Representations of groups with CAT(0) fixed point property

TL;DR

This paper investigates the representations of groups with the CAT(0) fixed point property, demonstrating finiteness of images in various algebraic and geometric contexts, especially over fields of positive characteristic.

Contribution

It establishes new rigidity results for representations of groups with CAT(0) fixed point property, including classical groups and automorphism groups, over different fields.

Findings

Representations over positive characteristic fields have finite image.

Low-dimensional complex representations with compact closure also have finite image.

Rigidity results apply to groups like SL_k(ℤ), SAut(F_k), and Mod(Σ_g).

Abstract

We show that certain representations over fields with positive characteristic of groups having CAT(0) fixed point property ${\rm F}\mathcal{B}_{\widetilde{A}_n}$ have finite image. In particular, we obtain rigidity results for representations of the following groups: the special linear group over $\mathbb{Z}$, $SL_k(\mathbb{Z})$, the special automorphism group of a free group, ${\rm SAut}(F_k)$, the mapping class group of a closed orientable surface, ${\rm Mod}(\Sigma_g)$, and many other groups. In the case of characteristic zero we show that low dimensional complex representations of groups having CAT(0) fixed point property ${\rm F}\mathcal{B}_{\widetilde{A}_n}$ have finite image if they always have compact closure.