Spaces of Graphs, Boundary Groupoids and the Coarse Baum-Connes Conjecture

TL;DR

This paper introduces a new boundary coarse Baum-Connes conjecture tailored for coarsely disconnected spaces, proves it for certain graph spaces, and relates it to the classical conjecture, providing insights into known counterexamples.

Contribution

The paper proposes a novel boundary conjecture, proves it for specific graph spaces with large girth, and connects it to the classical conjecture to explain counterexamples.

Findings

Proved the boundary conjecture for spaces of graphs with large girth.

Connected the boundary conjecture to the classical conjecture via homological methods.

Provided an elementary description of counterexamples to the coarse Baum-Connes conjecture.

Abstract

We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that are known to be counterexamples to the coarse Baum-Connes conjecture. In particular, we give a geometric proof of this conjecture for spaces of graphs that have large girth and bounded vertex degree. We then connect the boundary conjecture to the coarse Baum-Connes conjecture using homological methods, which allows us to exhibit all the current uniformly discrete counterexamples to the coarse Baum-Connes conjecture in an elementary way.