The universal relatively hyperbolic structure on a group and relative quasiconvexity for subgroups

TL;DR

This paper explores the universal relatively hyperbolic structure on groups, providing a framework to characterize relative hyperbolicity and examining the relationship with relative quasiconvexity of subgroups.

Contribution

It introduces the concept of a universal relatively hyperbolic structure and analyzes its connection to subgroup quasiconvexity, advancing understanding of group hyperbolicity.

Findings

Universal relatively hyperbolic structure characterizes all relative hyperbolic structures.

Relations established between hyperbolic structures and subgroup quasiconvexity.

Framework aids in classifying and understanding group hyperbolicity properties.

Abstract

We discuss the notion of the universal relatively hyperbolic structure on a group which is used in order to characterize relatively hyperbolic structures on the group. We also study relations between relatively hyperbolic structures on a group and relative quasiconvexity for subgroups of the group.