (q)”=ÿ and “ Õ 1 +“ Õ 2 >n,dim Õ < dim. 35 This follows easily by proving the existence of a classC inG of elements of prime power order r such that: (a) C(q)”=ÿ , (b) “ Æ n 2 , (c)dim C Õ Æ dim C, for all classes C Õ œC r such that C Õ (q)”=ÿ . First note if r | q, then the statement holds immediately, as all unipotent classes are defined in G F . Next assume r -q, so that the elements under consideration are semisimple. Note G F has order q ( n 2 ) (q 2 ≠ 1)(q 3 ≠ 1)....(q n ≠ 1). If r = 2 and C is a semisimple class of involutions, then the desired conditions are also immediately met. For convenience we exclude this case in what follows. Next, if r | q≠ 1 then all classes of order r in G are defined over G F . In particular, some class C of order r elements satisfies conditions (a)≠ (c) above. So assume r | q k ≠ 1 for some 1<k Æ n, but r - q j ≠ 1 for 1 Æ j<k. Then all elements in G F - classes of order r will have an irreducible composition factor of dimension at least k. In addition, ifjk |n for some jØ 1, therewillbeatleastoneG F -classwhoseelementshaveanirreduciblecompositionfactorof dimension at least jk. Let m =max{ jk | jk divides n}. This gives the largest dimension of an irreducible composition factor for a G F -class of order r elements. In other words, every class C Õ œC r such that C Õ (q)”=ÿ will consist of elements acting trivially on a space of dimension n≠ m< n 2 . Now let C be a class of order r elements in G with eigenspace decomposition V = E n≠ m 1 ü k i=1 E i – i , where each E i – i is as small as possible given r, and – i ”= 1. Then dim E 1 < n 2 . Furthermore, since C is not a class of involutions, dim E – i Æ m 2 < n 2 for each i.In particular, C(q)”=ÿ , and “ Æ n 2 . Let = ( n 1 ,...,n k ) be the partition ofn corresponding to the dimensions of the eigenspace decomposition of C. Let C Õ œC r be such that C Õ (q)”=ÿ . Assume the eigenspace decomposition of C Õ corresponds to the partition = ( m 1 ,...,m l ). By definition of C, n i Æ m i for each 1Æ iÆ k. Hence for xœ C and x Õ œ C Õ we have dim C G (x)Æ dim C G (x Õ ) and in particular, dimC Õ Æ dim C, with equality if and only if =. So we can find a class CœC r satisfying (a)≠ (c) above. Now pick classes C 1 ,C 2 of elements of orders r,s of G in the manner described above. Conditions (i) and (ii) are immediately satisfied. For (iii), pick classes C Õ 1 ,C Õ 2 of orders r, s such that C Õ 1 (q)◊ C Õ 2 (q)”=ÿ and “ Õ 1 +“ Õ 2 >n.Since “ 1 +“ 2 <n, we may assume without loss of generality that “ 1 <“ Õ 1 .ThendimC Õ 1 < dim C 1 and dimC Õ 2 Æ dim C 2 , so condition (iii) is also satisfied. 36 Finally, assume F : G æ G is a sequence of graph automorphisms such that G F = SU n (q). In this case, |G F | = q ( n 2 ) (q 2 ≠ 1)(q 3 +1)....(q n ≠ (≠ 1) n ). As above, if r | q k ±1 and r - q j ±1 fork<j Æ n,then k Æ n 2 ≠ 1. Hence the fixed space of a G F -class is at most slightly less than half, and we may repeat exactly the same construction given above to find classes C 1 ,C 2 possessing properties (i)≠ (iii). The following result is immediate from Lemma 19 and Theorem 3. Theorem 5. Let G = SL n (k),and (r,s) ”=(2,2) be prime powers. Assume F is a sequence of Steinberg endomorphisms such that G F contains elements of orders r and s. Then P r,s (G(q))æŒ ,as |G(q)|æŒ . Note Theorem 3 provides necessary conditions for using topological generation in order to ensure random (r,s)-generation of finite simple classical groups (in the fixed rank case). In fact, Theorem 3 can easily be extended to give necessary conditions for using topolog- ical generation to ensure random (r,s)-generation for all finite simple groups of Lie type. Hence if one can classify topological generation properties for simple algebraic groups in an appropriate manner, then one can also prove random (r,s)-generation for the correspond- ing groups of Lie type with rank independent of the orders of r,s.Thisisaprojectwe are currently undertaking. We now turn to our final application of topological generation concerning the representation theory of classical algebraic groups. 37 Chapter 5 Applications to generic stabilizers of classical algebraic groups In this section, let G be an algebraic group, C be a conjugacy class of G, and unless otherwise stated let V be a linear G-variety with no fixed points on G (i.e. V G = 0). Furthermore for any x œ G,let V x = {v œ V | xv = v} be the fixed points of x, and V(C)= {v œ V | xv = v for some g œ C} be the points in V fixed by some element in C. Recall a generic stabilizer G x = {gœ G| gx =x} is trivial if there exists a non-empty open subvariety V 0 of V such that G x is trivial for each x œ V 0 . For G = SL n (k), we show a generic stabilizer is trivial when the dimension of V is large enough. To begin, pick some conjugacy class C in G and assume d elements of C generate G topologically. The following lemma yields a bound on the dimension of V(C). Lemma 20. Let G be an algebraic group, and C be a conjugacy class in G.Assume d elements of C generate G topologically. Then dim V(C) Æ d d≠ 1 dim V + dim C.In particular, dim V(C)< dim V, when dimV>d·dim C. Proof. Pick x œ C, and let Ï : G◊ V x æ V be the map sending (g,v) ‘æ gv. For any v œ V x ,(gxg ≠ 1 )gv = gxv = gv,so im(Ï )= V(C). To achieve the desired inequality, we compute a bound on the dimension of a fiber Ï ≠ 1 (gv). Pick gv œ V(C). Note for any h œ C G (x), we have (hgh ≠ 1 )v = ghh ≠ 1 v = gv. Hence every fiber has dimension greater than or equal to the dimension of C G (x). In particular, dim V(C)Æ dim G+dim V x ≠ dim C G (x) Nowlet“ bethedimensionofthelargesteigenspaceofC.SincedelementsfromC generate G, itfollowsfromTheorem1thatd“ Æ (d≠ 1)n, andhenceinparticularthatdim V x Æ “ Æ d d≠ 1 ·dim V.Sincedim C=dim G≠ dim C G (g), we have dim V(C)Æ d d≠ 1 dim V +dim C. Finally, dimV>d · dim C … dimV> d d≠ 1 dim V +dim C,sodim V(C) < dim V whenever dimV>d·dim C. 38 LetC bethesetofconjugacyclassesofGcontainingelementsofprimeorder(orarbitrary unipotent elements if p = 0). The following lemma from the literature allows us to show a generic stabilizer is trivial when dim V(C)< dim V for all CœC . Lemma 21. Let G be an algebraic group with G ¶ reductive, and V be a linear G-variety with V G =0. Let V(C) and C be as above. If dim V(C) < dim V for all CœC , then a generic stabilizer is trivial. Proof. This follows from Proposition 2.10, [7], Lemma 10.2 in [11], and the fact a linear G-variety is also an irreducible G-variety. Finally, let C d be the following collection of conjugacy classes in C: C d := {CœC| d elements of C are required to generate G topologically} In other words, any class CœC d will have the property that no d≠ 1 elements from C will generate G topologically, but some elements x 1 ,...,x d œ C will. Furthermore, let – d :=max {dim C | CœC d }, and – :=max {– d ·d | 2Æ dÆ n}. Note for any non-central class C inSL n (k), it is always possible to generateSL n (k) by choosing n elements from C. This follows immediately from Theorem 1, as for any n non-central conjugacy classes in G, q n i=1 “ i Æ n(n≠ 1). NowassumedimV>– .ThendimV>d·C, foreveryprimeorderclass(andunipotent classes in characteristic p = 0). Applying Lemma 20, dim V(C)< dim V, for every CœC , and hence by Lemma 21 a generic stabilizer is trivial. So to find an upper bound for the dimension of a linear G-variety with a nontrivial generic stabilizer, it suces to find an upper bound for – . For G =SL n (k), this is not dicult to compute. Theorem 6. Let G = SL n (k),and V be an linear G-variety such that V G =0. Then a generic stabilizer is trivial when dimV> 9 4 n 2 . Proof. It suces to show 9 4 n 2 >– . To begin, assume d = 2 elements generate G topolog- ically. In this case, the largest dimensional conjugacy class CœC 2 will be regular, and it follows that – 2 =n 2 ≠ n. Next assume 2<dÆ n, and pick CœC d .Thenno d≠ 1 elements from C will generate G topologically, and it follows from Theorem 1 that “ Ø d≠ 2 d≠ 1 ·n,where “ is the dimension of the largest eigenspace of C (with“> d≠ 2 d≠ 1 ·n, ford> 3). First assumeC is a semisimple class in C d . To find an upper bound on the dimension of C,wecomputetheminimumdimensionofacentralizerC G (x),withxœ C.Since“ Ø d≠ 2 d≠ 1 ·n, 39 it follows that the centralizer of an element xœ C must contain a copy of GL — (k), where — =Á n(d≠ 2) d≠ 1 Ë. Furthermore, since elements inC have largest eigenspace of dimension “ ,the smallest conceivable centralizer occurs when one eigenspace has dimension “ = — , and all other eigenspaces are one dimensional. In this case, dim C G (x)= — 2 +(n≠ — )≠ 1. This implies that the largest possible dimension for a semisimple class CœC d is n 2 ≠ n≠ — 2 +— . Now assume C œC d is a unipotent class, and elements x œ C have Jordan forms involving Jordan blocks of size ⁄ 1 Ø ⁄ 2 Ø ...⁄ k . Let A be the Young diagram of [⁄ 1 ,...,⁄ k ], and [µ 1 ,...,µ j ] be the Young diagram of A T . Then for xœ C,dim C G (x)=( q j i=1 µ 2 j )≠ 1. SinceelementsinC havealargesteigenspaceofdimensionØ — , theJordanformofelements in C must have at least — Jordan blocks. Hence dim C G (x)=— 2 + j ÿ i=2 µ 2 i Ø — 2 +(n≠ — )≠ 1 Combined with the previous paragraph, this implies that – d Æ n 2 ≠ n≠ — 2 +— . Finally, we want to compute an upper bound for – = max {– d ·d | 2Æ dÆ n},where – 2 =n 2 ≠ n, and – d Æ n 2 ≠ — 2 ≠ (n≠ — )<n 2 ≠ — 2 Æ n 2 ≠ n 2 (d≠ 2) 2 (d≠ 1) 2 for 3Æ dÆ n. Note that: d(n 2 ≠ n 2 (d≠ 2) 2 (d≠ 1) 2 )= dn 2 ((d≠ 1) 2 ≠ (d≠ 2) 2 ) (d≠ 1) 2 =n 2 d(2d≠ 3) (d≠ 1) 2 Finally note d(2d≠ 3) (d≠ 1) 2 Æ 9 4 , for 3Æ dÆ n,so – Æ max (2(n 2 ≠ n),n 2 d(2d≠ 3) (d≠ 1) 2 )Æ 9 4 n 2 . 40 Chapter 6 Topological generation of symplectic algebraic groups In this final chapter, we consider topological generation problems for symplectic algebraic groups. Let G =Sp 2n (k), where k is an uncountable algebraically closed field not of even characteristic. Let C 1 ,...,C e be conjugacy classes of G, and up to conjugacy let M 1 ,...,M k be the maximal closed subgroups of G. As in Chapter 2, for each M i let: - i := r e j=1 (C j fl M i ) - X i := t gœ G M g i ◊ ...