VANISHING RESULTS FOR L2-BETTI NUMBERS AND L2-EULER CHARACTERISTICS AND THEIR APPLICATIONS

Abstract

A CW-complex X is called a [G,m]-complex if X is an m-dimensional complex with π1(X) ≅ G and the universal cover [Formula: see text] is (m - 1)-connected. We show that if G has an infinite amenable normal subgroup, then the asphericity of a [G,m]-complex X is equivalent to the vanishing of L2-Euler characteristic of [Formula: see text]. This result corresponds to a generalization and a variation of earlier several works. Also, we show that the L2-Betti numbers of a group which belongs to the class of groups K𝔉 eventually vanish. As a byproduct, we give an example of a group which belongs to the class of groups H𝔉 but does not belong to the class of groups K𝔉.