Tits Alternative in groups with proper product actions on proper Gromov-hyperbolic spaces

TL;DR

This paper investigates groups acting on products of hyperbolic spaces, establishing conditions under which their subgroups are either amenable or contain free groups, and characterizes hyperbolic groups with proper actions on trees.

Contribution

It introduces properties (PPH) and (PPT) for groups acting on hyperbolic spaces, providing new subgroup classifications and linking hyperbolic groups to actions on trees.

Findings

Finitely generated subgroups are either amenable or contain F_2.

Subgroups are virtually (locally-finite)-by-Z^n or contain F_2 under property (PPT).

Hyperbolic groups admit proper actions on products of trees iff they have property (PPT).

Abstract

In this paper, we study groups with property (PPH), i.e., there exist finitely many proper Gromov-hyperbolic spaces $X_1,\ldots, X_l$ on which $G$ acts cocompactly such that the diagonal action of $G$ on the $\ell^1$-product $\prod_{i=1}^lX_i$ is proper. We show that any finitely generated subgroup of a finitely generated group with property (PPH) either is amenable or contains $F_2$. Furthermore, we study groups with property (PPT), i.e., groups with property (PPH) so that $X_1,\cdots,X_l$ are all proper quasi-trees. We show that any finitely generated subgroup of a finitely generated group with property (PPT) either is virtually (locally-finite)-by-$\mathbb{Z}^n$ or contains $F_2$. Additionally, we establish that for a non-elementary hyperbolic group \(G\), \(G\) admits a proper diagonal action on a finite product of regular trees if and only if \(G\) has property (PPT). This result transforms a question posed by Button \cite{But19} into the problem of whether every non-elementary hyperbolic group has property (PPT).