◊ M g i - Ï i :G◊ i æ X i be the map (g,(x 1 ,...,x e ))‘æ (x g 1 ,...,x g e ). - Y i :=im(Ï i ). We again prove there is a tuple Ê œ topologically generating G in the relevant cases by showing that ”µ Y j , for 1Æ j Æ k. However there are now several added complications. First of all, it is quite inconvenient to prove the analogous generation results for arbitrary conjugacy classes, so we restrict to unipotent and semisimple classes C 1 ,...,C e , considered separately. Second, there is an additional condition that must be placed on the semisimple classes to ensure topological generation. Namely, if there is a tuple Ê œ topologically generating Sp 2n (k), then it must be the case that dim Ø dim G+rank G. When this condition is violated, every tupleÊ œ will generate a group having fixed points on the adjoint module, and hence not generate a Zariski dense subgroup of G (see Lemma 17 above). The list of cases where this new assumption is required is established in Lemma 28. Finally, even with this added assumption, there are still a few exceptions to topological generation involving semisimple classes of small rank groups. To describe these exceptions, let us fix some notation that will appear throughout the chapter. Let x œ Sp 2n (k)be semisimple, andV =ü k i=i E n i – i be the eigenspace decomposition ofx on the natural module, where E n i – i is an eigenspace of dimension n i , with corresponding eigenvalue – i . Recall the eigenvalues of x come in pairs – , – ≠ 1 , and the corresponding eigenspaces E – , E – Õ are 41 totally singular if – ”= ±1, and non-degenerate if – = ±1. In what follows, let ⁄ i denote an eigenvalue where ⁄ i ”= ±1. Furthermore, we write E – i instead of E 1 – i . The small rank exceptions to topological generation for symplectic groups are listed in the following table: Table 1: Exceptions to topological generation in small rank symplectic groups G Number of classes C 1 ,...,C e Description of classes Sp 4 (k) 3Æ eÆ 4 C 1 ,...,C e =E 2 1 ü E 2 ≠ 1 Sp 4 (k) 3 C 1 ,C 2 =E 2 1 ü E 2 ≠ 1 ; C 3 =E 2 ⁄ ü E 2 ⁄ ≠ 1 Sp 6 (k) 3 C 1 ,C 2 ,C 3 =E 4 ±1 ü E 2 û 1 Note the case of Sp 2 (k) ≥ = SL 2 (k) was treated previously in Chapter 3. The main topological generation results for symplectic groups are thus as follows. Theorem 2. Let G = Sp 2n (k), n Ø 2, where k is an uncountable algebraically closed field not of even characteristic. Let C 1 ,...,C e be non-central conjugacy classes of G such that C 1 ,...,C e are either all unipotent, or all semisimple. Then there exists a tuple Ê œ topologically generating G if and only if the following conditions hold: (i) q e i=1 “ i Æ 2n(e≠ 1); (ii)itisnotthecasethat e=2 and C 1 ,C 2 are quadratic. (iii) dim Ø dim G+rank (G); (iv) it is not the case that the classes C 1 ,...,C e and group G appear in Table 1. For any symplectic group in odd characteristic, the fact that conditions (i)≠ (iii) are required for topological generation follows from Lemmas 15,16, and 17 above. These argu- ments will not be repeated in what follows. The necessity of condition (iv) will be treated at the end of the following section. As with linear groups, the majority of the work lies in the converse direction; showing that conditions (i)≠ (iv) are sucient to ensure topological generation. Againthiswillbeprovedbyinduction,inthiscaseconsideringsemisimpleandunipotent classes separately. To use the inductive hypothesis, it will be necessary to restrict classes of Sp 2n (k) to classes of Sp 2n≠ 2 (k), which creates a few additional complications. First of all, to ensure the existence of tuple generating an irreducible subgroup, we need to consider the appropriate 1- and 2- dimensional subspaces of V. Second, the inductive argument we used for linear groups was based on the appearance of a tuple Ê œ generating a group containing strongly regular elements. When going down toSp 2n≠ 2 (k) the existence of such 42 elements no longer follows from the inductive hypothesis, and a slightly dierent method of proof must be devised. Recall in odd characteristic, symplectic groups preserve a skew-symmetric bilinear form (,):V ◊ V æ k. In other words, for any xœ Sp 2n (k), w,vœ V,wehave(xw,xv)=(w,v), and (w,v)= ≠ (v,w). A subspace W µ V is non-degenerate if (w,v) = 0 for all w œ W implies that v = 0. Similarly W µ V is totally singular if (w,v) = 0 for all v,w œ W. While SL n (k) acted transitively on the set of k-dimensional subspaces of V and hence it suced to consider a single type of k-dimensional subspace ofV, for symplectic groups two dierent types of subspaces of V must now be considered. To argue that generically tuples in generate an irreducible subgroup with other nice properties,itwillbehelpfultoshowthedimensionsoftuplesinfixingbothnon-degenerate and totally singular 1- and 2-spaces are strictly less than the dimension of . For 1- spaces, there is no added complication. Symplectic forms are alternating, and hence all 1-dimensional subspaces are totally singular. However, non-degenerate and totally singular 2-spaces both appear, and must be considered separately. For both the base and inductive cases, it will be helpful to record some general information about the stabilizers of these spaces. First assume W µ V is a non-degenerate subspace of dimension 2m, and let M i be the stabilizer of W. The stabilizer of W preserves the direct sum decomposition V =Wü W ‹ where W ‹ is also a non-degenerate space, so M i ≥ =Sp 2m (k)◊ Sp 2n≠ 2m (k). Hence dim G/M i =(2n 2 +n)≠ ((2m 2 +m)+(2(n≠ m) 2 +(n≠ m))) = 4m(n≠ m) In particular, the variety of non-degenerate 2-spaces G/M i has dimension 4n≠ 4. Now consider a totally singularm-spaceW µ V, for 1Æ mÆ n.ThenM i =QL =P i is aparabolicsubgroup,whereQistheunipotentradicalofM i ,andListheLevisubgroup. M i preserves the flag 0<W<W ‹ <V, and there is an induced non-singular form onW ‹ /W. By Witt’s lemma, we may choose a symplectic basis so that W is spanned by e 1 ,e 2 ,...,e m , andW ‹ is spanned bye 1 ,e 2 ,...,e l ,f m ,...,f l . HenceW ‹ /W has a basis ofe m ,...,e l ,f m ,..,f l withSp 2n≠ 2m (k) acting on it (see Section 3.5.4 in [37]). We also haveGL m (k) acting on the totallysingularspacesW andV/W ‹ , where theactiononV/W ‹ is thetranspose inverse of the action onW. Hence the Levi subgroup ofM i isL ≥ =Sp 2n≠ 2l (k)◊ GL l (k). Furthermore, the unipotent radical Q has the form: 43 S W W W U I l AB 0 I 2n≠ 2l C 00 I l T X X X V whereAisanymatrixofshape2l◊ (2n≠ 2m),C isamatrixofsize(2n≠ 2m)◊ 2ldetermined byA, andB isanysymmetricm◊ mmatrix. HencethedimQ=2l(2n≠ 2m)+m(m+1)/2. It follows that M i = QL has dimension 2n 2 ≠ 3n+5, and the variety of totally singular 2-spaces G/M i has dimension 4n≠ 5. 1 Semisimple classes In this section, we prove Theorem 2 when C 1 ,...,C e are non-central semisimple classes of G =Sp 2n (k). LetGactonthenaturalmoduleV. Forthebaseandinductivecasesitwillbe helpfultoprovethedimensionoftuplesinfixinga1-dimensionaltotallysingularsubspace of V, or 2-dimensional non-degenerate subspace of V is strictly less than the dimension of . Lemma 22. Let C 1 ,...,C e be non-central semisimple conjugacy classes of Sp 2n (k),with n Ø 2.Assume q e i=1 “ i Æ 2n(e≠ 1),anditisnotthecasethat e=2 and C 1 ,C 2 are quadratic. Then the dimension of tuples in that fix a totally singular 1-dimensional subspace of V, or fix a non-degenerate 2-dimensional subspace of V are strictly less than the dimension of . Proof. Let C 1 ,...,C e be non-central semisimple conjugacy classes of Sp 2n (k). First assume W is a totally singular 1-space, and M i is the stabilizer of W. Next, assume: e ÿ j=1 dim (G/M i ) x j Ø (e≠ 1)·dim G/M i Symplectic forms are alternating, so G acts transitively on totally singular 1-spaces and G/M i is isomorphic to projective space P 1 (V). Hence dim G/M i =2n≠ 1. In P 1 (V), we havev≥–v for all – œ k ◊ , and henceP 1 (V) x j = {v |x j ·v =–v } =fi r i=1 E – i ,whereE – i are the eigenspaces ofx j viewed inside projective space. It follows that dim (G/M i ) x j =“ j ≠ 1. Applying this information to the above inequality yields ( q e j=1 “ j )≠ e Ø (e≠ 1)(2n≠ 1). This in turn implies q e j=1 “ j Ø 2n(e≠ 1)+1, contradicting the assumption on the largest eigenspaces of C 1 ,...,C e . Hence q e j=1 dim (G/M i ) x j < (e≠ 1)· dim G/M i , and ”µ X i . Since not fixing a 1-dimensional subspace is an open condition, the statement immediately follows. 44 Next assume M i is the stabilizer of a non-degenerate 2-space W. Note there are two ways an element x j œ C j can fix W.Either W is contained in a single eigenspace E – of x j , or it is contained in a combination of two distinct eigenspaces E – and E – Õ. Assume E – , E – Õ are the two largest eigenspaces of x j ,with “ Õ j =dim E – Õ Æ dim E – = “ j .Thendim (G/M i ) x j is less than or equal to the dimension of the larger of the following two varieties: a) the variety of all 2-dimensional subspaces of the largest eigenspace E – of x j .Thisis the second Grassmanian of E – , which has dimension 2(“ j ≠ 2). b) the variety of all 1-dimensional subspaces of E – , plus the variety of all 1-dimensional subspaces ofE – Õ. This is projective space onE – times projective space onE – Õ,which has dimension (“ j ≠ 1)+(“ Õ j ≠ 1). Hence dim (G/M i ) x j Æ max (2“ j ≠ 4,“ j +“ Õ j ≠ 2). In particular, dim (G/M i ) x j Æ 2“ j ≠ 4, unless “ j =“ Õ j or “ j =“ Õ j +1. We determined above that the variety of non-degenerate 2-spaces G/M i has dimension 4n≠ 4, and hence by Lemma 4 it suces to show q e j=1 dim (G/M i ) x j < (e≠ 1)(4n≠ 4). This is easily proved by induction on the number of conjugacy classes e. For the base case, assume e = 2. First assume “ 1 ”= “ Õ 1 , “ 1 ”= “ Õ 1 +1, and “ 2 ”= “ Õ 2 , “ 2 ”= “ Õ 2 +1. Then by the condition on the largest eigenspaces, we have “ 1 Æ 2n≠ “ 2 , and dim (G/M i ) x 1 +dim(G/M i ) x 2 Æ (2(2n≠ “ 2 )≠ 4)+(2“ 2 ≠ 4) = 4n≠ 8< 4n≠ 4 Similarly if “ 1 = “ Õ 1 or “ 1 = “ Õ 1 + 1, and “ 2 ”= “ Õ 2 , “ 2 ”= “ Õ 2 + 1, then dim (G/M i ) x 1 + dim (G/M i ) x 2 Æ 4n≠ 6< 4n≠ 4. Finally assume that “ 1 = “ Õ 1 or “ 1 = “ Õ 1 +1, and “ 2 = “ Õ 2 , or “ 2 = “ Õ 2 +1. Since non- degenerate spaces