On the structure of asymptotic expanders

TL;DR

This paper investigates the structure of asymptotic expanders using geometric tools, revealing their properties, limitations in embedding into $L^p$-spaces, and providing new counterexamples to the coarse Baum--Connes conjecture.

Contribution

It introduces a uniform exhaustion by expanders for asymptotic expanders, characterizes them via their Roe algebra, and constructs new counterexamples to a major conjecture.

Findings

Asymptotic expanders admit a uniform exhaustion by expanders.

They cannot be coarsely embedded into any $L^p$-space.

Vertex-transitive asymptotic expanders are actual expanders.

Abstract

In this paper, we use geometric tools to study the structure of asymptotic expanders and show that a sequence of asymptotic expanders always admits a "uniform exhaustion by expanders". It follows that asymptotic expanders cannot be coarsely embedded into any $L^p$-space, and that asymptotic expanders can be characterised in terms of their uniform Roe algebra. Moreover, we provide uncountably many new counterexamples to the coarse Baum--Connes conjecture. These appear to be the first counterexamples that are not directly constructed by means of spectral gaps. Finally, we show that vertex-transitive asymptotic expanders are actually expanders. In particular, this gives a $C^*$-algebraic characterisation of expanders for vertex-transitive graphs.