Representations of groups with CAT(0) fixed point property

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.