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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 Γ i =ρ i (Γ) of a rank 1 Lie group G, with Γ finitely generated. We show that, for algebraically convergent sequences (Γ i ), unless Γ 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 (Γ i ) we show that the resulting action ΓT on a real tree satisfies certain semistability condition, which generalizes the notion of stability introduced by Rips. We then verify that the group Γ splits as an amalgam or HNN extension of finitely generated groups, so that the edge group has an amenable image in Isom(T).
MSC:
20H15Other geometric groups, including crystallographic groups
22E40Discrete subgroups of Lie groups
20E08Groups acting on trees
20E06Free products and generalizations (group theory)