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Generic Sharp Shift Embedding for Amenable Group Actions with the Uniform Rokhlin Property and Comparison
Abstract
Naryshkin recently proved that a free action of a countably infinite discrete amenable group G on a compact metrizable space X with the uniform Rokhlin property and comparison (URPC) admits an equivariant embedding into ([0,1]m)G whenever mdim(X,G)<m/2. We strengthen this result from existence to genericity by a substantially different method: the set of observables f∈ C(X,[0,1]m) whose orbit maps are equivariant embeddings into ([0,1]m)G is a dense Gδ subset of C(X,[0,1]m). The proof combines an adaptation of the Kakutani skyscraper construction used by Gutman--Tsukamoto with a new polynomial perturbation method, departing from the predominantly linear perturbation approaches used in earlier equivariant embedding literature.
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BibTeX
@misc{gutman2026generic,
title = {{Generic Sharp Shift Embedding for Amenable Group Actions with the Uniform Rokhlin Property and Comparison}},
author = {Yonatan Gutman and Chunlin Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31735},
eprint = {2609.31735},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31735},
}
RIS
TY - UNPB
TI - Generic Sharp Shift Embedding for Amenable Group Actions with the Uniform Rokhlin Property and Comparison
AU - Yonatan Gutman
AU - Chunlin Liu
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.31735
UR - https://arxiv.org/abs/2609.31735
AB - Naryshkin recently proved that a free action of a countably infinite discrete amenable group G on a compact metrizable space X with the uniform Rokhlin property and comparison (URPC) admits an equivariant embedding into ([0,1]^m)^G whenever mdim(X,G)<m/2. We strengthen this result from existence to genericity by a substantially different method: the set of observables f∈ C(X,[0,1]^m) whose orbit maps are equivariant embeddings into ([0,1]^m)^G is a dense G_δ subset of C(X,[0,1]^m). The proof combines an adaptation of the Kakutani skyscraper construction used by Gutman--Tsukamoto with a new polynomial perturbation method, departing from the predominantly linear perturbation approaches used in earlier equivariant embedding literature.
ER -