Joint work with M.Davis, T.Januszkiewicz
and B.Okun.
The goal is to calculate the cohomology spaces
of the complex of square-summable cochains on a building X.
These spaces are modules over the von Neumann algebra
of the automorphism group of X, and have finite
(von Neumann) dimensions: the $L^2$-Betti numbers of X.
Those Betti numbers can also be calculated using
weighted square-summable cochains on the Davis complex of
the Weyl group W of X and a Hecke-von Neumann
algebra of W. The weight comes from the thickness q of the
building. This reinterpretation allows to consider
non-integer real values of q and also facilitates
calculations. In particular, we get a complete answer for
q greater than the inverse of the logarithmic growth rate
of W.

Lecture #12580

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