are even dimensional, and totally singular spaces come in pairs of equal dimension, the condition on the largest eigenspace implies that “ 1 ,“ 2 Æ n. Hence by the above, dim (G/M i ) x 1 +dim(G/M i ) x 2 Æ 4n≠ 4, with equality only when “ 1 = “ 2 = n.In the latter case both C 1 and C 2 are quadratic. For induction, assume q e≠ 1 j=1 dim (G/M i ) x j < (e≠ 2)(4n≠ 4). Since C e is non-central, “ e Æ 2n≠ 2 and hence dim (G/M i ) xe Æ 4n≠ 8 < 4n≠ 4. So q e j=1 dim (G/M i ) x j < (e≠ 1)(4n≠ 4). To begin, we characterize topological generation for Sp 4 (k) and Sp 6 (k), showing when conditions(i)≠ (iii)ofTheorem2aremetthattheonlyexceptionstotopologicalgeneration 45 appear in Table 1. We then prove topological generation for Sp 8 (k), and use this as the base case for our inductive argument. Proposition2. LetG =Sp 4 (k),andC 1 ,...,C e be non-central semisimple conjugacy classes of G.Assume k does not have even characteristic. Then there exists a tuple Ê œ topo- logically generating G if and only if the following conditions hold: (i) q e i=1 “ i Æ 4(e≠ 1); (ii)itisnotthecasethat e=2 and C 1 ,C 2 are quadratic; (iii) dim Ø dim G+rank (G); (iv) it is not the case that 3Æ eÆ 4 andC 1 ,...C e have eigenspace decompositionsE 2 1 ü E 2 ≠ 1 on the natural module, or e=3 and C 1 ,C 2 have eigenspace decompositions E 2 1 ü E 2 1 and C 3 has eigenspace decomposition E 2 ⁄ ü E 2 ⁄ ≠ 1 on the natural module. To begin, it is helpful to record information about the conjugacy classes and maximal subgroups of Sp 4 (k). Lemma 23. Assume C is a non-central semisimple class of Sp 4 (k). Then dim C=4,6 or 8,anddim C=4 or 6 when C is quadratic. Proof. Pick a non-central semisimple conjugacy class C in G, and pick xœ C. (a) AssumexhaseigenspacedecompositionE 2 1 ü E 2 ≠ 1 .ThenC isquadratic, dimC G (x)= 6, and dim C = 10≠ 6 = 4. (b) AssumexhaseigenspacedecompositionE 2 ⁄ ü E 2 ⁄ ≠ 1 .ThenC isquadratic,dimC G (x)= 4, and dim C = 6. (c) Assume x has eigenspace decomposition E 2 ±1 ü E ⁄ ü E ⁄ ≠ 1.Thendim C G (x) = 4, and dim C = 6. (d) AssumexhaseigenspacedecompositionE ⁄ 1 ü E ⁄ ≠ 1 1 ü E ⁄ 2 ü E ⁄ ≠ 1 2 .ThendimC G (x) = 2, and dim C = 8. Lemma 24. Assume M i is a maximal closed subgroup of Sp 4 (k). Up to conjugacy M i has one of the following forms: (i) M i is finite 46 (ii) M i ≥ =Sp 2 (k)ÓS 2 (iii) M i ≥ =GL 2 (k).2 (iv) M i ≥ =SL 2 (k) (v) M i stabilizes a totally singular 1- or 2-space. Proof. This follows from Table 18.3 in [29], and Section 6 in [28]. Now pick conjugacy classesC 1 ,...,C e ofG =Sp 4 (k). Up to conjugacy, letM 1 ,...,M k be the maximal subgroups of G. As in Chapter 3, we show ”µ Y i for 1Æ iÆ k. By Lemma 2itsucestoshowdim > dim+dim G/M i when the hypothesis listed in Proposition 2 are satisfied. (i) AssumeM i isfinite. Thendim > 10whenconditions(ii)and(iii)fromProposition 2 hold. (ii) M i ≥ =Sp 2 (k)ÓS 2 . Note that dim G/M i = 4. We compute dim M i fl C j for the various classes C j . First assume C j is a class of involutions of type (a) listed in Lemma 23 above. The only positive dimensional conjugacy class of M i which has eigenspace decomposition E 2 1 ü E 2 ≠ 1 on the natural module is an outer class of involutions. This class centralizes of copy of Sp 2 (k), and hence dim M i fl C j =6≠ 3 = 3. For classes of types (b) and (d) in Lemma 23, the largest dimensional class inM i with the appropriate eigenspace decomposition will centralize a two dimensional torus, and hence dim M i fl C j = 4. Similarly for classes of type (c), dim M i fl C j = 2. Making use of the information on the dimensions of conjugacy classes C j provided in Lemma 23, it follows that dim Æ dim+dim G/M i if and only if 4a+6b+6c+8dÆ 3a+4b+2c+4e+4 This occurs only when c = d = 0. Next, assume it is not the case that e = 2 and C 1 ,C 2 are quadratic. Thena+2bÆ 4 only whenaÆ 4 andb = 0, oraÆ 2 andb = 1. These cases are excluded by condition (iv) of Proposition 2. (iii) M i ≥ =GL 2 (k).2. FirstassumeC j isaclassoftype(a)listedinLemma23. Thepositive dimensional classes of M i which have eigenspace decomposition E 2 1 ü E 2 ≠ 1 centralize either a two dimensional torus, or a copy of Sp 2 (k). Hence dim C j fl M i = 2. For 47 the remaining types of conjugacy classes (b)≠ (d), the largest class in M i with the appropriate eigenspace decomposition will centralize a two dimensional torus, and again dim C j fl M i = 2. Assumeconditions(ii)and(iii)ofProposition2aresatisfied. Then2a+4b+4c+6dÆ 6, only when a Æ 3 and b,c,d = 0. Again this case is excluded by condition (iv)in the statement of Proposition 2. (iv) M i ≥ = SL 2 (k). Note a regular class in M i is two dimensional. Assume it is not the case that e = 2 and C 1 ,C 2 are quadratic, or dim < 12. Then 2a+4b+4c+6dÆ 6 only when when aÆ 3 and b,c,d = 0. (v) M i isthe stabilizerofatotallysingular1-space or2-space. Applying Lemma 22,M i is not the stabilizer of a totally singular 1-space. AssumeM i is the stabilizer of a totally singular 2-space. Then dim G/M i = 3. Elements of type (a)≠ (d) listed in Lemma 23 will have at least a four dimensional centralizer inM i ,sodimC j fl M i Æ 7≠ 4 = 3. However, a+3b+3c+5dÆ 3 only when aÆ 3 and b,c,d = 0. It is checked at the end of the section that condition (iv) in Proposition 2 is necessary to ensure topological generation. ForG =Sp 6 (k) andG =Sp 8 (k), it is inconvenient to list the dimensions of all semisim- ple conjugacy classes in G as was done in Lemma 23 above. However, it will prove helpful to determine a lower bounds on the dimension of semisimple conjugacy classes in G whose largest eigenspace has dimension m (for each possible m). Let m (C) be the collection of all semisimple conjugacy classes in G whose largest eigenspaces are m dimensional, and ’ m =min{dim C |C œ m (C)}. The relevant values of ’ m forG =Sp 6 (k) andG =Sp 8 (k) are computed explicitly below. Note if xœ C has a nondegenerate eigenspace E 2m ±1 on the natural module, then C G (x) will contain a copy of Sp 2m , which has dimension 2m 2 +m. On the other hand, if x has a totally singular eigenspace E m ⁄ on the natural module, then it will have another totally singular eigenspace E m ⁄ ≠ 1 on the natural module which is determined by the first. Hence x will centralize a copy of GL m (k) of dimension m 2 , corresponding to eigenspaces of whose dimension is 2m. In particular, for any 1 Æ m Æ 2n, the value ’ m will correspond to the dimension of a conjugacy class in G whose nondegenerate eigenspaces E ±1 are as large as possible, and whose remaining totally singular spaces are then as large as possible. 48 Finally, let M i be a maximal closed subgroup of G.When C j is a semisimple conjugacy class of G, M i fl C j = D 1 fi ...fi D h where D j is a conjugacy class of M i In particular, dim C j fl M i =max{dim D j | 1Æ jÆ h}. Now let — m =min{dim C≠ dim (Cfl M i ) | C œ m (C)} To show ”µ Y i ,itsucestoprove q e j=1 (dim C j ≠ dim (C j fl M i ))> dim G/M i . Note for each C j with largest eigenspace of dimension “ j , that — “ j Æ dim C j ≠ dim (C j fl M i ). In particular, itsucestoshow q e j=1 — “ j > dim G/M i whenevertheconditionsfortopological generation listed in Theorem 2 are met. Computing the values of ’ m and establishing a lower bounds for dim C fl M j will typically suce to do this. In a few situations, more explicit calculations are required. Proposition 3. Let G =Sp 6 (k) and C 1 ,...,C e be non-central semisimple conjugacy classes of G.Assume k does not have even characteristic. Then there exists a tuple Ê œ topo- logically generating G if and only if the following conditions hold: (i) q e i=1 “ i Æ 6(e≠ 1); (ii) it is not the case that e=2 and C 1 ,C 2 are quadratic; (iii)dim Ø dim G+rank (G); (iv) it is not the case thate=3,andC 1 ,C 2 ,C 3 have eigenspace decompositionsE 4 ±1 ü E 2 û 1 . Again we record some information about maximal subgroups and semisimple conjugacy classes of Sp 6 (k). Lemma 25. Assume M i is a maximal closed subgroup of Sp 6 (k), where k does not have even characteristic. Up to conjugacy M i has one of the following forms: (i) M i is finite (ii) M i ≥ =Sp 4 (k)◊ Sp 2 (k) (iii) M i ≥ =Sp 2 (k)ÓS 3 (iv) M i ≥ =GL 3 (k).2 (v) M i ≥ =Sp 2 (k)¢ SO 3 (k) (vi) M i ≥ =P 1 ≠ P 3 49 (vii) M i ≥ =SL 2 (k) Proof. This follows from Table 18.3 in [29], and Section 6 in [28]. Finally, we record the values of ’ m for G =Sp 6 (k). - The smallest dimensional conjugacy class with a four dimensional eigenspace has eigenspace decomposition V = E 4 ±1 ü E 2 û 1 on the natural module, so ’ 4 = 21≠ (10+3) = 8. The next smallest conjugacy class C whose largest eigenspace is four dimensional eigenspace has decomposition V = E 4 ±1 ü E ⁄ ü E ⁄ ≠ 1. In this case dim C = 10. - The only conjugacy classes whose largest eigenspace are three dimensional have eigenspace decompositions E 3 ⁄ ü E 3 ⁄ ≠ 1 ,so ’ 3 = 21≠ 9 = 12. - The smallest class whose largest eigenspace is two dimensional has eigenspace decom- position E 2 ±1 ü E 2 û 1 ü E ⁄ 1 ü E 2 ⁄ ≠ 1 1 ,so ’ 2 = 21≠ (3+3+1) = 14. - The smallest class whose largest eigenspace is one dimensional is regular, so ’ 1 = 21≠ 3 = 18. Now pick conjugacy classesC 1 ,...,C e ofG =Sp 6 (k). Up to conjugacy, letM 1 ,...,M k be the maximal subgroups of G.Weprove q e j=1 — “ j > dim G/M i when the hypothesis given for topological generation in Proposition 3 are met. (i) AssumeM i isfinite. ThendimG/M i = 21and— j =’ j . Bythevaluesrecordedabove, q e j=1 — “ j > 21 when the conditions for topological generation listed in Proposition 3 are met. (ii) Assume M i ≥ =Sp 2 (k)ÓSp 4 (k). This case is covered by Lemma 22. (iii) Assume M i ≥ = Sp 2 (k)Ó S 3 .Thendim M i = 9, and dim G/M i = 12. The desired inequality follows by computing a lower bound for — j ,1 Æ j Æ 4. To compute — 4 , note all elements with a four dimensional eigenspace have at least a six dimensional centralizer in M i (the class with a six dimensional centralizer in M i is an outer class of involutions which swaps two copies of Sp 2 (k) and fixes the other copy), so — 4 = 8≠ (9≠ 3) = 5. All elements in M i have at least a three dimensional centralizer, so — 3 = 12≠ (9≠ 3)Ø 6, — 2 Ø 14≠ (9≠ 3) = 8 and — 1 Ø 18≠ (9≠ 3)≠ 12. It follows that q e j=1 — “ j > 12 when the hypotheses for Proposition 3 are met (noting that the only classes whose largest eigenspaces are three dimensional are quadratic). 