Higher index theory for certain expanders and Gromov monster groups I
By Rufus Willett, Guoliang Yu. 2010
117 citations recorded by the source corpus.
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.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/019479256a5e2452adeaff126df24ca7d2d3b03d
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In Geometry and Topology
- Rufus Willett
- Martin Finn-Sell
- Guoliang Yu
- Qin Wang
- Xiaoman Chen
- E. Guentner
- Romain Tessera
- Daniel A. Ramras
In Mathematical Physics
Citations in the corpus
- Fibred coarse embeddings, a-T-menability and the coarse analogue of the Novikov conjecture (2013, methodology): “This happens if and only if the ghost projection is compact [30].”
- Expanders, exact crossed products, and the Baum–Connes conjecture (2013, result): “On the other hand, the results of [Willett and Yu 2012a] show that the class Œp 2K0.”
- The maximal coarse Baum-Connes conjecture for spaces which admit a fibred coarse embedding into Hilbert space (2012, methodology): “[38]) if for any r > 0 there exists N ∈ N such that, for all n ≥ N , the covering map πn : X̃n → Xn is “r-isometric”, i.”
- Geometric property (T) (2013, result): “The treatment in [18] was quite brief; moreover, we expect that geometric property (T) will see some applications outside of K-theory and higher index theory.”