Mean Dimension, Mean Rank, and von Neumann-L\"uck Rank

We introduce an invariant, called mean rank, for any module M of the integral group ring of a discrete amenable group $\Gamma$, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced $\Gamma$-action on the Pontryagin dual of M, the mean rank of M, and the von Neumann-L\"uck rank of M all coincide. As applications, we establish an addition formula for mean dimension of algebraic actions, prove the analogue of the Pontryagin-Schnirelmnn theorem for algebraic actions, and show that for elementary amenable groups with an upper bound on the orders of finite subgroups, algebraic actions with zero mean dimension are inverse limits of finite entropy actions.