Cost, $\ell^2$-Betti numbers and the sofic entropy of some algebraic actions

Abstract

In 1987, Ornstein and Weiss discovered that the Bernoulli $2$-shift over the rank two free group factors onto the seemingly larger Bernoulli $4$-shift. With the recent creation of an entropy theory for actions of sofic groups (in particular free groups), their example shows the surprising fact that entropy can increase under factor maps. In order to better understand this phenomenon, we study a natural generalization of the Ornstein--Weiss map for countable groups. We relate the increase in entropy to the cost and to the first $\ell^2$-Betti number of the group. More generally, we study coboundary maps arising from simplicial actions and, under certain assumptions, relate $\ell^2$-Betti numbers to the failure of the Juzvinski{\u\i} addition formula. This work is built upon a study of entropy theory for algebraic actions. We prove that for actions on profinite groups via continuous group automorphisms, topological sofic entropy is equal to measure sofic entropy with respect to Haar measure whenever the homoclinic subgroup is dense. For algebraic actions of residually finite groups we find sufficient conditions for the sofic entropy to be equal to the supremum exponential growth rate of periodic points.