title: Geometric structures and representations of  surface groups
creator: Davalo, Colin
subject: ddc-510
subject: 510 Mathematics
description: Representations of hyperbolic groups into higher rank Lie groups has been an  active topic of study in recent years. In particular the character variety associ-  ated with a surface group for some semi-simple Lie group of non-compact type  admits remarkable connected components containing only discrete and faithful  representations. A union of such connected components is called a higher rank  Teichmüller space. In all the known cases, the representations in these compo-  nents all satisfy an Anosov property, which is a dynamical property stronger  than being discrete and faithful. Some of these spaces can be interpreted as  spaces of geometric structures: as for instance convex projective structures on  surfaces, or fibered photon structures.  In this thesis, we bring original contributions to this area, focusing in par-  ticular on the locally symmetric space and parabolic structures associated to  Anosov representations. The first part of this thesis is rather general, and dis-  cuss parabolic structures constructed using a domain of discontinuity as well  as their relation with the locally symmetric space for certain Anosov repre-  sentations. We study more precisely the domains of discontinuity that can be  interpreted as domains of proper Busemann functions.  The second part focuses on maximal representations in Spp2n, Rq, a particu-  lar class of higher rank Teichmüller spaces. We characterize maximal represen-  tations in terms of geometric structures that admit a special fibration. Finally  we study maximal representations that are also Borel Anosov, and show in par-  ticular that in Spp4, Rq these representations are Hitchin, answering a question  from Canary.  This thesis encompasses the results of the arxiv preprints Nearly geodesic  immersions and domains of discontinuity [Dav23] and Finite-sided Dirichlet  domains for Anosov subgroups [DR24] , a future preprint Geometric structures  for maximal representations and pencils , and finally the article Maximal and  Borel Anosov representations in Spp2n, Rq [Dav24]. The preprint [DR24] is joint  work with Max Riestenberg
date: 2024
type: Dissertation
type: info:eu-repo/semantics/doctoralThesis
type: NonPeerReviewed
format: application/pdf
identifier: https://archiv.ub.uni-heidelberg.de/volltextserverhttps://archiv.ub.uni-heidelberg.de/volltextserver/35082/1/Thesis%20DAVALO%20final.pdf
identifier: DOI:10.11588/heidok.00035082
identifier: urn:nbn:de:bsz:16-heidok-350820
identifier:   Davalo, Colin  (2024) Geometric structures and representations of surface groups.  [Dissertation]     
relation: https://archiv.ub.uni-heidelberg.de/volltextserver/35082/
rights: info:eu-repo/semantics/openAccess
rights: Please see front page of the work (Sorry, Dublin Core plugin does not recognise license id)
language: eng