On the structure of asymptotic 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 $C^{\ast }$-algebraic characterisation of expanders for vertex-transitive graphs.

To appear in "Advances in Mathematics"

References in corpus (4)

• Relative commutant pictures of Roe algebras
• Measure expanding actions, expanders and warped cones
• Superexpanders from group actions on compact manifolds
• A Markovian and Roe-algebraic approach to asymptotic expansion in measure