Relatively hyperbolic groups: geometry and quasi-isometric invariance

TL;DR

This paper proves that relative hyperbolicity is preserved under quasi-isometries and introduces simplified definitions, including a new one based on the existence of central cosets in quasi-geodesic triangles.

Contribution

It establishes the invariance of relative hyperbolicity under quasi-isometry and provides new, simplified definitions of the concept.

Findings

Relative hyperbolicity is a quasi-isometry invariant.

New simplified definitions of relative hyperbolicity are introduced.

A characterization involving central cosets in quasi-geodesic triangles is proposed.

Abstract

In this paper it is proved that relative hyperbolicity is an invariant of quasi-isometry. As a byproduct of the arguments, simplified definitions of relative hyperbolicity are obtained. In particular we obtain a new definition very similar to the one of hyperbolicity, relying on the existence for every quasi-geodesic triangle of a central left coset of peripheral subgroup.