A note on locally elliptic actions on cube complexes

TL;DR

This paper proves that groups acting locally elliptically on finite dimensional CAT(0) cube complexes must fix a point, and provides an example of a group with specific fixed point properties related to property (T).

Contribution

It establishes a fixed point result for locally elliptic actions and answers a question about fixed point properties of certain groups on CAT(0) cube complexes.

Findings

Groups acting locally elliptically on finite dimensional CAT(0) cube complexes fix a point.

Constructs an example of a group without property (T) with fixed point properties on cube complexes.

Addresses a question by Barnhill and Chatterji regarding fixed point actions.

Abstract

We deduce from Sageev's results that whenever a group acts locally elliptically on a finite dimensional CAT(0) cube complex, then it must fix a point. As an application, we give an example of a group G such that G does not have property (T), but G and all its finitely generated subgroups can not act without a fixed point on a finite dimensional CAT(0) cube complex, answering a question by Barnhill and Chatterji.