Anosov representations

Hitchin representations in lattices

Hitchin representations are a special kind of representations from the fundamental group of a closed surface to SL(n,R). Those are injective representations with discrete image that satisfy nice geometric properties. This talk is about finding Hitchin representations in lattices of SL(n,R) or of other split Lie groups. In first part, we define the Hitchin component and explain what lattices are. In the second part, we give motivations for this problem and state new results.

Paper: Zariski-dense surface groups in non-uniform lattices of split real Lie group 

Non-strictly convex divisible convex sets

In these talks, we give a construction of non-strictly convex, non-symmetric and irreducible divisible convex sets in any dimension at least 3, using a bending procedure.

Thurston's asymmetric metric and its generalization

We give an introduction of the Thurston's asymmetric metric on Teichmuller space and discuss some generalization of this metric to Hitchin components, as an important example of Anosov representations. 

Paper: Thurston's Asymmetric metrics for Anosov representations, with Leon Carvajales, Beatrice Pozzetti, and Anna Wienhard

Description of domains of discontinuity for Anosov representations

A geometry in the sense of Klein is given by a Lie group $G$ and a space $X$ on which $G$ acts transitively. Given a geometry $(G,X)$, he defines the notion of a $(G,X)$-structure on a manifold. Given a representation of a countable group into a Lie group, and a domain of discontinuity for it's action on a space $X$, one can define an associated geometric structure on the quotient of the domain.

For Anosov representations, domains of discontinuity on flag manifols have been constructed by Guichard,Wienhard and Kapovitch, Leeb, Porti. We will see how to understand the topology of the quotient of some of these domains for representation of a surface group that admits a nearly totally geodesic equivariant surface in the symmetric space.

Constructing proper affine actions via higher strip deformations

We use ideas of Danciger-Gueritaud-Kassel and the Margulis invariant to construct proper affine deformations of Fuchsian free groups in SO(2n,2n-1). Some of these ideas can also work for deformations away from the Fuchsian locus.

Paper: Constructing proper affine actions via higher strip deformations 

A Notion of Anosov Representations over Closed Subflows

The notion of Anosov representations of a hyperbolic group, introduced by Labourie, generalizes the notion of convex-cocompact representations to higher rank semisimple Lie groups. On the other hand, there are many non-discrete representations that have good geometric behaviors, such as the primitive-stable representations introduced by Minsky. In this talk, we introduce a weakening of the notion of Anosov representations by requiring the Anosov condition to hold only on a closed geodesic subflows. This weakening allows for such non-discrete representations to arise as examples. We discuss several properties of this type of representations that are analogous to the properties of classical Anosov representations, such as the stability of this type of representations and the Hölder continuity of the limit maps.

Paper: https://arxiv.org/abs/2112.11305