Volume 18, issue 2 (2018)

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Euler characteristics and actions of automorphism groups of free groups

Shengkui Ye

Algebraic & Geometric Topology 18 (2018) 1195–1204
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 ${SL}_{n}\left(ℤ\right)$) with $n\ge r+2$ on ${M}^{r}$ by homeomorphisms is trivial. This confirms a conjecture related to Zimmer’s program for these manifolds.

Keywords
Zimmer's program, Euler characteristics, matrix group actions
Primary: 57S20
Secondary: 57S17