A subgroup formula for f-invariant entropy

TL;DR

This paper establishes a subgroup formula for f-invariant entropy in free group actions, relating subgroup entropy ratios to subgroup indices and extending the concept to virtually free groups.

Contribution

It generalizes the subgroup entropy formula for free groups and extends the f-invariant entropy to virtually free groups, introducing a new virtual measure conjugacy invariant.

Findings

The ratio of f-invariant entropies equals the subgroup index.

Extended f-invariant entropy to virtually free groups.

Introduced a virtual measure conjugacy invariant.

Abstract

We study a measure entropy for finitely generated free group actions called f-invariant entropy. The f-invariant entropy was developed by Lewis Bowen and is essentially a special case of his measure entropy theory for actions of sofic groups. In this paper we relate the f-invariant entropy of a finitely generated free group action to the f-invariant entropy of the restricted action of a subgroup. We show that the ratio of these entropies equals the index of the subgroup. This generalizes a well known formula for the Kolmogorov--Sinai entropy of amenable group actions. We then extend the definition of f-invariant entropy to actions of finitely generated virtually free groups. We also obtain a numerical virtual measure conjugacy invariant for actions of finitely generated virtually free groups.