Coarse amenability versus paracompactness

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.

💡 Research Summary

The paper investigates the deep relationship between coarse amenability—a large‑scale analogue of group amenability introduced by G. Yu as Property A—and the classical topological notion of paracompactness. After recalling that Property A was later generalized to “exact spaces” via the existence of uniformly bounded partitions of unity, the authors propose a new coarse analogue of paracompactness that is explicitly modeled on the characteristic features of expander graphs.

The new definition, called coarse paracompactness, requires two ingredients for every prescribed large scale (R): (1) a uniformly bounded cover in which each (R)-ball meets only finitely many cover elements, and (2) a coarse partition of unity subordinate to that cover whose variation is uniformly small (controlled by a parameter (\varepsilon)). This formulation mirrors the classical paracompact condition—existence of locally finite refinements and subordinate partitions—while adapting it to the coarse category, where “local” is interpreted in terms of large‑scale balls rather than open neighborhoods.

Armed with this definition, the authors give a remarkably simple proof that three broad families of spaces fail to be coarsely amenable:

1. Expander sequences – By definition, expanders have a uniform Cheeger constant and bounded degree, yet their diameters grow without bound. Any attempt to produce a uniformly bounded (R)-cover for a fixed (R) would force some cover element to contain an entire graph, contradicting the finiteness condition on overlaps. Consequently, no coarse partition of unity can be constructed, and expanders are not coarsely paracompact.

2. Graph spaces with girth tending to infinity – When the girth (the length of the shortest cycle) diverges, the graphs contain arbitrarily large “holes”. These holes prevent the existence of a uniformly bounded cover that refines the large‑scale geometry without large overlaps, again violating the coarse paracompactness condition.

3. Unions of powers of a finite non‑trivial group – Consider the space formed by taking the disjoint union of the Cayley graphs of (G^n) for a finite group (G) and all (n\ge1). The exponential growth of the group’s word metric forces any uniform cover to have elements whose diameter grows with (n); no uniform bound exists, and a subordinate coarse partition of unity cannot be defined.

These arguments bypass the more involved analytic techniques traditionally used to show non‑amenability (e.g., spectral gap estimates, coarse cohomology). Instead, the failure of the two elementary coarse‑paracompact conditions suffices.

Beyond the three examples, the authors discuss the broader implications of their framework. Coarse paracompactness sits strictly between Property A and the absence of it: every space with Property A is coarsely paracompact, but the converse need not hold, opening a new hierarchy of large‑scale properties. Moreover, the definition is robust enough to be applied to box spaces of residually finite groups, spaces arising from measured groupoids, and metric spaces equipped with a coarse structure coming from a non‑uniformly expanding dynamical system.

In conclusion, the paper introduces a natural, expander‑inspired coarse analogue of paracompactness, demonstrates its utility by providing concise proofs of non‑amenability for several important classes of spaces, and suggests a fertile line of inquiry into how coarse partitions of unity can serve as a unifying tool linking large‑scale geometry, analysis, and topology.