Paths in graphs: bounded geometry and property A
V. Manuilov
TL;DR
This work identifies a natural class of discrete metric spaces—spaces of simple paths in graphs with controlled cycles, including Nowak’s coarse disjoint unions—where bounded geometry is equivalent to property A. By introducing the metric $d(\gamma,\delta)=|\gamma\Delta\delta|$ on simple paths and embedding path spaces into $(\mathbb Z/2\mathbb Z)^E$ via an incidence map $f$, the authors connect coarse-geometric properties to cycle-structure: a precise criterion states that $P(\Gamma)$ has bounded geometry iff, for every $n$, any path intersects at most $m$ cycles of length at most $n$. They prove the equivalence of bounded geometry, property A, and the Higson–Roe condition for graphs with controlled cycles, using Ostrovsky’s and Nowak–Yu’s frameworks to handle non-bounded cases. This clarifies when coarse amenability notions are governed by cycle interactions in path spaces and provides a robust setting that includes and extends known examples, with implications for coarse disjoint unions and cactus-like graphs.
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.