Euler characteristics and actions of automorphism groups of free groups

Abstract

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

The article arXiv:1609.07699 is split into two parts. This is the first part. As suggested by the referee, the main result has been generalized to topological actions of automorphism group of free groups. The title is also changed. To appear in Algebraic and Geometric Topology

References in corpus (2)

• Groups acting on manifolds: around the Zimmer program
• Invariant measures and measurable projective factors for actions of higher-rank lattices on manifolds

Cited by in corpus (5)

• Symmetries of flat manifolds, Jordan property and the general Zimmer program
• Property FW, differentiable structures, and smoothability of singular actions
• A survey of topological Zimmer's program
• Zimmer's conjecture for lattice actions: the $\mathrm{SL}(n,\mathbb{C})$-case
• Topological symmetries of simply-connected four-manifolds and actions of automorphism groups of free groups