Expansive algebraic actions of discrete residually finite amenable groups and their entropy

Abstract

We prove an entropy formula for certain expansive actions of a countable discrete residually finite group F by automorphisms of compact abelian groups in terms of Fuglede-Kadison determinants. This extends an earlier result proved by the first author under somewhat more restrictive conditions. The main tools for this generalization are a representation of the Γ-action by means of a 'fundamental homoclinic point' and the description of entropy in terms of the renormalized logarithmic growth rate of the set of Γn-fixed points, where (Γn, n ≥ 1) is a decreasing sequence of finite index normal subgroups of Γ with trivial intersection. © 2007 Cambridge University Press.