Metric mean dimension for algebraic actions of Sofic groups

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We prove that if Γ \Gamma is a sofic group and A A is a finitely generated Z ( Γ ) \mathbb {Z}(\Gamma ) -module, then the metric mean dimension of Γ ↷ A ^ , \Gamma \curvearrowright \widehat {A}, in the sense of Hanfeng Li, is equal to the von Neumann-Lück rank of A . A. This partially extends the results of Hanfeng Li and Bingbing Liang from the case of amenable groups to the case of sofic groups. Additionally we show that the mean dimension of Γ ↷ A ^ \Gamma \curvearrowright \widehat {A} is the von Neumann-Lück rank of A A if A A is finitely presented and Γ \Gamma is residually finite. It turns out that our approach naturally leads to a notion of p p -metric mean dimension, which is in between mean dimension and the usual metric mean dimension. This can be seen as an obstruction to the equality of mean dimension and metric mean dimension. While we cannot decide if mean dimension is the same as metric mean dimension for algebraic actions, we show in the metric case that for all p p the p p -metric mean dimension coincides with the von Neumann-Lück rank of the dual module.