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 * -algebraic characterisation of expanders for vertex-transitive graphs.

Asymptotic expansion in measure and strong ergodicity

“…Even if they are of the same spirit, these cases are treated separately because they differ significantly in the details. Note that these structural results are dynamical analogues of a structure theory first developed for asymptotic expanders [15]. 4.1.…”

“…We refer the reader to Section 6 for the relevant definitions and a detailed explanation. As a concluding remark, we wish to point out that Theorem A can be regarded as the dynamic analogue to a structure result for asymptotic expanders [15]. Moreover, both structure results have applications to the coarse Baum-Connes conjecture [15,16,17].…”

“…The proof of the structure results relies on the existence of "maximal" Følner sets. This is an adaptation of a technique introduced in [15] to study finite metric spaces. It is worth pointing out that the existence of such sets is less obvious in the current dynamical setting.…”

“…In fact, our starting point to introduce the notion of asymptotic expansion in measure is that it provides an efficient and unified way to construct asymptotic expanders. These are sequences of finite metric spaces whose expansion properties "asymptotically resembles" those of expander graphs (these metric spaces were introduced and studied in [15,16]). More precisely, we can translate between properties of dynamical systems and sequences of finite metric spaces of sequences via the discretisation procedure developed in [26] to prove the following: Theorem E. Let (X, d, ν) be a locally compact metric space with a Radon probability measure, and {P n } n∈N a sequence of measurable partitions of X with uniformly bounded measure ratios and mesh(P n ) → 0.…”

Remarks on partitions into expanders

“…This sort of maximality argument is also used fairly often in the theory of von Neumann algebras and it was also a key ingredient in [5]. The proof of Lemma 2.1 is completely elementary and can also be found in [5,Lemma 3.1].…”