Asymptotic expansion for groupoids and Roe type algebras

TL;DR

This paper introduces a new notion of asymptotic expansion for groupoids, linking it to classical expanders and Roe algebras, and characterizes when these algebras contain block-rank-one projections, impacting the coarse Baum-Connes conjecture.

Contribution

It develops an asymptotic expansion framework for groupoids and establishes structural theorems connecting expansion to operator algebra properties, generalizing previous results.

Findings

Asymptotic expansion can be approximated by domains of expansion.

Characterization of Roe algebra projections via asymptotic expansion.

Counterexamples to the coarse Baum-Connes conjecture derived from these properties.

Abstract

In this paper, we introduce a notion of expansion for groupoids, which recovers the classical notion of expander graphs by a family of pair groupoids and expanding actions in measure by transformation groupoids. We also consider an asymptotic version for expansion and establish structural theorems, showing that asymptotic expansion can be approximated by domains of expansions. On the other hand, we introduce dynamical propagation and quasi-locality for operators on groupoids and the associated Roe type algebras. Our main results characterise when these algebras possess block-rank-one projections by means of asymptotic expansion, which generalises the crucial ingredients in previous works to provide counterexamples to the coarse Baum-Connes conjecture.