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

We prove an entropy formula for certain expansive actions of a countable discrete residually finite group $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$-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 ${\mathrm{\Gamma }}_n$-fixed points, where $({\mathrm{\Gamma }}_n,n\ge 1)$ is a decreasing sequence of finite index normal subgroups of $\mathrm{\Gamma }$ with trivial intersection.