For finitely generated discrete groups with polynomial growth, product entropy of induced automorphisms on reduced group C*-algebras is bounded above and below by algebraic and geometric entropies; for exponential growth the metric dimension is generically infinite.
Gromov,Groups of polynomial growth and expanding maps, Publications mathématiques de l’I.H.É.S
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Metric dimension and product entropy of group $C^{\ast}$-algebras
For finitely generated discrete groups with polynomial growth, product entropy of induced automorphisms on reduced group C*-algebras is bounded above and below by algebraic and geometric entropies; for exponential growth the metric dimension is generically infinite.