APPROXIMATION OF CENTER-VALUED BETTI-NUMBERS

A useful tool to calculate L-2-Betti-numbers is an approximation theorem which is proved in its original version by W. Luck. It shows that L-2-Betti-numbers beta((2))(n) ((X) over tilde) of the universal covering (X) over tilde of a CW complex X, with residually finite fundamental group pi, can be approximated by the Betti-numbers of the finite subcovers beta((2))(n) ((X) over tilde/pi(i)). Since then, the approximation theorem has been generalized in several steps and is now proven for a large class of groups containing e. g. sofic groups, all extensions of residually finite groups with amenable quotients, all residually amenable groups and free products of these. However, there is also a finer invariant than L-2-Betti-numbers: the so called universal or center-valued Betti-numbers beta(u). They measure the dimension of the homology, using the center-valued trace tr(N(G))(u) of the finite von Neumann algebra N(G), instead of the usual C-valued trace tr(N(G))(C). The major advantage of center-valued Betti-numbers is that they fully classify the L-2-homology modules. In this paper we generalize the ordinary approximation theorem to an approximation theorem for universal Betti-numbers.