A non-quasiconvex embedding of relatively hyperbolic groups

TL;DR

This paper constructs a group extension where the original group is not relatively quasiconvex, expanding understanding of subgroup embeddings in relatively hyperbolic groups.

Contribution

It demonstrates the existence of a group extension where the original group is not relatively quasiconvex, generalizing prior results for hyperbolic groups.

Findings

Existence of a group extension with non-quasiconvex embedding

Construction of a malnormal, quasiconvex subgroup within the extension

Generalization of Kapovich's result to relatively hyperbolic groups

Abstract

For any finitely generated, non-elementary, torsion-free group $G$ that is hyperbolic relative to $\mathbb P$, we show that there exists a group $G^*$ containing $G$ such that $G^*$ is hyperbolic relative to $\mathbb P$ and $G$ is not relatively quasiconvex in $G^*$. This generalizes a result of I. Kapovich for hyperbolic groups. We also prove that any torsion-free group $G$ that is non-elementary and hyperbolic relative to $\mathbb P$, contains a rank 2 free subgroup $F$ such that the group generated by "randomly" chosen elements $r_1,...,r_m$ in $F$ is aparabolic, malnormal in $G$ and quasiconvex relative to $\mathbb P$ and therefore hyperbolically embedded relative to $\mathbb P$.