Gromov hyperbolic groups and the Macaev norm

Rui Okayasu

Vol. 223 (2006), No. 1, 141–157

DOI: 10.2140/pjm.2006.223.141

Abstract

Let Γ be a Gromov hyperbolic group with a finite set A of generators. We prove that h_top(Σ(∞)) ≤ k_∞⁻(λ_A) ≤ gr(Γ,A), where gr(Γ,A) is the growth entropy, h_top(Σ(∞)) is the Coornaert–Papadopoulos topological entropy of the subshift Σ(∞) associated with (Γ,A), and k_∞⁻(λ_A) is Voiculescu’s numerical invariant, which is an obstruction to the existence of quasicentral approximate units relative to the Macaev norm for a tuple of unitary operators λ_A = (λ_a)_a∈A in the left regular representation of Γ. We also prove that these three quantities are equal for a hyperbolic group splitting over a finite group.

Keywords

perturbation theory, Macaev ideal, hyperbolic groups

Mathematical Subject Classification 2000

Primary: 47B10

Secondary: 47A30, 37B10, 20F65

Milestones

Received: 23 September 2003

Revised: 7 May 2005

Accepted: 27 May 2005

Published: 1 January 2006