On some $p$-approximation properties of discrete groups and ${\ell }^p$ uniform Roe algebras

Abstract

We study $p$-approximation properties of ${\ell }^p$ 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 $1<p<\infty$, 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 $S{Q}_p$ 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.

arXiv comment58 pages. Substantially expanded and reorganized; title revised. New results include coefficient exactness and p-complete quotient results, Gevrey rapid-decay criteria, stronger permanence properties for p-ITAP, arbitrary SQ_p coefficient results, and a group-extension theorem for p-ITAP and p-AP