Higher index theory for certain expanders and Gromov monster groups, I

Abstract

In this paper, the first of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs is an expander and the girth of the graphs tends to infinity, then the coarse Baum-Connes assembly map is injective, but not surjective, for the associated metric space X.Expanders with this girth property are a necessary ingredient in the construction of the so-called 'Gromov monster' groups that (coarsely) contain expanders in their Cayley graphs. We use this connection to show that the Baum-Connes assembly map with certain coefficients is injective but not surjective for these groups. Using the results of the second paper in this series, we also show that the maximal Baum-Cones assembly map with these coefficients is an isomorphism.

Spaces of graphs, boundary groupoids and the coarse Baum–Connes conjecture

“…From this, we then construct an operator that is not a ghost operator, but is ghostly on certain parts of the boundary. A tracelike argument, similar to those of [Hig99,WY12] then allows us to conclude that the boundary coarse Baum-Connes conjecture fails to be surjective for the space Y .…”

“…K for any compact operator [WY12]. As these differ under T r, they cannot possibly be equal in K 0 (I G ).…”

“…It is well known that this conjecture has counterexamples [HR00, Hig99,WY12] in the class of coarsely disconnected spaces.…”

“…Using this machinery we give elementary proofs of many of the counterexample arguments from [HLS02,Hig99,WY12] as well as many results concerning classes of expander graphs present in the literature [CTWY08,GTY11,OOY09]; in particular in Section 4 we give an elementary proof of results of Willett and Yu [WY12] concerning large girth sequences:…”

Group approximation in Cayley topology and coarse geometry, III: Geometric property (T)

“…This problem is related to a question by J. Roe [Roe03] to define 'coarse property (T). ' R. Willett and G. Yu [WY12a], [WY12b] have studied the maximal coarse BaumConnes conjecture and introduced the notion of geometric property (T) for a (coarse) disjoint union of uniformly locally finite and finite graphs, which is stronger than being an expander sequence. They have showed that this property is an obstruction to the surjectivity of the maximal coarse Baum-Connes assembly map, and that a box space G has geometric property (T) if and only if G possesses property (T).…”