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 $\Gamma$ 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 $\Gamma$-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 ${\Gamma }_n$-fixed points, where $({\Gamma }_n,n\ge 1)$ is a decreasing sequence of finite index normal subgroups of $\Gamma$ with trivial intersection.

Using a recent result of Y. Choi the discussion of expansiveness was strengthened and shortened. The paper will appear in: Ergodic Theory and Dynamical Systems