# 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.

## Key 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.