Curvature-Free Margulis Lemma for Gromov-Hyperbolic Spaces

TL;DR

This paper extends the Margulis Lemma to Gromov-hyperbolic spaces without curvature assumptions, providing quantitative estimates and broadening applicability to more general metric spaces.

Contribution

It introduces curvature-free versions of the Margulis Lemma applicable to Gromov-hyperbolic spaces, with explicit, computable bounds on invariants.

Findings

Established curvature-free Margulis Lemma for Gromov-hyperbolic spaces

Provided quantitative estimates for algebraic and geometric invariants

Extended classical results to more general metric spaces

Abstract

We prove curvature-free versions of the celebrated Margulis Lemma. We are interested by both the algebraic aspects and the geometric ones, with however an emphasis on the second and we aim at giving quantitative (computable) estimates of some important invariants. Our goal is to get rid of the pointwise curvature assumptions in order to extend the results to more general spaces such as certain metric spaces. Essentially the upper bound on the curvature is replaced by the assumption that the space is hyperbolic in the sense of Gromov and the lower bound of the curvature by an upper bound on the entropy which we recall the definition.