Random Complexes and l^2-Betti Numbers

By Russell Lyons. 2008

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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.

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4b069f8b3dea7a90bd18a7a3a0342878a09deefa