Random graphs, weak coarse embeddings, and higher index theory

TL;DR

This paper investigates higher index theory for random sequences of bounded degree graphs with increasing diameter, revealing properties of the coarse Baum-Connes assembly map and expansion characteristics.

Contribution

It demonstrates almost sure injectivity and non-surjectivity of the coarse Baum-Connes assembly map, and that the maximal map is an isomorphism for such graphs, linking to expansion properties.

Findings

Coarse Baum-Connes assembly map is injective almost surely.

The assembly map is not surjective almost surely.

Maximal coarse Baum-Connes assembly map is an isomorphism almost surely.

Abstract

This paper studies higher index theory for a random sequence of bounded degree, finite graphs with diameter tending to infinity. We show that in a natural model for such random sequences the following hold almost surely: the coarse Baum-Connes assembly map is injective; the coarse Baum-Connes assembly map is not surjective; the maximal coarse Baum-Connes assembly map is an isomorphism. These results are closely tied to issues of expansion in graphs: in particular, we also show that such random sequences almost surely do not have geometric property (T), a strong form of expansion. The key geometric ingredients in the proof are due to Mendel and Naor: in our context, their results imply that a random sequence of graphs almost surely admits a weak form of coarse embedding into Hilbert space.