Notes on relatively hyperbolic groups and relatively quasiconvex subgroups

TL;DR

This paper explores the properties of relatively quasiconvex subgroups within relatively hyperbolic groups, establishing their expected characteristics and clarifying equivalent definitions of relative hyperbolicity.

Contribution

It introduces a definition of relatively quasiconvex subgroups in the sense of Osin and proves their key properties, also clarifying multiple equivalent definitions of relatively hyperbolic groups.

Findings

Relatively quasiconvex subgroups have expected properties.

Several definitions of relatively hyperbolic groups are shown to be equivalent.

The paper clarifies the structure of relatively hyperbolic groups and their subgroups.

Abstract

We define relatively quasiconvex subgroups of relatively hyperbolic groups in the sense of Osin and show that such subgroups have expected properties. Also we state several definitions equivalent to the definition of relatively hyperbolic groups in the sense of Osin.