Cyclic hyperbolicity in CAT(0) cube complexes

TL;DR

The paper introduces and studies cyclic hyperbolicity for groups acting on CAT(0) cube complexes, proving that a virtually cocompact special group is cyclically hyperbolic precisely when it does not contain $\mathbb{F}_2 \times \mathbb{F}_2$. It establishes a structure theorem, showing such a group virtually splits as a direct product $A \times H$ with $A$ free abelian and $H$ acylindrically hyperbolic, and a strong Tits alternative: every subgroup is virtually abelian or has a finite-index subgroup whose commutator is acylindrically hyperbolic. The authors develop a hyperbolicity criterion for cone-offs, construct horomorphisms via the Roller boundary to enable a devissage argument, and derive obstruction criteria using transverse trees of hyperplanes. Applications to cocompact special groups yield concrete characterizations for right-angled Coxeter and Artin groups, graph products, graph braid groups, and BMW-groups, illustrating where cyclic hyperbolicity holds or fails. Overall, the work advances understanding of negative curvature phenomena in cube complexes and provides practical tools to verify cyclic hyperbolicity in important geometric group theory contexts.

Abstract

It is known that a cocompact special group $G$ does not contain $\mathbb{Z} \times \mathbb{Z}$ if and only if it is hyperbolic; and it does not contain $\mathbb{F}_2 \times \mathbb{Z}$ if and only if it is toric relatively hyperbolic. Pursuing in this direction, we show that $G$ does not contain $\mathbb{F}_2 \times \mathbb{F}_2$ if and only if it is weakly hyperbolic relative to cyclic subgroups, or cyclically hyperbolic for short. This observation motivates the study of cyclically hyperbolic groups, which we initiate in the class of groups acting geometrically on CAT(0) cube complexes. Given such a group $G$, we first prove a structure theorem: $G$ virtually splits as the direct sum of a free abelian group and an acylindrically hyperbolic cubulable group. Next, we prove a strong Tits alternative: every subgroup $H \leq G$ either is virtually abelian or it admits a series $H=H_0 \rhd H_1 \rhd \cdots \rhd H_k$ where $H_k$ is acylindrically hyperbolic and where $H_i/H_{i+1}$ is finite or free abelian. As a consequence, $G$ is SQ-universal and it cannot contain subgroups such that products of free groups and virtually simple groups.

Figures (7)

• Figure 1: Shifting the hyperplane $J$ by modifying the disc diagram.
• Figure 2: A comparison between coningoff and Lemma \ref{['lem:Interval']}.
• Figure 3: Decomposition of the staircase $E$ as a union of flat rectangles.
• Figure 4: How to construct a ray of hyperplanes with no bad hyperplane.
• Figure 5: Tree of hyperplanes constructed by Lemma \ref{['lem:TreeHyp']}.

Theorems & Definitions (101)

• Theorem 1.1
• Theorem 1.2: Structure theorem
• Corollary 1.3
• Theorem 1.4: Strong Tits alternative
• Corollary 1.5
• proof
• Corollary 1.6
• proof
• Conjecture 1.7
• Corollary 1.8