  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 $F \wr \mathbb{F}_2$ over a free group does not act properly
on a finite-dimensional CAT(0) cube complex (although it acts properly on a
infinite-dimensional CAT(0) cube complex). We also deduce from this general
criterion that, roughly speaking, given a group $G$ acting on a CAT(0) cube
complex of finite dimension and an infinite torsion subgroup $L \leq G$, either
the normaliser $N_G(L)$ is close to be free abelian or, for every $k \geq 1$,
$N_G(L)$ contains a non-abelian free subgroup commuting with a subgroup of $L$
of size $\geq k$.
