\(L^{2}\)-Betti numbers of infinite configuration spaces (Q863785)

Let \(\Gamma_ X\) denote the space of all locally finite subsets (configurations) in a complete, connected, oriented Riemannian manifold \(X\) of infinite volume with a lower bounded curvature. The authors study the de Rham complex of square-integrable differential forms over the configuration space \(\Gamma_ X\) equipped with the Poisson measure, the corresponding de Rham cohomology, and the spaces of harmonic forms, in the case where \(X\) is an infinite covering of a compact manifold. Also, the authors consider a natural von Neumann algebra containing the projection onto the space of harmonic forms and obtain explicit formulas for the corresponding trace. Moreover, a regularized index of the Dirac operator associated with the de Rham differential on the configuration space of an infinite covering is considered. Finally, they introduce \(L^ 2\)-Betti numbers of configuration spaces over infinite covers.