Fixed points for actions of Aut(Fn) on CAT(0) spaces

TL;DR

This paper investigates fixed point properties of automorphism groups of free groups acting on CAT(0) spaces, establishing conditions under which these groups must have a fixed point, with implications for their representations.

Contribution

It proves fixed point theorems for Aut(Fn) and SAut(Fn) acting on low-dimensional CAT(0) spaces, extending understanding of their geometric actions.

Findings

Aut(Fn) fixes a point on certain low-dimensional CAT(0) spaces

Results apply to irreducible representations of Aut(Fn)

Similar fixed point results hold for SAut(Fn)

Abstract

For n greater or equal 4 we discuss questions concerning global fixed points for isometric actions of Aut(Fn), the automorphism group of a free group of rank n, on complete CAT(0) spaces. We prove that whenever Aut(Fn) acts by isometries on complete d-dimensional CAT(0) space with d is less than 2 times the integer function of n over 4 and minus 1, then it must fix a point. This property has implications for irreducible representations of Aut(Fn), which are also presented here. For SAut(Fn), the unique subgroup of index two in Aut(Fn), we obtain similar results.