Virtual Amalgamation of Relatively Quasiconvex Subgroups

TL;DR

This paper explores conditions under which the subgroup generated by two relatively quasiconvex subgroups in a relatively hyperbolic group remains relatively quasiconvex and isomorphic to their amalgamated free product, extending known results.

Contribution

It generalizes existing theorems from hyperbolic groups to the broader context of relatively hyperbolic groups, providing new insights into subgroup structure.

Findings

Conditions for relative quasiconvexity of generated subgroups

Extension of amalgamation results to relatively hyperbolic groups

Generalization of hyperbolic group subgroup theorems

Abstract

For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup generated by two relatively quasiconvex subgroups $Q_1$ and $Q_2$ is relatively quasiconvex and isomorphic to $Q_1 \ast_{Q_1 \cap Q_2} Q_2$. The main theorem extends results for quasiconvex subgroups of word-hyperbolic groups, and results for discrete subgroups of isometries of hyperbolic spaces.