Tits Alternative in groups with proper product actions on proper Gromov-hyperbolic spaces
Jiaqi Cui, Renxing Wan

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 on which acts cocompactly such that the diagonal action of on the -product is proper. We show that any finitely generated subgroup of a finitely generated group with property (PPH) either is amenable or contains . Furthermore, we study groups with property (PPT), i.e., groups with property (PPH) so that 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- or contains . 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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
