A note on torsion subgroups of groups acting on finite-dimensional CAT(0) cube complexes

Abstract

In this article, we state and prove a general criterion which prevents some groups from acting properly on finite-dimensional CAT(0) cube complexes. As an application, we show that, for every non-trivial finite group F, the lamplighter group F≀F2 over a free group does not act properly on a finite-dimensional CAT(0) cube complex (although it acts properly on an infinite-dimensional CAT(0) cube complex). We also deduce from this general criterion that, given a group G acting on a CAT(0) cube complex of finite dimension and an infinite torsion subgroup L≤G, either the normaliser NG(L) is virtually (locally finite)-by-(free abelian) or, for every k≥1, NG(L) contains a non-abelian free subgroup commuting with a subgroup of L of size ≥k.