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HYPERBOLIC GEOMETRY AND CANONICAL TRIANGULATIONS IN DIMENSION
THREE
by
Franc¸ois Gu´eritaud
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MATHEMATICS)
December 2006
Copyright 2006 Franc¸ois Gu´eritaud
Object Description
| Title | Hyperbolic geometry and canonical triangulations in dimension three |
| Author | Guéritaud, François |
| Author email | Francois.Gueritaud@normalesup.org |
| Degree | Doctor of Philosophy |
| Document type | Dissertation |
| Degree program | Mathematics |
| School | College of Letters, Arts and Sciences |
| Date defended/completed | 2006-10-31 |
| Date submitted | 2006 |
| Restricted until | Unrestricted |
| Date published | 2006-11-15 |
| Advisor (committee chair) | Bonahon, Francis |
| Advisor (committee member) |
Ekholm, Tobias Jonckheere, Edmond |
| Abstract | We explicitly construct the unique hyperbolic metric carried by the following three -- dimensional spaces: punctured--torus bundles over the circle with pseudo -- Anosov monodromy, and convex cores of quotients of quasifuchsian punctured -- torus groups with given boundary pleating data. The method is to choose a particular decomposition of the space into topological ideal tetrahedra, and to endow the tetrahedra with dihedral angles. In the space of all dihedral angle assignments that satisfy certain linear constraints, we then find a critical point for the functional defined as the sum of the volumes of hyperbolic ideal tetrahedra with the given angles. By a theorem of Rivin, these geometric tetrahedra defined by the critical point define the complete hyperbolic metric. As a consequence, we prove that, in the resulting metric, the triangulation initially chosen is canonical in a purely geometric sense, which can be defined either in terms of convex hulls of discrete isotropic orbits in Minkowski space or, in the bundle case, in terms of the Ford--Vorono\"{\i} domain with respect to the cusp. Our particular choice of triangulation is dictated only by its apparent combinatorial convenience, so the result can be paraphrased by saying that, for Kleinian punctured--torus groups, the hyperbolic geometry chooses the most convenient combinatorics. In the quasifuchsian case, our construction provides a new proof of a theorem of C. Series which says that the two measured pleating laminations on the boundary of the convex core determine the Kleinian group, continuously and uniquely up to conjugacy. We also summarize some applications to knot theory. In the third and last chapter, we prove a related number--theoretic result: a certain family of Laurent polynomials, which generalize the Markoff numbers, has only positive coefficients. |
| Keyword | hyperbolic geometry; punctured torus; pleating laminations |
| Language | English |
| Part of collection | University of Southern California dissertations and theses |
| Publisher (of the original version) | University of Southern California |
| Place of publication (of the original version) | Los Angeles, California |
| Publisher (of the digital version) | University of Southern California. Libraries |
| Type | texts |
| Legacy record ID | usctheses-m147 |
| Rights | Guéritaud, François |
| Repository name | Libraries, University of Southern California |
| Repository address | Los Angeles, California |
| Repository email | http://www.usc.edu/isd/libraries/services/ask_a_librarian/email/ |
| Filename | etd-Gueritaud-20061115 |
| Archival file | uscthesesreloadpub_Volume44/etd-Gueritaud-20061115.pdf |
Description
| Title | Page 1 |
| Full text | HYPERBOLIC GEOMETRY AND CANONICAL TRIANGULATIONS IN DIMENSION THREE by Franc¸ois Gu´eritaud A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MATHEMATICS) December 2006 Copyright 2006 Franc¸ois Gu´eritaud |
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