Cyclic hyperbolicity in CAT(0) cube complexes

TL;DR

This paper explores cyclic hyperbolicity in groups acting on CAT(0) cube complexes, establishing structural theorems and a strong Tits alternative that classify subgroups and reveal the group's algebraic properties.

Contribution

It introduces the concept of cyclically hyperbolic groups within CAT(0) cube complexes and proves a structure theorem and Tits alternative for these groups, advancing understanding of their subgroup structure.

Findings

Groups virtually split as a direct sum of free abelian and acylindrically hyperbolic groups

Every subgroup is either virtually abelian or admits a series ending in an acylindrically hyperbolic subgroup

The group is SQ-universal and excludes certain subgroup configurations

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.