The Extension of Möbius Groups

Abstract

Let G be a non-elementary subgroup of SL(2,Г n ) containing hyperbolic elements. We show that G is the extension of a subgroup of SL(2,C) if and only if that G is conjugate in SL(2,Г n ) to a group G' with the following properties: (1) There are g 0, h ∈ G', where g 0 and h are hyperbolic, such that fix(g 0) = {0,∞}, fix(h)∩fix(g 0) = ∅ and fix(h) ∩ C ≠ ∅; (2) tr(g) ∈ C for each g ∈ G'. As an application, we show that if G contains only hyperbolic elements and uniformly parabolic elements, then G is the extension of a subgroup of SL(2,C), which also yields the discreteness of G.