Vanishing results for L<SUP>2</SUP>-Betti numbers and L<SUP>2</SUP>-Euler characteristics and their applications

초록

A CW-complex X is called a [G, m]-complex if X is an m-dimensional complex with pi(1)(X) congruent to G and the universal cover (X) over tilde 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 L-2-Euler characteristic of (X) over tilde. This result corresponds to a generalization and a variation of earlier several works. Also, we show that the L-2-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.

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