On sequences of finitely generated discrete groups.

*(English)* Zbl 1234.20060
Bonk, Mario (ed.) et al., In the tradition of Ahlfors-Bers, V. Proceedings of the 4th triennial Ahlfors-Bers colloquium, Newark, NJ, USA, May 8–11, 2008. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4732-9/pbk). Contemporary Mathematics 510, 165-184 (2010).

Summary: We consider sequences of discrete subgroups ${{\Gamma}}_{i}={\rho}_{i}\left({\Gamma}\right)$ of a rank 1 Lie group $G$, with ${\Gamma}$ finitely generated. We show that, for algebraically convergent sequences $\left({{\Gamma}}_{i}\right)$, unless ${{\Gamma}}_{i}$’s are (eventually) elementary or contain normal finite subgroups of arbitrarily high order, their algebraic limit is a discrete nonelementary subgroup of $G$. In the case of divergent sequences $\left({{\Gamma}}_{i}\right)$ we show that the resulting action ${\Gamma}\u293bT$ on a real tree satisfies certain semistability condition, which generalizes the notion of stability introduced by Rips. We then verify that the group ${\Gamma}$ splits as an amalgam or HNN extension of finitely generated groups, so that the edge group has an amenable image in $\text{Isom}\left(T\right)$.

##### MSC:

20H15 | Other geometric groups, including crystallographic groups |

22E40 | Discrete subgroups of Lie groups |

20E08 | Groups acting on trees |

20E06 | Free products and generalizations (group theory) |