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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 \(\Gamma_i=\rho_i(\Gamma)\) of a rank 1 Lie group \(G\), with \(\Gamma\) finitely generated. We show that, for algebraically convergent sequences \((\Gamma_i)\), 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 \((\Gamma_i)\) we show that the resulting action \(\Gamma\curvearrowright T\) 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 \(\text{Isom}(T)\).
For the entire collection see [Zbl 1185.30001].

MSC:

20H15 Other geometric groups, including crystallographic groups
22E40 Discrete subgroups of Lie groups
20E08 Groups acting on trees
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
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