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 5th 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 of a rank 1 Lie group , with finitely generated. We show that, for algebraically convergent sequences , unless ’s are (eventually) elementary or contain normal finite subgroups of arbitrarily high order, their algebraic limit is a discrete nonelementary subgroup of . In the case of divergent sequences we show that the resulting action on a real tree satisfies certain semistability condition, which generalizes the notion of stability introduced by Rips. We then verify that the group splits as an amalgam or HNN extension of finitely generated groups, so that the edge group has an amenable image in .
|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)|