Geometric structures and representations of surface groups

Abstract:

Representations of hyperbolic groups into higher rank Lie groups has been an active topic of study in recent years. In particular the character variety associated 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 components 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 particular on the locally symmetric space and parabolic structures associated to Anosov representations. The first part of this thesis is rather general, and discuss parabolic structures constructed using a domain of discontinuity as well as their relation with the locally symmetric space for certain Anosov representations. 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 \(Sp(2n, \mathbb{R})\), a particular class of higher rank Teichmüller spaces. We characterize maximal representations in terms of geometric structures that admit a special fibration. Finally we study maximal representations that are also Borel Anosov, and show in particular 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 \(Sp(2n, \mathbb{R})\)" [Dav24]. The preprint [DR24] is joint work with Max Riestenberg.

[Dav23] Colin Davalo, "Nearly geodesic immersions and domains of discontinuity", 2023, 2303.11260.
[Dav24] Colin Davalo, "Maximal and borel anosov representations into \(sp(4,r)\)", Advances in Mathematics 442 (2024), 109578.
[DR24] Colin Davalo and Max Riestenberg, "Finite-sided dirichlet domains for anosov representations", 2024.