On relative hyperbolicity for a group and relative quasiconvexity for a subgroup

TL;DR

This paper explores the relationships between relative hyperbolicity of a group with respect to two subgroup families and the relative quasiconvexity of its subgroups, providing new insights into subgroup structures.

Contribution

It establishes conditions linking relative hyperbolicity and quasiconvexity for groups and their subgroups concerning two subgroup families.

Findings

Relations between relative hyperbolicity and quasiconvexity are characterized.

Subgroups contained in one family are related to hyperbolic structures.

Conditions for subgroup quasiconvexity relative to different families are identified.

Abstract

We consider two families of subgroups of a group. Each subgroup which belongs to one family is contained in some subgroup which belongs to the other family. We then discuss relations of relative hyperbolicity for the group with respect to the two families, respectively. If the group is supposed to be hyperbolic relative to the two families, respectively, then we consider relations of relative quasiconvexity for a subgroup of the group with respect to the two families, respectively.