On projective Anosov subgroups of symplectic groups

TL;DR

This paper establishes restrictions on hyperbolic groups admitting projective Anosov representations into symplectic groups, linking boundary topology to algebraic properties of the groups.

Contribution

It proves new constraints on hyperbolic groups with Anosov representations into symplectic groups, connecting boundary topology with group structure.

Findings

Hyperbolic groups with boundary containing a 2-sphere cannot have such representations.

Groups with these representations are virtually free or surface groups.

Results align with independent findings by Dey-Greenberg-Riestenberg.

Abstract

We prove that a word hyperbolic group whose Gromov boundary properly contains a $2$-sphere cannot admit a projective Anosov representation into $\mathsf{Sp}_{2m}(\mathbb{C})$, $m\in \mathbb{N}$. We also prove that a word hyperbolic group which admits a projective Anosov representation into $\mathsf{Sp}_{2m}(\mathbb{R})$ is virtually a free group or virtually a surface group, a result established indepedently by Dey-Greenberg-Riestenberg.