Combination of fully quasiconvex subgroups and its applications

TL;DR

This paper establishes new combination theorems for relatively quasiconvex subgroups in relatively hyperbolic groups, leading to applications in subgroup separability and the existence of surface subgroups.

Contribution

It introduces two novel combination theorems for relatively quasiconvex subgroups and applies them to important problems in subgroup separability and surface subgroup existence.

Findings

Proved two combination theorems for relatively quasiconvex subgroups.

Applied the theorems to show separability of double cosets.

Demonstrated the existence of closed surface subgroups in certain relatively hyperbolic groups.

Abstract

In this paper, we state two combination theorems for relatively quasiconvex subgroups in a relatively hyperbolic group. Applications are given to the separability of double cosets of certain relatively quasiconvex subgroups and the existence of closed surface subgroups in relatively hyperbolic groups.