Two necessary and sufficient conditions for the extension of Möbius groups

Abstract

Let SL(2, Γn) be the n-dimensional Clifford matrix group and G ⊂ SL(2, Γn) be a non-elementary subgroup. We show that G is the extension of a subgroup of SL(2, ℂ) if and only if G is conjugate in SL(2, Γn) to a group G′ which satisfies the following properties:(1) there exist loxodromic elements g0, h ∈ G′ such that fix(g0) = {0, ∞}, fix(g0) ∩ fix(h) = ∅ and fix(h) ∩ ℂ ≠ ∅;(2) tr(g) ∈ ℂ for each loxodromic element g ∈ G′.Further G is the extension of a subgroup of SL(2, ℝ) if and only if G is conjugate in SL(2, Γn) to a group G′ which satisfies the following properties:(1) there exists a loxodromic element g0 ∈ G′ such that fix(g0) ∩ {0, ∞} ≠ ∅;(2) tr(g) ∈ ℝ for each loxodromic element g ∈ G′.The discreteness of subgroups of SL(2, Γn) is also discussed.