URP, comparison, mean dimension, and sharp shift embeddability

For a free action \(G \curvearrowright X\) of an amenable group on a compact metrizable space, we study the Uniform Rokhlin Property (URP) and the conjunction of Uniform Rokhlin Property and comparison (URPC). We give several equivalent formulations of the latter and show that it passes to extensions. We introduce technical conditions called property FCSB and property FCSB in measure, both of which reduce to the marker property if \(G\) is abelian. Our first main result is that for any amenable group \(G\) property FCSB in measure is equivalent to URP, and for a large class of amenable groups property FCSB is equivalent to URPC. In the latter case, it follows that if the action is moreover minimal then the C\(^*\)-crossed product \(C(X) \rtimes G\) has stable rank one, satisfies the Toms-Winter conjecture, and is classifiable if \(\mathrm{mdim}(G \curvearrowright X) = 0\). Our second main result is that if a system \(G \curvearrowright X\) has URPC and \(\mathrm{mdim}(G \curvearrowright X) < M/2\), then it is embeddable into the \(M\)-cubical shift \(\left([0, 1]^M\right)^G\). Combined with the first main result, we recover the Gutman-Qiao-Tsukamoto sharp shift embeddability theorem as a special case. Notably, the proof avoids the use of either Euclidean geometry or signal analysis and directly extends the theorem to all abelian groups. Finally, we show that if \(G\) is a nonamenable group that contains a free subgroup on two generators and \(G \curvearrowright X\) is a topologically amenable action, then it is embeddable into \([0, 1]^G\).