Furstenberg entropy realizations for virtually free groups and lamplighter groups

TL;DR

This paper constructs stationary actions with arbitrarily small positive Furstenberg entropy for virtually free and lamplighter groups, contrasting with groups having property (T) where entropy is either zero or bounded below.

Contribution

It introduces a method to realize a dense set of Furstenberg entropies for virtually free and lamplighter groups, expanding understanding of entropy behavior in these groups.

Findings

Constructed stationary actions with arbitrarily small positive entropy.

Realized a dense set of entropies between zero and the Poisson boundary entropy.

Contrasted entropy properties with groups having property (T).

Abstract

Let $(G,\mu)$ be a discrete group with a generating probability measure. Nevo shows that if $G$ has property (T) then there exists an $\epsilon>0$ such that the Furstenberg entropy of any $(G,\mu)$-stationary ergodic space is either zero or larger than $\epsilon$. Virtually free groups, such as $SL_2(\mathbb{Z})$, do not have property (T), and neither do their extensions, such as surface groups. For these, we construct stationary actions with arbitrarily small, positive entropy. This construction involves building and lifting spaces of lamplighter groups. For some classical lamplighters, these spaces realize a dense set of entropies between zero and the Poisson boundary entropy.