Elementary subgroups of relatively hyperbolic groups and bounded generation

Details Elementary subgroups of relatively hyperbolic groups and bounded generation Elementary subgroups of relatively hyperbolic groups and bounded generation

2004-04-06

arxiv.org/abs/math/0404118v1

Mathematics

Group Theory

21 pages

Scientific paper

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 $.

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