Cusped Hitchin representations and Anosov representations of geometrically finite Fuchsian groups

TL;DR

This paper develops a theory of Anosov representations for geometrically finite Fuchsian groups in SL(d,R), showing that cusped Hitchin representations are Borel Anosov and analyzing their stability and deformation properties.

Contribution

It introduces a new framework for Anosov representations of geometrically finite Fuchsian groups, connecting cusped Hitchin representations with existing theories.

Findings

Cusped Hitchin representations are Borel Anosov.

Anosov representations are stable under deformations.

Limit maps vary analytically.

Abstract

We develop a theory of Anosov representation of geometrically finite Fuchsian groups in SL(d,R) and show that cusped Hitchin representations are Borel Anosov in this sense. We establish analogues of many properties of traditional Anosov representations. In particular, we show that our Anosov representations are stable under type-preserving deformations and that their limit maps vary analytically. We also observe that our Anosov representations fit into the previous frameworks of relatively Anosov and relatively dominated representations developed by Kapovich-Leeb and Zhu.