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Hanfeng Li
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We introduce mean dimensions for continuous actions of countable sofic groups on compact metrizable spaces. These generalize the Gromov-Lindenstrauss-Weiss mean dimensions for actions of countable amenable groups, and are useful for distinguishing continuous actions of countable sofic groups with infinite entropy.
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We refine two results in the paper entitled ``Sofic mean dimension by Hanfeng Li, improving two inequalities with two equalities, respectively, for sofic mean dimension of typical actions. On the one hand, we study sofic mean dimension of full shifts, for which, Li provided an upper bound which however is not optimal. We prove a more delicate estimate from above, which is optimal for sofic mean dimension of full shifts over arbitrary alphabets (i.e. compact metrizable spaces). Our refinement, together with the techniques (in relation to an estimate from below) in the paper entitled ``Mean dimension of full shifts by Masaki Tsukamoto, eventually allows us to get the exact value of sofic mean dimension of full shifts over any finite dimensional compact metrizable spaces. On the other hand, we investigate finite group actions. In contrast to the case that the acting group is infinite (and amenable), Li showed that if a finite group acts continuously on a finite dimensional compact metrizable space, then sofic mean dimension may be different from (strictly less than) the classical (i.e. amenable) mean dimension (an explicitly known value in this case). We strengthen this result by proving a sharp lower bound, which, combining with the upper bound, gives the exact value of sofic mean dimension for all the actions of finite groups on finite dimensional compact metrizable spaces. Furthermore, this equality leads to a satisfactory comparison theorem for those actions, deciding when sofic mean dimension would coincide with classical mean dimension. Moreover, our two results, in particular, verify for a typical class of sofic group actions that sofic mean dimension does not depend on sofic approximation sequences.
For discrete measured groupoids preserving a probability measure we introduce a notion of sofic dimension that measures the asymptotic growth of the number of sofic approximations on larger and larger finite sets. In the case of groups we give a formula for free products with amalgamation over an amenable subgroup. We also prove a free product formula for measure-preserving actions.
We introduce some notions of conditional mean dimension for a factor map between two topological dynamical systems and discuss their properties. With the help of these notions, we obtain an inequality to estimate the mean dimension of an extension system. The conditional mean dimension for $G$-extensions are computed. We also exhibit some applications in the dynamical embedding problems.
Given a length function L on the R-modules of a unital ring R, for each sofic group $Gamma$ we define a mean length for every locally L-finite $RGamma$-module relative to a bigger $RGamma$-module. We establish an addition formula for the mean length. We give two applications. The first one shows that for any unital left Noetherian ring R, $RGamma$ is stably direct finite. The second one shows that for any $ZGamma$-module M, the mean topological dimension of the induced $Gamma$-action on the Pontryagin dual of M coincides with the von Neumann-L{u}ck rank of M.
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-Luck 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.
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