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Uniform spanning trees on finite graphs and their analogues on infinitegraphs are a well-studied area. On a Cayley graph of a group, we show that theyare 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 onfinite and infinite CW-complexes, which relate to the higher $\ell^2$-Bettinumbers. One consequence is a uniform isoperimetric inequality extending workof Lyons, Pichot, and Vassout. We also present an enumeration similar to recentwork of Duval, Klivans, and Martin.

RANDOM COMPLEXES AND ℓ2-BETTI NUMBERS

RANDOM COMPLEXES AND ℓ²-BETTI NUMBERS