Free Action of Amenable Groups: Uniform Rokhlin Property, Comparison, Mean Dimension, and Sharp Shift Embeddability

For free actions of amenable groups on compact metrizable spaces, the Uniform Rokhlin Property (URP) and the conjunction of URP and comparison (URPC) are shown to be equivalent to technical conditions called property FCSB and property FCSB in measure. These properties are then used to establish a sharp shift embeddability theorem and characterize when the associated crossed product C*-algebras have desirable properties.