Weak Notions of Normality and Vanishing up to Rank in L2-Cohomology

An important application of the algebraic theory of L2-Betti numbers [10] (see Farber [8] for an alternative approach) is that the L2-Betti numbers β i (Γ ) of a group Γ vanish if it has a normal subgroup whose L2-Betti numbers vanish. With regard to the first L2-Betti number, one can significantly relax the normality condition to obtain similar vanishing results [14]. Peterson and Thom prove in [14] that the first L2-Betti number of a group vanishes if it has a s-normal subgroup (defined below) with vanishing first L2-Betti number. The aim of this article is to extend such vanishing results to arbitrary degrees and to present some applications. Next, we describe the main notions and results in greater detail.