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RANDOM COMPLEXES AND ℓ²-BETTI NUMBERS

RUSSELL LYONS

Department of Mathematics, Indiana University, Bloomington, IN 47405–5701, USA

https://doi.org/10.1142/S1793525309000072Cited by:21 (Source: Crossref)

Abstract

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 ℓ²-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 ℓ²-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.

Keywords:

AMSC: Primary 60B05, Primary 20F65, Primary 60C05, Secondary 05A15, Secondary 05C05

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