On some p-approximation properties of discrete groups and ℓp uniform Roe algebras
Abstract
We study p-approximation properties of ℓp uniform Roe algebras and their connections to coarse geometry and group theory. For bounded geometry metric spaces, property A implies p-nuclearity of Bup(X) for 1<p<∞, while Bu1(X) is always $1$-nuclear. Property A also yields coefficient characterizations and p-complete exactness results for Bup(X,−); at p=1 these coefficient results hold without property A. We introduce the p-invariant translation approximation property and its operator space version. The p-AP implies a coefficient fixed-point identity for arbitrary SQp coefficients, and for exact groups it is equivalent to the p-operator ITAP. At p=1, every discrete group satisfies the full coefficient fixed-point identity. We also establish coefficient permanence under subgroups, finite-index supergroups, and group extensions with p-AP quotient, including direct and semidirect products. In particular, the class of exact groups with the p-AP is closed under group extensions. We also obtain higher-rank lattices as examples failing the p-operator ITAP.
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BibTeX
@misc{chung2026some,
title = {{On some $p$-approximation properties of discrete groups and $\ell^p$ uniform Roe algebras}},
author = {Yeong Chyuan Chung},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.27230},
eprint = {2606.27230},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.27230},
}
RIS
TY - UNPB
TI - On some p-approximation properties of discrete groups and ℓ^p uniform Roe algebras
AU - Yeong Chyuan Chung
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2606.27230
UR - https://arxiv.org/abs/2606.27230
AB - We study p-approximation properties of ℓ^p uniform Roe algebras and their connections to coarse geometry and group theory. For bounded geometry metric spaces, property A implies p-nuclearity of B_u^p(X) for 1<p<∞, while B_u^1(X) is always $1$-nuclear. Property A also yields coefficient characterizations and p-complete exactness results for B_u^p(X,−); at p=1 these coefficient results hold without property A. We introduce the p-invariant translation approximation property and its operator space version. The p-AP implies a coefficient fixed-point identity for arbitrary SQ_p coefficients, and for exact groups it is equivalent to the p-operator ITAP. At p=1, every discrete group satisfies the full coefficient fixed-point identity. We also establish coefficient permanence under subgroups, finite-index supergroups, and group extensions with p-AP quotient, including direct and semidirect products. In particular, the class of exact groups with the p-AP is closed under group extensions. We also obtain higher-rank lattices as examples failing the p-operator ITAP.
ER -