2014/04/25 by Rufus Willett, Willett, Rufus
Mathematics · #FOS: Mathematics #K-Theory and Homology (math.KT) #Metric Geometry (math.MG) #Operator Algebras (math.OA) #math.KT #math.MG #math.OA
paper · pdf · doi:10.48550/arxiv.1404.6456
arxiv created 2014/04/25 · arxiv updated 2014/04/28
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.