Random graphs, weak coarse embeddings, and higher index theory
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.
Cite
@article{arxiv.1404.6456,
title = {Random graphs, weak coarse embeddings, and higher index theory},
author = {Rufus Willett},
journal= {arXiv preprint arXiv:1404.6456},
year = {2014}
}