A condition that prevents groups from acting fixed point free on cube complexes

TL;DR

This paper introduces a group-theoretic condition guaranteeing fixed points for group actions on CAT(0) cube complexes, with applications to automorphism groups of free groups.

Contribution

It identifies a new condition ensuring fixed points in cube complex actions and applies it to automorphism groups of free groups.

Findings

Automorphism group of free groups satisfies the fixed point condition.

Subgroup of index two in automorphism group also satisfies the condition.

Fixed point criterion applies broadly to groups acting on cube complexes.

Abstract

We describe a group theoretic condition which ensures that any cellular action of a group satisfying this condition on a CAT(0) cube complex has a global fixed point. In particular, we show that this fixed point criterion is satisfied by the automorphism group of a free group of rank n. For the unique subgroup of index two in the automorphism group of a free group, we obtain a similar result.