50 (iv) M i ≥ = GL 3 (k).2. Then dim G/M i = 12. Again we compute a lower bound for — j , 1Æ j Æ 4. To compute a lower bound for — 4 , note elements with a four dimensional eigenspace appear in the in the commutator subgroup of M i . These elements have at least a five dimensional centralizer inGL 3 (k), so — 4 Ø 8≠ (9≠ 5) = 4. All elements in M i will have at least a three dimensional centralizer, so — 3 Ø 6, — 2 Ø 8, and — 1 Ø 12. Furthermore, by an easy computation if the largest eigenspace of a class C is two dimensional, then dim C fl M i = 8 if and only if C has eigenspace decomposition E 2 ±1 ü E 2 ⁄ ü E 2 ⁄ ≠ 1 , or E 2 ±1 ü E 2 û 1 ü E ⁄ ü E ⁄ ≠ 1. In both cases dim C = 14. Now assume conditions (i) and (ii) from Proposition 3 are met. Then q e j=1 — “ j > 12 unless e = 3 and C 1 ≠ C 3 have eigenspace decomposition E 4 ±1 ü E 2 û 1 , or e = 2 and dim C 1 = 8, dim C 2 = 14. In first case, condition (iv) of Proposition 3 is violated. In the second case, condition (iii) is violated. (v) AssumeM i ≥ =SL 2 (k)¢ SO 3 (k). Then dimM i = 6, and dimG/M i = 15. Note in this setting many conjugacy classes of G do not appear in M i . Elements with eigenspace decomposition E 4 ±1 ü E 2 û 1 appearing in M i will have eigenspace decompositions E 2 ±1 inSL 2 (k), andE 1 ü E 2 û 1 inSO 3 (k). Suchelementshaveafourdimensionalcentralizer inM i , and hence— 4 = 6. Elements whose largest eigenspace is three dimensional have eigenspace decompositions E ⁄ ü E ⁄ ≠ 1 in SL 2 (k), and E 3 1 in SO 3 (k), and so have a four dimensional centralizer. So — 3 = 8. Elements in the smallest dimensional class in M i whose largest eigenspace is two dimensional have eigenspace decomposition E 2 1 ü E 2 ⁄ ü E 2 ⁄ ≠ 1 . In this case dim C = 14, and elements in this class have a four dimensional centralizer in M i ,so — 2 = 12. Similarly, — 1 = 18≠ (6≠ 2) = 14. So q e j=1 — “ j > 15, when the appropriate hypotheses are met. (vi) Assume M i is a parabolic subgroup. M i = P 1 is handled by Lemma 22. Assume M i = P 2 .Thendim M i = 13, and dim G/M i = 8. The elements with the smallest dimensional centralizer in M i having eigenspace decomposition E 4 ±1 ü E 2 û 1 will cen- tralize a copy of GL 1 (k)◊ GL 1 (k)◊ Sp 2 (k) on the Levi subgroup, and have a four dimensionalcentralizerontheunipotentradical. Inthiscase, dimC = 13≠ 9≠ 4. Ele- ments in the class E 4 ±1 ü E ⁄ ü E ⁄ ≠ 1 will have at least an eight dimensional centralizer in M i . Hence — 4 = 4. Elements with eigenspace decomposition E 3 ⁄ ü E 3 ⁄ ≠ 1 will have at least a seven dimen- sional centralizer in M i ,so — 3 = 6. Similarly elements whose largest eigenspace is 51 two dimensional will have at least a four dimemsional cetntralizer in M i ,so — 2 = 5. Finally, — 1 = 8. So q e j=1 — “ j > 8, when the appropriate hypothesis are met. Although M i ≥ =P 3 is essentially treated by paragraph (iv) above, we check it for the sake of completeness. In this case dim M i = 15, and dim G/M i = 6. Elements with eigenspacedecompositionE 4 ±1 ü E 2 û 1 willhaveatleastaninedimensionalcentralizerin M i , andelementswitheigenspacedecompositioneigenspacedecompositionE 4 ±1 ü E 2 û 1 will have at least an eight dimensional centralizer in M i .So — 4 = 2. Elements with eigenspace decomposition E 3 ⁄ ü E 3 ⁄ ≠ 1 will have at least an eight dimensional centralizer in M i so — 3 = 4. Similarly elements in classes C whose largest eigenspace is two dimensional will have at least a ten dimensional centralizer in M i ,so — 2 = 4. Furthermore, by an easy computation dimCfl M i = 4 if and only ifC has eigenspace decompositionE 2 ±1 ü E 2 ⁄ ü E 2 ⁄ ≠ 1 , orE 2 ±1 ü E 2 û 1 ü E ⁄ ü E ⁄ ≠ 1. InbothcasesdimC = 14. Finally — 1 = 6. So q e j=1 — “ j > 6, when the hypotheses for Proposition 3 are not met. (vii) Assume M i ≥ = SL 2 (k). Then dim M i = 3, and dim G/M i = 18. All elements in M i have at least one dimensional centralizer. So — 4 Ø 6, and dimC>6if C is a non-quadratic class containing a four-dimensional eigenspace. Furthermore, — 3 Ø 10, — 2 Ø 12, — 1 Ø 16. So q e j=1 — “ j > 18, when the hypotheses from Proposition 3 are met. Finally we turn our attention to Sp 8 (k), the base case of the induction. Proposition 4. Let G =Sp 8 (k) and C 1 ,...,C e be non-central semisimple conjugacy classes of G.Assume k does not have even characteristic. Then there exists a tuple Ê œ topo- logically generating G if and only if the following conditions hold: (i) q e i=1 “ i Æ 8(e≠ 1); (ii)dim Ø dim G+rank(G); (iii) it is not the case that e=2 and C 1 ,C 2 are quadratic. Again we record information about the maximal subgroups and semisimple conjugacy classes of Sp 8 (k). Lemma 26. Assume M i is a maximal closed subgroup of Sp 8 (k), where k does not have even characteristic. Up to conjugacy M i has one of the following forms: (i) M i is finite 52 (ii) M i ≥ =Sp 6 (k)◊ Sp 2 (k) (iii) M i ≥ =Sp 4 (k)ÓS 2 (iv) M i ≥ =Sp 2 (k)ÓS 4 (v) M i ≥ =GL 4 (k).2 (vi) M i ≥ =Sp 2 (k)¢ SO 4 (k) (vii) M i ≥ = (¢ 3 i=1 Sp 2 (k)).S 3 (viii) M i ≥ =P 1 ≠ P 4 (ix) M i ≥ =SL 2 (k) Proof. This follows from Table 18.3 in [29], and Section 6 in [28]. Now we record the values of ’ m for G =Sp 8 (k). - The smallest dimensional conjugacy class with an six dimensional eigenspace has eigenspace decomposition V = E 6 ±1 ü E 2 û 1 on the natural module, so ’ 6 = 36≠ (21+3) = 12. The next smallest class C with a six dimensional eigenspace has form V =E 4 ±1 ü E ⁄ ü E ⁄ ≠ 1. In this case, dim C = 14. - The smallest class whose largest eigenspace is four dimensional has eigenspace decom- position V = E 4 ±1 ü E 4 û 1 ,so ’ 4 = 16. The next smallest class C whose largest eigenspace is four dimensional is quadratic, and contains elements of the form E 4 ⁄ ü E 4 ⁄ ≠ 1 . In this case, dim C = 20. - Thesmallestclasswhoselargesteigenspaceisthreedimensionalhaseigenspacedecom- position E 3 ⁄ ü E 3 ⁄ ≠ 1 ü E 2 ±1 ,so ’ 3 = 24. The next smallest class C whose largest eigenspace is three dimensional has form E 3 ⁄ 1 1 ü E 3 ⁄ ≠ 1 1 ü E ⁄ 2 ü E ⁄ ≠ 1 2 , and dim C = 26. - The smallest class whose largest eigenspace is two dimensional has eigenspace decom- position E 2 ±1 ü E 2 û 1 ü E 2 ⁄ ü E 2 ⁄ ≠ 1 ,so ’ 2 = 26. - The smallest class whose largest eigenspace is one dimensional is regular, so ’ 1 = 32. Pick conjugacy classes C 1 ,...,C e of G =Sp 8 (k). Up to conjugacy, let M 1 ,...,M k be the maximal subgroups ofG. Again we show q e j=1 — “ j > dim G/M i , when the hypotheses from Proposition 4 are met. 53 (i) Assume M i is finite. Then dim G/M i = 36, and — j =’ j for 1Æ j Æ 6. By the values listed above, q e j=1 — “ j > 36 when the conditions from Proposition 4 are met. (ii) Assume M i ≥ =Sp 6 (k)ÓSp 2 (k). This case is covered by Lemma 22. (iii) AssumeM i ≥ =Sp 4 (k)ÓS 2 .ThendimM i = 20, and dimG/M i = 16. All elements with asixdimensionaleigenspacehaveatleastafourteendimensionalcentralizerinM i (no elements in the outer class of involutions of M i have a six dimensional eigenspace), so — 6 = 6. All involutions in M i with a four dimensional eigenspace have at least a ten dimensional centralizer (the class of outer involutions centralizes a copy ofSp 4 (k) in M i ), and all other elements in M i with a four dimensional eigenspace will have a centralizer that is at least six dimensional. So — 4 = 6. Elements whose largest eigenspace is three dimensional have at least an eight dimen- sionalcentralizeriftheyhavetheformE 3 ⁄ ü E 3 ⁄ ≠ 1 ü E 2 ±1 , andatleastasixdimensional centralizer otherwise. So — 3 = 12. Similarly, — 2 ,— 1 Ø 12. Assume the conditions for Proposition 4 are met. Then q e j=1 — “ j > 16, except when e = 2 and at least one of C 1 ,C 2 is a non-quadratic classes whose largest eigenspace is four dimensional. By an easy computation, for any non-quadratic class C j whose largest eigenspace is four dimensional, dim C j ≠ dim C j fl M i Ø 16. (iv) M i ≥ = Sp 2 (k)Ó S 4 .Thendim M i = 12, and dim G/M i = 24. Elements with a six dimensionaleigenspacehaveatleastaninedimensionalcentralizerinM i (theelements with a nine dimensional centralizer in M i are involutions that swap two copies of Sp 2 (k) and fix the other two other copies of Sp 2 (k)). So — 6 = 9. Quadratic elements with a four dimensional eigenspace have at least a four dimensional centralizer, and all other elements with a four dimensional eigenspace have at least a six dimensional centralizer, so — 4 = 10. Elements whose largest eigenspace is three dimensional have at least a four dimensional centralizer, so — 3 Ø 16. Finally, all elements in M i have at least a three dimensional centralizer (the elements with the smallest centralizer in M i cyclically permute four copies of Sp 4 (k)), so “ 1 ,“ 2 Ø 17. It again follows that q e j=1 — “ j > 24 when the hypotheses for Proposition 4 are met, except when e=2 at least one of C 1 ,C 2 are a non-quadratic classes whose largest eigenspaces is four dimensional. However if C j is a non-quadratic class whose largest eigenspace is four dimensional, then dim C j ≠ dim C j fl M i Ø 16. (v) M i ≥ = GL 4 (k).2. Then dim M i = 16, and dim G/M i = 20. Elements with a six dimensional eigenspace live in the commutator subgroup of M i and have at least a 54 tendimensionalcentralizer. So— 6 = 6. Involutionswithafourdimensionaleigenspace have at least a ten dimensional centralizer in M i (the smallest dimensional class cen- tralizes a copy ofSp 4 (k)inM i ). A quadratic class with a four dimensional eigenspace hasatleastasixdimensionalcentralizerinM i ,so— 4 = 10. Similarly,anyelementwith a three dimensional eigenspace has at least a six dimensional centralizer, so — 3 Ø 14. Every element has at least a four dimensional centralizer in M i so — 1 ,— 2 Ø 14. Again q e j=1 — “ j > 20, when the hypothesis for Proposition 4 are met and except whene=2 at least one of C 1 ,C 2 are a non-quadratic classes whose largest eigenspaces is four dimensional. However if C j is a