Euler characteristics and actions of automorphism groups of free groups

TL;DR

This paper proves that for certain manifolds with non-zero Euler characteristic modulo 6, any action by the special automorphism group of a free group or the special linear group is trivial, confirming a conjecture in Zimmer's program.

Contribution

It establishes the triviality of actions of specific automorphism groups on manifolds with particular Euler characteristics, confirming a conjecture in Zimmer's program.

Findings

Any action of ut(F_n) on the manifold is trivial.

The result applies to the subgroup ut(F_n) and the group SL_n(al Z).

Confirms a conjecture related to Zimmer's program for these manifolds.

Abstract

Let $M^{r}$ be a connected orientable manifold with the Euler characteristic $\chi(M)\not \equiv 0\operatorname{mod}6$. 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 $\mathrm{SL}_{n}(\mathbb{Z})$) $(n\geq r+2$) on $M^{r}$ by homeomorphisms is trivial. This confirms a conjecture related to Zimmer's program for these manifolds.