\[s\].ent/. Both $\fsf$ and $\wsf$ are important in their own right; see \[b\].Lyons:bird/ for a survey and  for a comprehensive treatment.

Suppose now that $\gp$ is an amenable group that acts quasi-transitively on a graph $\gh$. Since $\fsf_\gh$ is defined as a limit of uniform measures, it is natural to expect that it has maximum (metric) entropy in an appropriate class of $\gp$-invariant measures. Since the set of essential spanning forests is closed, one generally considers the class of $\gp$-invariant measures on this set. Furthermore, one might expect that the entropy of $\fsf_\gh$ is related in a simple way to the exponential growth rate of the complexity of the finite subgraphs $\gh_n$ exhausting $\gh$. Finally, one might wonder whether $\fsf_\gh$ is unique as a measure of maximal entropy on the essential spanning forests.

The case of $\gp = \Z^d$ was considered by \[b\].BurPem/. Their Theorem 6.1 gave a positive answer to all three questions implicit in the preceding paragraph. However, it turns out that one of the claims, the uniqueness of the measure of maximal entropy, received an incorrect proof. The correct proofs of the other claims relied to some extent on the natural tiling of $\Z^d$ by large boxes, 