On vanishing of L²-Betti numbers for groups

Abstract

We show that if a group G admits a finite dimensional contractible G-CW-complex X then the vanishing of the L²-Betti numbers for all stabilizers G_σ of X determines that of the L²-Betti numbers for G. We also give a relation among the L²-Euler characteristics for X as a G-CW-complex and those for X as a G_σ-CW-complex under certain assumptions. Finally, we present a new class of groups satisfying the Chatterji-Mislin conjecture which amounts to putting Brown’s formula within the framework of L²-homology.