Ping pong on CAT(0) cube complexes

TL;DR

This paper proves a ping-pong type result for groups acting on CAT(0) cube complexes, showing the existence of elements that generate free products with given subgroups, and explores implications for group properties and operator algebras.

Contribution

It introduces a new ping-pong argument for groups on CAT(0) cube complexes and applies it to establish property P_{naive} and conditions for simplicity of reduced C*-algebras.

Findings

Existence of infinite order elements forming free products with inessential subgroups.

Groups acting on CAT(0) cube complexes have property P_{naive}.

Criteria for simplicity of the group's reduced C*-algebra.

Abstract

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.