On sequences of finitely generated discrete groups

We consider sequences of discrete subgroups Γi = ρi(Γ) of a rank 1 Lie group G, with Γ finitely generated. We show that, for algebraically convergent sequences (Γi), unless Γi’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 (Γi) we show that the resulting action Γ y T 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 ).