Some groups whose reduced [formula omitted]-algebras have stable rank one

Abstract

It is proved that, for the following classes of groups, Γ, the reduced group C ∗ -algebra C ∗ λ(Γ) has stable rank 1: 1. hyperbolic groups which are either torsion-free and non-elementary or which are cocompact lattices in a real, noncompact, simple, connected Lie group of real rank 1 having trivial center; 2. amalgamated free products of groups, Γ=G 1∗ HG 2 , where H is finite and there is γ∈ Γ such that γ −1 Hγ∩ H={1}. The proofs involve some analysis of the free semigroup property, which is one way of saying that a group Γ has an abundance of free sub-semigroups, and of the ℓ 2-spectral radius property, which says that spectral radius of appropriate elements in C ∗ λ(Γ) may be computed with the 2-norm.