Radius of comparison and mean cohomological independence dimension

Abstract

We introduce a notion of mean cohomological independence dimension for actions of discrete amenable groups on compact metrizable spaces, as a variant of mean dimension, and use it to obtain lower bounds for the radius of comparison of the associated crossed product C⁎-algebras. Our general theory, gives the following for the minimal subshifts constructed by Dou in 2017. For any countable amenable group G and any polyhedron Z, Dou's subshift T of ZG with density parameter ρ satisfiesrc(C(X)⋊TG)>12mdim(T)(1−1−ρρ)−2.If k=dim⁡(Z) is even and Hˇk(Z;Q)≠0, thenrc(C(X)⋊TG)>12mdim(T)−1,regardless of what ρ is.