Schreier’s Formula for Some Free Probability Invariants

Abstract Let $G\stackrel{\alpha }{\curvearrowright }(M,\tau )$ be a trace-preserving action of a finite group $G$ on a tracial von Neumann algebra. Suppose that $A \subset M$ is a finitely generated $\alpha $-invariant unital *-subalgebra. We give a formula relating the von Neumann dimension of the space of derivations on $A$ valued on its coarse bimodule to the von Neumann dimension of the space of derivations on $A \rtimes ^{\textrm{alg}}_\alpha G$ valued on its coarse bimodule, which is reminiscent of Schreier’s formula for finite index subgroups of free groups. This formula induces a formula for the free Stein dimension (defined by Charlesworth and Nelson), $\dim \textsf{Der}_{c,1\otimes 1}(A,\tau )$, and $\Delta $ (defined by Connes and Shlyakhtenko). The latter is done by establishing that $\Delta $ is equal to the von Neumann dimension of a certain subspace of the derivation space of $A$, similar to that of the free Stein dimension. Using the formula for $\Delta $, we strengthen the recent results of Shlyakhtenko on the microstates free entropy dimension.