On the Girth of Groups acting on CAT(0) cube complexes

TL;DR

This paper establishes conditions under which groups acting on finite-dimensional CAT(0) cube complexes have infinite girth, and explores the Girth Alternative, providing counterexamples in broader contexts.

Contribution

It introduces a sufficient condition for infinite girth in lattices acting on CAT(0) cube complexes and demonstrates the limits of the Girth Alternative with counterexamples.

Findings

Lattices can have infinite girth under certain conditions.

The Girth Alternative holds for groups acting geometrically.

Counterexamples show the alternative fails in some cocompact actions.

Abstract

We obtain a sufficient condition for lattices in the automorphism group of a finite dimensional CAT(0) cube complex to have infinite girth. As a corollary, we get a version of Girth Alternative for groups acting geometrically: any such group is either {locally finite}-by-{virtually abelian} or it has infinite girth. We produce counterexamples to show that the alternative fails in the general class of groups acting cocompactly on finite dimensional CAT(0) cube complexes by obtaining examples of non virtually solvable groups which satisfy a law.