Hyperbolic hyperbolic-by-cyclic groups are cubulable

TL;DR

This paper proves that certain hyperbolic-by-cyclic groups, including those with torsion, are cubulable, providing new proofs and extending existing theorems in geometric group theory.

Contribution

It establishes cubulability of hyperbolic hyperbolic-by-cyclic groups and extends known results to groups with torsion, offering new proofs and broader applicability.

Findings

Hyperbolic hyperbolic-by-cyclic groups are cubulable.

An alternative proof for hyperbolic free-by-cyclic groups cubulability.

Extension of hyperbolic-by-cyclic group characterization to groups with torsion.

Abstract

We show that the mapping torus of a hyperbolic group by a hyperbolic automorphism is cubulable. Along the way, we (i) give an alternate proof of Hagen and Wise's theorem that hyperbolic free-by-cyclic groups are cubulable, and (ii) extend to the case with torsion Brinkmann's thesis that a torsion-free hyperbolic-by-cyclic group is hyperbolic if and only if it does not contain $\mathbb{Z}^2$-subgroups.