Paths in graphs: bounded geometry and property A

TL;DR

This paper characterizes a class of discrete metric spaces where bounded geometry is equivalent to property A, including certain path spaces in graphs like cactus graphs, and explores the implications of unbounded geometry.

Contribution

It introduces a new class of graph path spaces where bounded geometry and property A are equivalent, and analyzes the structure of spaces lacking bounded geometry.

Findings

Bounded geometry is equivalent to property A in the studied class.

Includes spaces formed by simple paths in cactus graphs.

Unbounded geometry spaces contain bounded geometry subspaces without property A.

Abstract

We expose a class of discrete metric spaces, for which bounded geometry is equivalent to the property A of G. Yu. This class includes the coarse disjoint union of $(\mathbb Z/2\mathbb Z)^n$, $n\in\mathbb N$, and consists of spaces of simple paths in a class of graphs that includes cactus graphs, with the metric defined as the number of edges in the symmetric difference of the paths. We also show that if a space in this class does not have bounded geometry then it contains a subspace of bounded geometry without property A.