On Borel Anosov representations in even dimensions

TL;DR

This paper proves that certain hyperbolic groups with specific Anosov representations are virtually free or surface groups, providing an affirmative answer to a question about Borel Anosov representations.

Contribution

It establishes a classification result for hyperbolic groups admitting particular Borel Anosov representations into high-dimensional projective linear groups.

Findings

Hyperbolic groups with $P_{2q+1}$-Anosov representations are virtually free or surface groups.

Confirmed Sambarino's question for Borel Anosov representations into $ ext{SL}(4q+2, r)$.

Provides structural insights into hyperbolic groups with specific Anosov representations.

Abstract

We prove that a word hyperbolic group which admits a $P_{2q+1}$-Anosov representation into $\mathsf{PGL}(4q+2, \mathbb{R})$ contains a finite-index subgroup which is either free or a surface group. As a consequence, we give an affirmative answer to Sambarino's question for Borel Anosov representations into $\mathsf{SL}(4q+2,\mathbb{R})$.