Ping Pong on CAT(0) cube complexes

Abstract

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