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Metric Mean Dimension for Algebraic Actions of Sofic Groups
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.Comment: 36 pages. Fixed numerous typographical errors. Additionally I fixed
some issues related to left and right multiplication in the paper which
confused matters in previous versions. Accepted to Transactions of the AM
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