non-quadratic class whose largest eigenspace is four dimensional, dim C j ≠ dim C j fl M i Ø 12. (vi) M i ≥ = Sp 2 (k)¢ SO 4 (k). Then dim M i = 9, and dim G/M i = 27. No elements with a six dimensional eigenspace appear in M i . Involutions with a four dimensional eigenspacewillhaveatleastafivedimensionalcentralizerinM i ,andnonon-quadratic classeswhoselargesteigenspaceisfourdimensionalappear. Furthermore, allelements have at least a three dimensional centralizer in M i .So — 4 = 12, — j Ø 18 forj< 4, and q e j=1 — “ j > 27 when the hypotheses for Proposition 4 are met. (vii) M i ≥ = (¢ 3 i=1 Sp 2 (k)).S 3 .Thendim M i = 9, and dim G/M i = 27. No elements with a six dimensional eigenspace appear in M i . Involutions with a four dimensional eigenspace will have at least a six dimensional centralizer in M i , and no quadratic elementswhoselargesteigenspaceisfourdimensionalappear. So— 4 Ø 12,and— j Ø 18 forj< 4. So — “ j > 27, when the hypotheses for Proposition 4 are met. (viii) M i ≥ = P 1 ≠ P 4 . P 1 and P 4 are handled by Lemma 22, and paragraph (v) above. So assume M i ≥ = P 2 .Thendim M i = 25, and dim G/M i = 11. Involutions with a six dimensional eigenspace have at least a seventeen dimensional centralizer in M i (elementswiththesmallestdimensionalcentralizerinM i centralizeacopyofGL 2 (k)◊ Sp 2 (k)◊ Sp 2 (k)) on the Levi subgroup, a three dimensional space on the symmetric matrices contained in the unipotent radical, and a four dimensional space on the remaining matrices in the unipotent radical). Other elements with a six dimensional eigenspace have at least a fifteen dimensional centralizer in M i ,so — 6 = 21≠ 17 = 4. Involutions with whose largest eigenspace is four dimensional have at least a four- teen dimensional centralizer in M i , and quadratic elements with a four dimensional eigenspace have at least a twelve dimensional centralizer, so — 4 = 5. Elements whose largest eigenspace is three dimensional dimensional have at least an eight dimensional 55 centralizer in M i ,so — 3 Ø 7. Elements whose largest eigenspace is two dimensional have at least six dimensional centralizer in M i ,so — 2 Ø 7. Similarly — 1 Ø 7. So — “ j > 27, when the hypothesis for Proposition 4 are met, except when e = 2 at least oneofC 1 ,C 2 areanon-quadraticclasseswhoselargesteigenspacesisfourdimensional. However if C j is a non-quadratic class whose largest eigenspace is four dimensional, dim C j ≠ dim C j fl M i Ø 8. Now assume M i ≥ = P 3 .Thendim M i = 21, and dim G/M i = 15. Involutions with a six dimensional eigenspace have at least a sixteen dimensional centralizer in M i . Other elements with a six dimensional eigenspace have at least a fourteen dimensional centralizer in M i .So — 6 = 7. Involutions with whose largest eigenspace is four dimensional have at least a fourteen dimensional centralizer in M i , and quadratic elementswithafourdimensionaleigenspacehaveatleastatendimensionalcentralizer, so — 4 = 9. Elements whose largest eigenspace is three dimensional dimensional have at least an nine dimensional centralizer in M i ,so — 3 = 12. Similarly, — 2 ,— 1 Ø 12. So — “ j > 27, when the hypothesis for Proposition 4 are met, except when e = 2 at least oneofC 1 ,C 2 areanon-quadraticclasseswhoselargesteigenspacesisfourdimensional. Again in this situation dim C 1 +dim C 2 > 21. (ix) M i ≥ = SL 2 (k). Then dim M i = 3, and dim G/M i = 33. Every element in M i has at least a one dimensional centralizer, so — 6 Ø 10,— 4 Ø 14,— 3 Ø 22,— 2 Ø 24, and — 1 Ø 30. So — “ j > 33, when the hypothesis for Proposition 4 are met, except when e = 2 at least one of C 1 ,C 2 are a non-quadratic classes whose largest eigenspaces is four dimensional. However if C j is a non-quadratic class whose largest eigenspace is four dimensional, dim C j ≠ dim C j fl M i Ø 22. This completes the proof of the base case. We now argue by induction that there is a tuple Ê œ topologically generating a large rank subgroup with nice properties. To do this, it is first necessary to restrict the classes C 1 ,...,C e of Sp 2n (k) to classes C Õ 1 ,...,C Õ e of Sp 2n≠ 2 (k) and check that the inequality q e i=1 “ Õ i Æ (2n≠ 2)(e≠ 1) holds, where “ Õ i is the dimension of the largest eigenspace in C Õ i . Pick a semisimple class C i in Sp 2n (k), and assume that xœ C i has eigenspace decom- position V = ü m j=1 E n j – j on the natural module. Without loss of generality assume that n 1 Æ n 2 Æ ... Æ n m , so that – m is the eigenvalue corresponding to the largest eigenspace E nm – m of C i . This largest eigenspace of x is either totally singular or nondegenerate (if there aretwolargesteigenspacessuchthatoneis totallysingularandthe otherisnon-degenerate, assume E nm – m is non-degenerate). 56 To begin, assume that the largest eigenspaceE nm – m isd-dimensional and totally singular. Then every element xœ C i has eigenspaces E d ⁄ ,E d ⁄ ≠ 1 on the natural module. Let C Õ i be the conjugacy class whose elements have eigenspace decomposition ü m≠ 2 j=1 E n j – j ü E d≠ 1 ⁄ ü E d≠ 1 ⁄ ≠ 1 . Note C Õ i is a class of Sp 2n≠ 2 (k) as desired. Next assume the largest eigenspace of C i is non-degenerate, so that E nm – m = E nm ±1 ,with n m even. Let C Õ i be the class whose elements have eigenspace decomposition ü m≠ 1 j=1 E n j – j ü E nm≠ 2 ±1 . Performing the restrictions outlined above for C 1 ,...,C e , yields conjugacy classes C Õ 1 ,...,C Õ e of Sp 2n≠ 2 (k). Lemma 27. Assume C 1 ,...,C e are non-central semisimple conjugacy classes of Sp 2n (k) such that q e i=1 “ i Æ 2n(e≠ 1),anditisnotthecasethat e=2,and C 1 ,C 2 are quadratic. Furthermore, assumeC Õ 1 ,...,C Õ e are classes ofSp 2n≠ 2 (k) as described above. Then q e i=1 “ Õ i Æ (2n≠ 2)(e≠ 1). Proof. We argue by induction on the number of classes e. To begin, assume e = 2. Note that “ Õ i = “ i if and only if “ i Æ n, with equality if and only if C i is quadratic. Clearly if “ Õ 1 <“ 1 and “ Õ 2 <“ 2 ,then “ Õ 1 +“ Õ 2 Æ 2n≠ 2. Similarly, if C 1 and C 2 are both non-quadratic the desired inequality still holds. So assume C 1 is a class where “ Õ 1 <“ 1 , and C 2 is a quadratic class where “ Õ 2 =“ 2 . Note if the largest eigenspace ofC 2 is totally singular, then“ Õ 2 <“ 2 so we may assume the largest eigenspaceisnon-degenerate. Inparticular,thisimpliesthatC 2 isaclassofinvolutions,and n is even. Now if the largest eigenspace of C 1 were totally singular, then as totally singular eigenspaces come in pairs, it would follow that “ 1 Æ n≠ 1, and “ Õ 1 Æ n≠ 2, again implying the desired inequality. So assume that the largest eigenspace ofC 1 is non-degenerate. If the next largest eigenspace were totally singular, it would then follow that “ 1 Æ 2 3 n as totally singular eigenspaces come in pairs. This again would imply that “ Õ 1 +“ Õ 2 Æ 2n. Finally, assume the second largest eigenspace of C 1 is non-degenerate. As C 1 is not quadratic, there must also be a totally singular composition factor of dimension at least 2. By the hypothesis on the largest eigenspaces “ 1 Æ 2n≠ “ 2 . Hence the next largest eigenspace ofC 1 has dimension less than or equal to “ 2 ≠ 2. It follows that “ Õ 1 +“ Õ 2 Æ 2n≠ 2. For the induction assume q e≠ 1 i=1 “ i Æ (2n≠ 2)(e≠ 2). As C e is non-central, “ e Æ 2n≠ 2. This implies q e i=1 “ 1 Æ (2n≠ 2)(e≠ 1). We must also check that condition (iii) of Theorem 2 holds when we restrict to Sp 2n≠ 2 (k). Again we prove this fact for all semisimple classes C 1 ,...,C e of Sp 2n (k). 57 Lemma 28. Assume C 1 ,C 2 are conjugacy classes of Sp 2n (k) such that “ 1 +“ 2 Æ n,and it is not the case that e=2 and C 1 , C 2 are quadratic. Then dim C 1 +dim C 2 < dim G + rank(G) if and only if (i) n is even, and elements in C 1 ,C 2 have eigenspace decompositions E n ±1 ü E n û 1 ,and E n ±1 ü E n≠ 2 û 1 ü E 1 ⁄ ü E 1 ⁄ ≠ 1 ,or (ii) n is odd, and elements in C 1 ,C 2 have eigenspace decompositions E n+1 ±1 ü E n≠ 1 û 1 ,and E n≠ 1 ±1 ü E n≠ 1 û 1 ü E 1 ⁄ ü E 1 ⁄ ≠ 1 . Proof. Assume it is not the case that both C 1 and C 2 are not quadratic, and furthermore assume that “ 1 +“ 2 Æ n. As mentioned above, the two conjugacy classes with this property whosesumisofminimaldimensionwillhavenon-degenerateeigenspacesaslargeaspossible, and exactly one of the two classes will have a totally singular composition factor (with dimension as large as possible). Without loss of generality assumeC 1 is the quadratic class. If “ 1 ”= n, then the two classes whose sum have minimal dimension will have eigenspace decompositions E n+k ±1 ü E n≠ k û 1 , and E n≠ k ±1 ü E n≠ k û 1 ü E k ⁄ ü E k ⁄ ≠ 1 with k Ø 1. By an easy computation, it follows that dimC 1 +dimC 2 < 2n 2 +2n if and only ifk = 1. On the other hand, if “ 1 = n, then the two conjugacy classes whose sum have minimal dimension will have eigenspace decompositions E n ±1 ü E n û 1 , and E n ±1 ü E n≠ 2 û 1 ü E 1 ⁄ ü E 1 ⁄ ≠ 1 . In this case dim C 1 +dim C 2 =2n 2 +2n≠ 3< 2n 2 +2n. The following lemma establishes the final condition necessary for induction. Lemma 29. Assume C 1 ,C 2 are conjugacy classes of G =Sp 2n (k) such that “ 1 +“ 2 Æ n, and it is not the case that e=2 and C 1 , C 2 are quadratic. Assume C Õ 1 ,C Õ 2 are the classes of G Õ =Sp 2n≠ 2 (k) described above. Then dim C 1 +dim C 2 < dim G + rank(G) if and only if dim C Õ 1 +dim C Õ 2 < dim G Õ + rank(G Õ ) Proof. Pick C 1 ,C 2 in G such that “ 1 +“ 2 Æ n, and it is not the case that e = 2 and C 1 , C 2 are quadratic. By Lemma 27, “ Õ 1 +“ Õ 2 Æ 2n≠ 2 when we restrict to classes C Õ 1 and C Õ 2 in G Õ . Applying Lemma 28, the only classes C Õ 1 ,C Õ 2 ofSp 2n≠ 2 (k)wheredim C Õ 1 +dim C Õ 2 < dim G + rank(G) are of types (i) and (ii) listed in the above lemma. Hence it suces to consider which classes C 1 ,C 2 ofSp 2n (k) restrict to classes C Õ 1 ,C Õ 2 of types (i) and (ii)inSp 2n≠ 2 (k). Fortunately, elements in C 1 ,C 2 inSp 2n (k) have form (i)if and only if elements in C Õ 1 ,C Õ 2 in Sp 2n (k) have form (ii), and vice versa. ByLemmas27, 29andtheinductivehypothesis, thereexistsatupleÊ œ topologically generating a group containing Sp 2n≠ 2 (k). Rceall for linear groups, the existence of a tuple 58 generating