Cubulating a free-product-by-cyclic group

TL;DR

This paper proves that certain automorphisms of free-product groups lead to their mapping tori acting on hyperbolic CAT(0) cube complexes, extending previous results to more general group structures.

Contribution

It generalizes Hagen and Wise's result by showing that fully irreducible, atoroidal automorphisms of free-product groups produce mapping tori acting on hyperbolic CAT(0) cube complexes.

Findings

Mapping tori act relatively geometrically on hyperbolic CAT(0) cube complexes.

Generalizes previous results from free-by-cyclic to free-product-by-cyclic groups.

Extends understanding of group actions on non-positively curved spaces.

Abstract

Let $G = H_1 * ... * H_k * F_r$ be a torsion-free group and $\phi$ an automorphism of $G$ that preserves this free factor system. We show that when $\phi$ is fully irreducible and atoroidal relative to this free factor system, the mapping torus $\Gamma = G \rtimes_{\phi} \mathbb{Z}$ acts relatively geometrically on a hyperbolic CAT(0) cube complex. This is a generalisation of a result of Hagen and Wise for hyperbolic free-by-cyclic groups.