Coarse Geometry and Index Theory

Coarse Geometry and P. A. Smith Theory

We define a generalization of the fixed point set, called the bounded fixed set, for a group acting by isometries on a metric space. An analogue of the P. A. Smith theorem is proved for metric spaces of finite asymptotic dimension, which relates the coarse homology of the bounded fixed set to the coarse homology of the total space.

Coarse Geometry and Randomness

1 Introductory graph and metric notions 5 1.1 The Cheeger constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Expander graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Isoperimetric dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Rough isome...

Random graphs, weak coarse embeddings, and higher index theory

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 BaumConnes assembly map is an isomorphism. These res...

Coarse geometry and asymptotic dimension

We prove that two spaces, whose coarse structures are induced by metrisable compactifications, are coarsely equivalent if and only if their (Higson) coronas are homeomorphic. We introduce translation C∗-algebras for coarse spaces which admit a countable, uniformly bounded cover using projection-valued measures. This was already done in [Roe03]. Here we give a more complete exposition on the sub...

Coarse Geometry and K-homology