Random Complexes and l^2-Betti Numbers

Details Random Complexes and l^2-Betti Numbers Random Complexes and l^2-Betti Numbers

2008-11-18

arxiv.org/abs/0811.2933v2

J. Top. Anal. 1, no. 2 (2009), 153-175.

Mathematics

Probability

25 pages, 1 figure

Scientific paper

Uniform spanning trees on finite graphs and their analogues on infinite graphs are a well-studied area. On a Cayley graph of a group, we show that they are related to the first $\ell^2$-Betti number of the group. Our main aim, however, is to present the basic elements of a higher-dimensional analogue on finite and infinite CW-complexes, which relate to the higher $\ell^2$-Betti numbers. One consequence is a uniform isoperimetric inequality extending work of Lyons, Pichot, and Vassout. We also present an enumeration similar to recent work of Duval, Klivans, and Martin.

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