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 $$\mathrm{F}\mathcal {B}_{\widetilde{A}_n}$$ have finite image. In particular, we obtain rigidity results for representations of the following groups: the special linear group over $${\mathbb {Z}}$$ , $${\mathrm{SL}}_k({\mathbb {Z}})$$ , the special automorphism group of a free group, $$\mathrm{SAut}(F_k)$$ , the mapping class group of a closed orientable surface, $$\mathrm{Mod}(\Sigma _g)$$ , 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 $$\mathrm{F}\mathcal {B}_{\widetilde{A}_n}$$ have finite image if they always have compact closure.