Schreier's Formula for some Free Probability Invariants

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 unital $*$-subalgebra which is globally invariant under $\alpha$. 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 $A \rtimes_\alpha G$ 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) $\dim \text{Der}_c(A,\tau)$ (defined by Shlyakhtenko) 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, and assuming that $G$ is abelian group. Using the formula for $\Delta$, we recover recent results of Shlyakhtenko on the microstates free entropy dimension.