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.