TY  - GEN
CY  - Heidelberg
A1  - Davalo, Colin
TI  - Geometric structures and representations of
surface groups
N2  - 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
ID  - heidok35082
AV  - public
UR  - https://archiv.ub.uni-heidelberg.de/volltextserver/35082/
Y1  - 2024///
ER  -