Elementary subgroups of relatively hyperbolic groups and bounded generation

Abstract

Let $G$ be a group hyperbolic relative to a collection of subgroups $\{H_{\lambda },\lambda \in \Lambda \}$. We say that a subgroup $Q\le G$ is hyperbolically embedded into $G$, if $G$ is hyperbolic relative to $\{H_{\lambda },\lambda \in \Lambda \}\cup \{Q\}$. In this paper we obtain a characterization of hyperbolically embedded subgroups. In particular, we show that if an element $g\in G$ has infinite order and is not conjugate to an element of $H_{\lambda }$, $\lambda \in \Lambda$, then the (unique) maximal elementary subgroup contained $g$ is hyperbolically embedded into $G$. This allows to prove that if $G$ is boundedly generated, then $G$ is elementary or $H_{\lambda }=G$ for some $\lambda \in \Lambda$.

21 pages