Higher index theory for certain expanders and Gromov monster groups II

TL;DR

This paper advances higher index theory for expanders, proving the maximal coarse Baum-Connes assembly map is an isomorphism for graphs with increasing girth, and introduces a new property to distinguish expanders.

Contribution

It establishes the isomorphism of the maximal coarse Baum-Connes assembly map for certain expanders and introduces geometric property (T) as an obstruction.

Findings

Maximal coarse Baum-Connes assembly map is an isomorphism for graphs with girth tending to infinity.

Introduction of geometric property (T) as an obstruction to the assembly map being an isomorphism.

Differentiation between expanders with girth tending to infinity and those from property (T) groups.

Abstract

In this paper, the second of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs has girth tending to infinity, then the maximal coarse Baum-Connes assembly map is an isomorphism for the associated metric space $X$. As discussed in the first paper in this series, this has applications to the Baum-Connes conjecture for `Gromov monster' groups. We also introduce a new property, `geometric property (T)'. For the metric space associated to a sequence of graphs, this property is an obstruction to the maximal coarse assembly map being an isomorphism. This enables us to distinguish between expanders with girth tending to infinity, and, for example, those constructed from property (T) groups.