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 $.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 April 2004
 arXiv:
 arXiv:math/0404118
 Bibcode:
 2004math......4118O
 Keywords:

 Mathematics  Group Theory;
 20F65;
 20F67;
 20F69
 EPrint:
 21 pages