Finitely generated groups acting uniformly properly on hyperbolic space

TL;DR

This paper constructs an uncountable family of groups acting properly on hyperbolic spaces, revealing many are not virtually torsion-free and are not subgroups of hyperbolic groups, thus expanding the known landscape of such group actions.

Contribution

It introduces a new uncountable sequence of groups with proper hyperbolic actions, demonstrating the existence of non-virtually torsion-free groups outside hyperbolic subgroup classifications.

Findings

Uncountable groups acting properly on hyperbolic spaces are constructed.

Only countably many of these groups are virtually torsion-free.

Some groups cannot be embedded as subgroups of hyperbolic groups.

Abstract

We construct an uncountable sequence of groups acting uniformly properly on hyperbolic spaces. We show that only countably many of these groups can be virtually torsion-free. This gives new examples of groups acting uniformly properly on hyperbolic spaces that are not virtually torsion-free and cannot be subgroups of hyperbolic groups.