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GEOMETRIC PROPERTIES OF ANOSOV REPRESENTATIONS
by
Guillaume Dreyer
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Ful llment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MATHEMATICS)
August 2012
Copyright 2012 Guillaume Dreyer
Object Description
| Title | Geometric properties of Anosov representations |
| Author | Dreyer, Guillaume |
| Author email | dreyfactor@gmail.com;dreyfactor@gmail.com |
| Degree | Doctor of Philosophy |
| Document type | Dissertation |
| Degree program | Mathematics |
| School | College of Letters, Arts And Sciences |
| Date defended/completed | 2012-05-17 |
| Date submitted | 2012-07-28 |
| Date approved | 2012-07-30 |
| Restricted until | 2012-07-30 |
| Date published | 2012-07-30 |
| Advisor (committee chair) | Bonahon, Francis |
| Advisor (committee member) |
Honda, Ko Asok, Aravind Haydn, Nicolai Däppen, Werner |
| Abstract | Let S be a connected, closed, oriented surface of negative Euler characteristic, we consider the PSLn(R)-character variety Rep_{PSLn(R)}(S). An interesting connected component of the latter space is the Hitchin space H_{PSLn(R)}(S): it contains a copy of the Teichm uller space T(S), and hence is regarded as the higher rank Teichm uller space in the case of PSLn(R). In order to study the elements in the Hitchin space H_{PSLn(R)}(S), F.Labourie introduced the notion of Anosov representation. In particular, he proved that every Hitchin representation is discrete and injective, some properties already shared by Teichm uller representations. In this dissertation, we extend to Anosov representations several classic tools from hyperbolic geometry designed to study Teichm uller representations: we generalize Thurston's length function and cataclysm deformation, and analyze how these two notions relate to each other. We then explain how these techniques illuminate the geometry of Anosov representations, and provide crucial information about a new system of coordinates on the Hitchin space H_{PSLn(R)}(S). |
| Keyword | surface group representation; Anosov representation; Hitchin representation; Hitchin component; Anosov flow; Hölder geodesic current; length function; cataclysm deformation; geodesic lamination; Thurston's intersection number |
| 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 |
| Provenance | Electronically uploaded by the author |
| Type | texts |
| Legacy record ID | usctheses-m |
| Rights | Dreyer, Guillaume |
| 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 author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright. The original signature page accompanying the original submission of the work to the USC Libraries is retained by the USC Libraries and a copy of it may be obtained by authorized requesters contacting the repository e-mail address given. |
| Repository name | University of Southern California Digital Library |
| Repository address | USC Digital Library, University of Southern California, University Park Campus MC 7002, 106 University Village, Los Angeles, California 90089-7002, USA |
| Repository email | cisadmin@usc.edu |
| Archival file | uscthesesreloadpub_Volume4/etd-DreyerGuil-1055-0.pdf |
Description
| Title | Page 1 |
| Full text | GEOMETRIC PROPERTIES OF ANOSOV REPRESENTATIONS by Guillaume Dreyer A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Ful llment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MATHEMATICS) August 2012 Copyright 2012 Guillaume Dreyer |
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