Representations of groups with CAT(0) fixed point property

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.