Coarse amenability versus paracompactness

TL;DR

This paper explores the deep analogies between coarse amenability and paracompactness, introducing a new coarse analog of paracompactness based on expanders to identify non-amenable spaces.

Contribution

It defines a novel coarse analog of paracompactness inspired by expanders, providing simpler proofs for classes of spaces that are coarsely non-amenable.

Findings

Expander sequences are coarsely non-amenable.

Graph spaces with increasing girth are coarsely non-amenable.

Unions of powers of finite groups are coarsely non-amenable.

Abstract

Recent research in coarse geometry revealed similarities between certain concepts of analysis, large scale geometry, and topology. Property A of G.Yu is the coarse analog of amenability for groups and its generalization (exact spaces) was later strengthened to be the large scale analog of paracompact spaces using partitions of unity. In this paper we go deeper into divulging analogies between coarse amenability and paracompactness. In particular, we define a new coarse analog of paracompactness modelled on the defining characteristics of expanders. That analog gives an easy proof of three categories of spaces being coarsely non-amenable: expander sequences, graph spaces with girth approaching infinity, and unions of powers of a finite non-trivial group.