  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.
