Sofic Mean Length

TL;DR

This paper introduces a mean length concept for modules over group rings of sofic groups, establishing an addition formula and applying it to show stable finiteness and relate mean topological dimension to von Neumann-Lück rank.

Contribution

It defines a new mean length for modules over sofic group rings, proves an addition formula, and applies it to stable finiteness and dimension-rank relations.

Findings

RΓ is stably direct finite for unital left Noetherian rings R.

Mean topological dimension equals von Neumann-Lück rank for induced actions.

Addition formula for mean length of modules established.

Abstract

Given a length function L on the R-modules of a unital ring R, for each sofic group $\Gamma$ we define a mean length for every locally L-finite $R\Gamma$-module relative to a bigger $R\Gamma$-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, $R\Gamma$ is stably direct finite. The second one shows that for any $Z\Gamma$-module M, the mean topological dimension of the induced $\Gamma$-action on the Pontryagin dual of M coincides with the von Neumann-L\"{u}ck rank of M.