SAut(Fn) cannot act on small tori

TL;DR

This paper proves that the group SAut(Fn), a subgroup of automorphisms of a free group, cannot non-trivially act on small-dimensional tori, supporting aspects of the Zimmer program.

Contribution

It establishes that all smooth actions of SAut(Fn) on low-dimensional tori are trivial, extending understanding of group actions in geometric group theory.

Findings

SAut(Fn) actions on low-dimensional tori are trivial

Supports the generalized Zimmer program

Provides new constraints on automorphism group actions

Abstract

We study smooth actions of SAut(Fn), the unique subgroup of index two in the automorphism group of a free group of rank n, as a part of the generalized 'Zimmer program'. In particular, we show that every action of SAut(Fn) on a low dimensional torus is trivial.