On sequences of finitely generated discrete groups

TL;DR

This paper studies the limits of sequences of finitely generated discrete subgroups of rank 1 Lie groups, showing conditions under which their limits are discrete and describing the structure of divergent sequences' actions.

Contribution

It establishes new conditions for algebraic limits of such groups to be discrete and describes the semistability and splitting properties of divergent sequences.

Findings

Algebraically convergent sequences have discrete limits unless they are elementary or contain large finite normal subgroups.

Divergent sequences have limits acting on real trees satisfying a generalized semistability condition.

Groups in the limit split as amalgams or HNN extensions with amenable edge groups.

Abstract

We consider sequences of finitely generated discrete subgroups Gamma_i=rho_i(Gamma) of a rank 1 Lie group G, where the representations rho_i are not necessarily faithful. 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 limiting action on a real tree T 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 the isometry group of T.