a rankn≠ 1 subgroup was sucient to ensure the existence of a tuple generating a strongly regular element. Unfortunately for symplectic groups this is no longer true. The new approach will be to show that if µ X i and there is a tuple Ê œ topologically generating a reductive group of rank n≠ 1, then M i must have rankØ n≠ 3. Lemma 30. Let G be the symplectic group Sp 2n (k), and let T be a maximal torus of Sp 2n≠ 2 (k) viewed in G . Then dim t gœ G T g = dim G≠ 3. Proof. Consider the map Ï :G◊ T æ G sending (g,t)æ t g . Note that im(Ï )= t gœ G T g . Pick a generic element yœ T, and note that y is regular semisimple. It suces to compute the dimension of the fiber Ï ≠ 1 (y). There are only finitely manysœ T that are conjugate to y so if Ï (h,s)=y,then C G (y)=T 2n≠ 2 ◊ Sp 2 (k). In particular, dim C G (y)=n+2. Hence dim t gœ G T g =dim G≠ 3. Lemma 31. Assume G is simple algebraic group of rank n and H is a connected reductive group of rank s. Then dim t gœ G H g Æ dim G≠ (n≠ s). Proof. ThisisLemma3.3from[6]. SinceH isreductive,thesetofsemisimpleelementsinH containsanopendensesubsetofH. LetT beamaximaltorusofH.Then t gœ G T g isdense in H,so t gœ G T g = t gœ G H g . Now consider the usual morphism Ï : G◊ T æ t gœ G H g sending (g,t) æ t g . In this case, the map is dominant and every fiber has dimension at least n,sodim t gœ G H g Æ dim G≠ (n≠ s). Combining the previous two lemmas shows if µ X i ,then M i must have large rank. Lemma 32. Let G =Sp 2n (k) and assume there is a tuple Ê œ topologically generating a group containing Sp 2n≠ 2 (k).If µ X i for some 1Æ iÆ k, then rank M i Ø n≠ 3. Proof. Assume µ X i , and let s =rank(M i ). Let W be the free group in e letters, and Z := {w(x 1 ,...,x e ) | wœ W,(x 1 ,...,x e )œ } First note since there is a tuple Ê œ X i topologically generatingSp 2n≠ 2 (k), that generically tuples in generate a group having composition factors of dimension at least 2 n≠ 2. Furthermore, by Lemma 22, we have that generically tuples in fix no totally singular 1-subspace, and hence we may assume M i is reductive. Now let T be the maximal torus of M i . Note a generic element w(x 1 ,...,x e ) œ M i is semisimple, and hence is contained in a conjugate of a maximal torus of M i . So an open 59 subset of Z is contained in t gœ G T g , and Z µ t gœ G T g µ t gœ G M g i . Applying Lemmas 29 and 30, dim G≠ 3Æ dim Z Æ dim € gœ G M g i Æ dim G≠ (n≠ s) We now show there is a tuple Ê œ generating a large rank group with several nice properties. By Lemma 22 generic tuples in fix no 1-dimensional totally singular space. Since there is a tuples Ê œ topologically generating a group containing Sp 2n≠ 2 (k), gener- ically tuples in generate a group having a composition factor of dimension at least 2 n≠ 2. It follows that generically tuples in will not fix a totally singular 2-space, and hence that generically tuples will not fix a mixed 2-space (for otherwise it would fix a totally singular 1-space). However by Lemma 22, the dimension of tuples in fixing a non-degenerate 2-space is strictly less than the dimension of. Hence there is a tuple in not fixing any 2-dimensional subspace. By Lemma 5, fixing a 2-space is a closed condition and there- fore generically tuples in fix no 2-dimensional subspace. This in turn implies that some tuple Ê œ generates a group acting irreducibly on the natural module, and hence that generically tuples in share this property. Applying Lemmas 11 and 12, generically tuples in generate an infinite, imprimitive, irreducible subgroup. By Lemma 32, if µ X i we may assume rank M i Ø n≠ 3. So by Table 18.3 in [29] and Section 6 in [28], it suces to check ”µ X i when M i permutes two totally singular n-spaces (class type C 3 ), acts tensor decomposably on the natural module (class type C 4 ), or acts irreducibly and tensor indecomposably (class typeC 6 ) on the natural module. Repeating the argument given in Lemma 12, it immediately follows that ”µ X i ,when M i œC 3 . Next, tensor decomposable maximal subgroups having rank at least n≠ 3occur when n Æ 8. They have the form M i ≥ = Sp 2n 1 (k)¢ SO n 2 (k)with n = n 1 n 2 , or M i ≥ = (¢ 3 i=1 Sp 2 (k)).S 3 . In these cases, it can easily be checked that > dim i +dim G/M i when the hypothesis for Theorem 2 are met. For instance, if M i ≥ = Sp 2 (k)¢ SO 3 (k)in Sp 6 (k), then dim G/M i = 15. The smallest dimensional conjugacy class C in M i has eigenspace decomposition E 4 ±1 ü E 2 û 1 in G, and the second smallest class D occurring in M i has eigenspace decomposition E 2 ±1 ü E 2 û 1 ü E 1 ⁄ ü E 1 ⁄ ≠ 1 . In the first case, dim C=8 and dim Cfl M i = 2. In the second case, dim D = 14, and dim Dfl M i = 2. Hence dim > dim i +dim G/M i when the appropriate hypotheses are satisfied. 60 For maximal subgroups M i œC 6 with rank M i Ø n≠ 3, we must check M i ≥ = Sp 4 (k) in G ≥ = Sp 10 (k). In this case, dim M i = 10, and dim G/M i = 45, — 8 Ø 12, — 6 Ø 16, — 5 Ø 22, — 4 Ø 26, and — m Ø 34 for mÆ 3. Again q e i=1 — “ j > 52, in the appropriate cases. This proves that satisfying conditions (i)≠ (iv) listed in Theorem 2 is sucient to ensure topological generation. To complete the proof of Theorem 2 for semisimple classes, it remains to be shown that condition (iv) is necessary to ensure topological generation. Lemma 33. Let C 1 ,...,C e be conjugacy classes of G ≥ =Sp 2n (k) such that C 1 ,...C e and G are listed in Table 1 of Theorem 2. Then there is no tuple Ê œ topologically generating G. Proof. FirstassumeG ≥ =Sp 4 (k),andC 1 ≠ C 4 areinvolutionswitheigenspacedecomposition E 2 1 ü E 2 ≠ 1 . Makinguseofexceptionalisomorphisms,wecanviewSp 4 (k)asthegroupSO 5 (k). Inthissetting, therelevantclassofelementshaseigenspacedecompositionE 4 ≠ 1 ü E 1 1 , andin particular q 4 i=1 “ i > 12. Hence tuples in will generate a reducible subgroup, and so will not generateG topologically. Similarly, ife = 3 andC 1 ,C 2 have eigenspace decompositions E 2 1 ü E 2 ≠ 1 and C 3 has eigenspace decomposition E 2 ⁄ ü E 2 ⁄ ≠ 1 , then by viewing these classes in SO 5 (k), we again have that q 3 i=1 “ i > 10. Now assume G ≥ =Sp 6 (k), and C 1 ≠ C 3 have eigenspace decompositions E 4 ±1 ü E 2 û 1 .We would like to show all tuples Ê œ topologically generate a group preserving a pair of totally singular 3-spaces. Let Let C = C 1 , H = GL 3 (k), = r 3 i=1 (Cfl H), and consider the map Ï : G◊ æ sending( g,x 1 ,x 2 ,x 3 ) ‘æ (x g 1 ,x g 2 ,x g 3 ). An open subset of tuples in will topologically generate a group either living inside H, or inside the conjugate of H which swaps the two totally singular 3-spaces. Now pick a tuple (x 1 ,x 2 ,x 3 ) œ which topologically generates a group containing SL 3 (k). We would like to determine the dimension of the fiber Ï ≠ 1 ((x 1 ,x 2 ,x 3 )). Note Èx 1 ,x 2 ,x 3 Í fixes the two totally singular 3- spaces that are fixed by H. Furthermore for any (g,x Õ 1 ,x Õ 2 ,x Õ 3 )inthefiber,Èx Õ 1 ,x Õ 2 ,x Õ 3 Í will also fix only those two totally singular 3-spaces. It follows that g œ N G (H) ≥ = H.2, and hence that dim Ï ≠ 1 ((x 1 ,x 2 ,x 3 )) = 9. This implies the generic fiber is at most nine dimensional. On the other hand, every fiber is at least nine dimensional, so a generic fiber is nine dimensional. Since dim (Cfl H) = 4, the dimension of im(Ï )mustbe3 ·4+dim G≠ dim H = 12 + 21≠ 9 = 24. On the other hand, dim = 24, so Ï is dominant. Hence an open dense subset of tuples in fix a pair of totally singular 3 spaces. Since G/P 3 is a projective variety, the set of triples fixing a totally singular 3-space is closed. Hence the set of tuples in preserving a totally singular 61 3-space is both open and closed, and thus every tuple in generates a group living in a parabolic subgroup. 2 Unipotent classes The proof of Theorem 2 for unipotent classes is easier, as no small rank exceptions appear. For ease of reference, let us record some information about unipotent classes in Sp 2n (k) that will used in the following. Recall if x œ Sp 2n (k) has Jordan form ü J r i i then r i must be even for odd i (see Theorem 3.1 [33]). Furthermore: dim C G (x)= 1 2 ÿ ir 2 i + ÿ i<j ir i r j + 1 2 ÿ iodd r i This formula can be used to show that condition (iii) from Theorem 2 may be omitted for unipotent classes. Lemma 34. Let G = Sp 2n (k) where k is a field of odd characteristic. Assume C 1 ,...,C e are unipotent classes of G.If q e i=1 dim C i < dim G+ rank G, then e=2 and C 1 ,C 2 are quadratic. Proof. Note dim G+ rank G=2n 2 +2n. The smallest dimensional quadratic class C in Sp 2n (k) has Jordan form J n 2 . By the above, elements in this class have a n 2 dimensional centralizer, and hence dim C =n 2 +n. Any non-quadratic class will be of strictly smaller dimension. Againforthebaseandinductioncasesitwillbehelpfultoprovethatgenericallytupesin fix no 1-dimensional totally singular spaces, or 2-dimensional non-degenerate subspaces. Lemma 35. Let C 1 ,...,C e be unipotent classes of Sp 2n (k), nØ 2.Assumethat q e i=1 “ i Æ 2n(e≠ 1),anditisnotthecasethat e=2 and C 1 ,C 2 are quadratic. Then generic tuples in fix no totally singular 1-dimensional subspace, or non-degenerate 2-dimensional subspace. Proof. The argument given in Lemma 22 for totally singular 1-spaces does not depend on the classes being semisimple, and hence generic tuples in will not fix a totally singular 1-space. AssumeM i is the stabilizer of a non-degenerate 2-spaceW. There are two waysx j œ C j can fix W.Either W is contained in the fixed space E 1 , or there is some v œ V such that W is generated as a vector space byÈv,x j (v)Í, and (x j ≠ I)v”= 0, but (x j ≠ I) 2 v = 0. In the first case, dim (G/M i ) x j Æ 2(“ j ≠ 2), the second Grassmanian of E 1 . Now let Ï j : V æ V 62 be the map v‘æ (x j ≠ I) 2 v. Then in the second case, dim (G/M i ) x j is less than or equal to the dimension of projective space in ker Ï j . Again we show q e j=1 dim (G/M i ) x j < (e≠ 1)(4n≠ 4) by induction on the number of conjugacy classes e. To begin, assume C 1 ,C 2 are classes such that at least one of the two classes is not quadratic. Pick a class C j . Note that ker Ï j has dimension 2n if and only all Jordan blocks have size at most two, which occurs if and only if C j is quadratic. Hence C j is quadratic if and only if dim (G/M i ) x j =2n≠ 1. Without loss of generality assume C 1 is not quadratic. Note if C 1 has a Jordan block of size four or greater, then dim ker Ï 1 Æ 2n≠ 3 and dim (G/M i ) x 1 Æ 2n≠ 4. In this case, dim (G/M i ) x 1 +dim(G/M i ) x 2 < 4n≠ 4. SoassumeC 1 containsnoJordanblocksofsizegreaterthanthree. IfC 1 containsatleast three Jordan blocks of size three, then by the previous calculation the desired inequality is satisfied. However if C 1 contains exactly two Jordan blocks of size two, we have that dim (G/M i ) x 1 +dim(G/M i ) x 2 =4n≠ 4. In this case we need an additional argument to show there is a tuple Ê œ which topologically generates a group not fixing a non-degenerate 2-space. We use the fact that for symplectic groups in odd characteristic the closure of a group containing unipotents element of a given Jordan form will contain all unipotent elements with Jordan blocks of smaller sizes. More precisely, let = ( n 1 ,...,n r ) be a partition of n (with n 1 Ø n 2 Ø ... Ø n r ), and let = ( m 1 ,...,m s ) be a partition of n (again where m 1 Ø m 2 Ø ...