Combination of quasiconvex subgroups of relatively hyperbolic groups

TL;DR

This paper extends combination theorems for quasiconvex subgroups from hyperbolic groups to relatively hyperbolic groups, providing conditions under which the subgroup generated by two quasiconvex subgroups remains quasiconvex and has a specific algebraic structure.

Contribution

It generalizes known results for hyperbolic groups to the broader class of relatively hyperbolic groups, offering new conditions for subgroup quasiconvexity and isomorphism.

Findings

Identifies conditions for quasiconvexity of subgroup generated by two quasiconvex subgroups

Shows the subgroup is isomorphic to an amalgamated free product

Provides applications of the generalized combination theorems

Abstract

For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup generated by two quasiconvex subgroups $Q$ and $R$ is quasiconvex and isomorphic to $Q \ast_{Q\cap R} R$. Our results generalized known combination theorems for quasiconvex subgroups of word-hyperbolic groups. Some applications are presented.