Euler characteristics and actions of automorphism groups of free groups

Abstract

Let $M^r$ be a connected orientable manifold with the Euler characteristic $\chi \left(M\right)\not\equiv 0mod6$. Denote by $SAut\left(F_n\right)$ the unique subgroup of index two in the automorphism group of a free group. Then any group action of $SAut\left(F_n\right)$ (and thus the special linear group ${SL}_n\left(\mathbb{Z}\right)$) with $n\ge r+2$ on $M^r$ by homeomorphisms is trivial. This confirms a conjecture related to Zimmer’s program for these manifolds.