Metric Mean Dimension for Algebraic Actions of Sofic Groups

Abstract: Recently Bingbing Liang and Hanfeng Li computed the mean dimension and metric mean dimension for algebraic actions of amenable groups. We show how to extend their computation of metric mean dimension to the case of sofic groups, provided that the dual module is finitely generated. Additionally, we show that when the dual module is finitely presented that the mean dimension is the von Neumann rank. The proof also goes through introducing \ell^{p}-analogues of metric mean dimension, which may be seen as an obstruction to the equality of mean dimension and metric mean dimension.