A non-quasiconvex embedding of 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 $G^{\ast }$ containing $G$ such that $G^{\ast }$ is hyperbolic relative to $\mathbb{P}$ and $G$ is not relatively quasiconvex in $G^{\ast }$. 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 $\mathbb{P}$.