Sofic Mean Length

Abstract

Given a length function L on the R-modules of a unital ring R, for each sofic group $Γ$ we define a mean length for every locally L-finite $RΓ$-module relative to a bigger $RΓ$-module. We establish an addition formula for the mean length. We give two applications. The first one shows that for any unital left Noetherian ring R, $RΓ$ is stably direct finite. The second one shows that for any $ZΓ$-module M, the mean topological dimension of the induced $Γ$-action on the Pontryagin dual of M coincides with the von Neumann-Lück rank of M.