Ø m s ) dominated by . In other words, n i Ø m i for 1Æ iÆ r. If a group contains a unipotent element whose Jordan blocks correspond to , then its closure will also contain unipotent elements whose Jordan blocks correspond to. In the specific case under consideration, the elements x 1 œ C 1 and x 2 œ C 2 have Jordan formsJ n 2 andJ n≠ 3 2 ü J 2 3 , fornØ 3. Letk = n≠ 2 2 andl = n≠ 3 2 . Fromthepreviousparagraph, the closure of Èx 1 ,x 2 Í will contain elements with Jordan form (J k 2 ü J 1 1 )ü (J k 2 ü J 1 1 ), and (J l 2 ü J 1 3 )ü (J l 2 ü J 1 3 ). Consider an n-dimensional totally singular subspace of V, and note thatJ k 2 ü J 1 1 andJ l 2 ü J 1 3 correspond to unipotent elements of orders 2 and 3 inSL n (k)with “ 1 +“ 2 =n. In particular, using the topological generation results given in Theorem 1 we knowthereisatupleÊ œ generatingagroupcontaining SL n (k). Hencegenericallytuples inC 1 ◊ C 2 willgenerateagroupwhosesmallestcompositionfactorisatleastn-dimensional, and in particular generically tuples in will not fix a non-degenerate 2-space. This proves the base case. For the induction, assume that q e≠ 1 j=1 dim (G/M i ) x j < (e ≠ 2)(4n ≠ 4). As dim (G/M i ) xe < 2n≠ 1< 4n≠ 4, the desired result follows immediately. 63 To prove the base case it will be helpful to record some information about the unipotent classes in Sp 4 (k). Recall the maximal subgroups of Sp 4 (k) listed in Lemma 24. Lemma 36. Assume C is a nontrivial unipotent class of Sp 4 (k). Then dim C=4,6 or 8, and dim C=4 or 6 when C is quadratic. Proof. Pick a unipotent class C and xœ C. (a) Assume x has Jordan form J 2 1 ü J 1 2 .Then C is quadratic, the centralizer C G (x) has dimension 6, and dim C = 10≠ 6 = 4. (b)If x has Jordan form J 2 2 ,then C is quadratic, the centralizer C G (x) has dimension 4, and dim C = 10≠ 4 = 6. (c)If x has Jordan form J 1 4 ,then C is quadratic, the centralizer C G (x) has dimension 2, and dim C = 10≠ 2 = 8. Lemma 37. Let C 1 ,...,C e be non-trivial unipotent conjugacy classes of Sp 4 (k), where k is an uncountable algebraically closed field not of even characteristic. Assume (i) q e i=1 “ i Æ 4e≠ 4,and (ii) it is not the case that e=2 and C 1 ,C 2 are quadratic. Then there is a tuple Ê œ topologically generating Sp 4 (k). Proof. Sincetherearenotmanyclassestocheck,itissimplesttoconsiderthevariousunipo- tent classes of Sp 4 (k) in combination with the generation properties of Sp 2 (k) established in Lemma 14. Recall it was shown that two non-central classesC 1 ,C 2 ofSp 2 (k) will contain a tuple topologically generating Sp 2 (k)unless C 1 ,C 2 are classes of involutions modulo the center. To start, assume C is the conjugacy class of transvections inSp 4 (k), so that the largest eigenspace of an element x œ C has dimension “ = n≠ 1. According to the lemma, we should be able to generateSp 4 (k) with four elements x 1 ,x 2 ,x 3 ,x 4 œ C.Byrestrictingtoa class of transvectionsC Õ inSp 2 (k) we know that we can generateSp 2 (k)withtwoelements x Õ 1 ,x Õ 2 œ C Õ . Hence with four transvections in C we can generate M = Sp 2 (k)◊ Sp 2 (k). This rules out the possibility that µ Y i for M i of types (i),(iii),(iv) and (v)listedin Lemma24. Nownotethatatransvectionfixesathreedimensionspace, soinparticularfour tranvections willnotpermutetwonon-degenerate2-dimensionalspaces. Hence if µ Y i for M i ≥ =Sp 2 (k)ÓS 2 , wemayassumethatM i ≥ =Sp 2 (k)◊ Sp 2 (k). SinceM i isareductivegroup, 64 Cfl M i is the union of finitely many classes in M, and dim Cfl M i = 2, dim G/M i = 4. Then dim = 16 > 8+4, so ”µ Y i . Now let C be the class of double transvections in G (elements with Jordan form J 2 2 ). According to the lemma, it should be possible to generate with three elements from this class. We know two elements from this class will generate a copy ofSp 2 (k)◊ Sp 2 (k), which is maximal in Sp 4 (k) modulo an outer involution. So with a third element which is not an involution, we will be able to generate Sp 4 (k) topologically. Finally, let C be the class of regular unipotent elements inSp 4 (k). The hypothesis says we should be able to generate with one element from C and another unipotent element. As discussed earlier, the closure of a group containing a unipotent element with a given Jordan formcontainsallunipotentelementswithJordanblocksofsmallersizes. Sotoprovewecan generate with one regular unipotent element and any other unipotent element, it suces to check that we can generate with one regular unipotent element and one transvection. Theonlypositivedimensionalmaximalsubgroupscontainingregularunipotentelements are parabolic subgroups and the copy ofSL 2 (k) coming from an irreducible representation. NowuptoconjugacyaregularunipotentelementisinauniqueBorelsubgroup,andhencein a unique stabilizer of a totally singular 1-space, and a unique stabilizer of a totally singular 2-space. Note the irreducible copy of SL 2 (k) contains no transvections. Furthermore, we can pick an element from the class of transvections that does not stabilize the totally singular 1-space, or totally singular 2-space fixed by the regular unipotent element. We can also pick two elements that generate a group of infinite order. Hence there will be a tuple consisting of a regular unipotent element and any other type of unipotent element that will topologically generate Sp 4 (k). We now argue by induction that there is a tuple Ê œ topologically generating a large rank group with nice properties. First we restrict the classesC 1 ,...,C e ofSp 2n (k) to classes C Õ 1 ,...,C Õ e of Sp 2n≠ 2 (k), and check the inequality q “ Õ i Æ (2n≠ 2)(e≠ 1) still holds, where “ Õ i is the dimension of the largest eigenspace in C Õ i . Pick a class C i , and assume x œ C i has Jordan form ü k i=1 J r i i . Note odd sized Jordan blocks come in pairs. Let l =min {i | i”=0, 1Æ iÆ k}.If l = 1, then remove two Jordan blocks of size one from C i .If l = 2, remove a Jordan block of size two from C i .Ifl> 2, replace one Jordan block of size J l with a Jordan block of size J l≠ 2 .Thisprocedureyields a class C Õ i of Sp 2n≠ 2 (k). 65 Lemma 38. Assume C 1 ,...,C e are non-trivial unipotent conjugacy classes of Sp 2n (k) such that q e i=1 “ i Æ 2n(e≠ 1),and C Õ 1 ,...,C Õ e are classes of Sp 2n≠ 2 (k) as described above. Then q e i=1 “ Õ i Æ (2n≠ 2)(e≠ 1). Proof. We argue by induction on the number of classes of e. For the base case, assume e = 2. Note when restricting elements of Jordan formJ n 2 to elements of the formJ n≠ 1 2 that thedimensionofthefixedspacedropsfromnton≠ 1. Henceweareinoneoftwosituations: either “ Õ i <“ i , or “ i <n. In either case, we have “ Õ 1 +“ Õ 2 Æ 2n≠ 2. For the induction, note that non-trivial classes of Sp 2n (k) have dimension less than or equal to 2n≠ 2. Now by the inductive hypothesis there is a tuple Ê œ topologically generating a group containing Sp 2n≠ 2 (k). By Lemma 35 generically tuples in will not fix a 1-dimensional totally singular space or non-degenerate 2-space. Since generically tuples in generate a group having a composition factor of dimensionØ 2n≠ 2, it follows that generically tuples in will not fix a totally singular 2-space (and hence generically tuples will not fix a mixed 2-space). In particular, there is a tuple in not fixing any 2-space. Again by Lemma 5, generically tuples in will fix no 2-dimensional subspace. This implies that some tuple Ê œ generates a group acting irreducible on the natural module, and hence generically tuples in share this property. Applying Lemmas 11 and 12, generically tuples in will generate an infinite, imprimi- tive, irreducible subgroup. By Lemma 32, if µ X i we may assume rankM i Ø n≠ 3. So by Table 18.3 in [29] and Section 6 in [28], it suces to show ”µ X i where M i permutes two totally singular n-spaces (class type C 3 ), acts tensor decomposably on the natural module (class type C 4 ), or acts irreducibly and tensor indecomposably (class typeC 6 ) on the natural module. Repeating the arguments for the semismiple classes given above, and using the fact that the dimensions of these maximal subgroups is quite small in comparison with the dimension of G, these cases are again easily excluded. It follows that there is a tuple Ê œ topologically generating G. 66 Bibliography [1] M. Aschbacher, On the maximal subgroups of the finite classical groups,M.Invent Math (1984), 76: 469-514. [2] M. Bhargava and B.H. Gross, Arithmetic invariant theory, in Symmetry: Representa- tionTheoryanditsApplications, ProgressinMathematics, vol.257, Birkhauser(2014), 33-54. [3] M. Bhargava and B.H. Gross, The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point, in Automorphic Representa- tions and L-functions, TIFR Studies in Math., vol. 22 (2013), 23-91. [4] M. Bhargava, B.H. Gross and X. Wang, Arithmetic invariant theory II ,preprint, arXiv:1310.7689. [5] M. Bhargava and W. 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Abstract (if available)

