Hyperbolic hyperbolic-by-cyclic groups are cubulable

TL;DR

The paper establishes that the mapping torus $G\rtimes_\phi \mathbb{Z}$ of a hyperbolic group $G$ by a hyperbolic automorphism is cubulable whenever both $G$ and the extension are hyperbolic. The authors develop a relative cubulation bootstrap, combining techniques from Groves–Manning and Hsu–Wise with a careful analysis of free factor systems, train-track dynamics, and peripheral suspensions, to prove cubulability in the torsion-free case and then extend to torsion via Dunwoody–Stallings decompositions and Brinkmann-type arguments. As a result, they provide an alternative approach to Hagen–Wise-type cubulations for hyperbolic free-by-cyclic groups and obtain corollaries such as virtual specialness, linearity, separability properties, and Anosov representations for the hyperbolic-by-cyclic setting, while also posing an algebraic characterization problem for hyperbolic groups admitting hyperbolic automorphisms. The work thus advances the understanding of when hyperbolic extensions act geometrically on CAT(0) cube complexes and connects free-factor technology with modern cubulation techniques.

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.

Theorems & Definitions (20)

• Theorem : \ref{['cor;main_result']}
• Corollary
• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Theorem 3.2: cf. DK23
• Lemma 3.3: Guirardel
• proof
• Theorem 3.4: cf. GM