Euler characteristics and actions of automorphism groups of free groups

Let $M^r$ be a connected orientable manifold with the Euler characteristic $\chi (M)\equiv ̸0\mathrm{mod}6$. Denote by $\mathrm{S}\mathrm{A}\mathrm{u}\mathrm{t}(F_n)$ the unique subgroup of index two in the automorphism group of a free group. Then any group action of $\mathrm{S}\mathrm{A}\mathrm{u}\mathrm{t}(F_n)$ (and thus the special linear group ${\mathrm{S}\mathrm{L}}_n(\mathbb{Z})$) $(n\ge r+2$) on $M^r$ by homeomorphisms is trivial. This confirms a conjecture related to Zimmer’s program for these manifolds.