  Let $G$ be a group acting properly and essentially on an irreducible,
non-Euclidean finite dimensional CAT(0) cube complex $X$ without fixed points
at infinity. We show that for any finite collection of simultaneously
inessential subgroups $\{H_1, \ldots, H_k\}$ in $G$, there exists an element
$g$ of infinite order such that $\forall i$, $\langle H_i, g\rangle \cong H_i *
\langle g\rangle$. We apply this to show that any group, acting faithfully and
geometrically on a non-Euclidean possibly reducible CAT(0) cube complex, has
property $P_{naive}$ i.e. given any finite list $\{g_1, \ldots, g_k\}$ of
elements from $G$, there exists $g$ of infinite order such that $\forall i$,
$\langle g_i, g\rangle \cong \langle g_i \rangle *\langle g\rangle$. This
applies in particular to the Burger-Moses simple groups that arise as lattices
in products of trees. The arguments utilize the action of the group on its
Poisson boundary and moreover, allow us to summarise equivalent conditions for
the reduced $C^*$-algebra of the group to be simple.

Group Theory;Metric Geometry