Abstract We consider a particular type of generation problem for classical algebraic groups, and use the solution of this question to provide applications to the representation theory of classical algebraic groups, as well as random generation of finite groups of Lie type. Let C₁,…,Cₑ be a specified collection of conjugacy classes of a linear or symplectic algebraic group G defined over an uncountable algebraically closed field. We answer the following question: what are necessary and sufficient conditions for the existence of elements x₁,…,xₑ ∈ C₁,…,Cₑ such that 〈x₁,…,xₑ〉 is Zariski dense in G? As a consequence, we strengthen existing results on random (r,s)-generation of finite groups of Lie type, and prove a new result concerning generic stabilizers of classical algebraic groups.

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University of Southern California Dissertations and Theses

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Asset Metadata

Creator Gerhardt, Spencer (author)

Core Title Topological generation of classical algebraic groups

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Publication Date 06/21/2017

Defense Date 05/01/2017

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag generic stabilizers,OAI-PMH Harvest,random generation of finite groups of Lie type,simple algebraic groups,Zariski dense subgroups

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Guralnick, Robert (committee chair), Kamienny, Sheldon (committee member), Zanardi, Paolo (committee member)

Creator Email sgerhard@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-389529

Unique identifier UC11264244

Identifier etd-GerhardtSp-5448.pdf (filename),usctheses-c40-389529 (legacy record id)

Legacy Identifier etd-GerhardtSp-5448.pdf

Dmrecord 389529

Document Type Dissertation

Rights Gerhardt, Spencer

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA