Notions of Anosov representation of relatively hyperbolic groups

TL;DR

This paper establishes a connection between divergent geometrically finite representations and restricted Anosov representations in the context of relatively hyperbolic groups, demonstrating their stability under deformations.

Contribution

It introduces a new interpretation of certain geometrically finite representations as restricted Anosov representations and proves their stability under small deformations.

Findings

Representations can be interpreted as restricted Anosov over specific flow spaces.

Such representations are stable under small type-preserving deformations.

Galois coverings of geometrically finite representations retain divergence and geometric finiteness.

Abstract

We prove that divergent, extended geometrically finite (in the sense of Weisman arXiv:2205.07183) representations can be interpreted as restricted Anosov (in the sense of Tholozan--Wang arXiv:2307.02934) representations over certain flow spaces. We also show that the representations of this type are stable under small type preserving deformations. As an example, we show that a representation induced from a geometrically finite one through a Galois covering, constructed in Tholozan--Wang arXiv:2307.02934, is divergent and extended geometrically finite with a non-homeomorphic boundary extension.