On vanishing criteria of $L^{2}$-Betti numbers of groups

Abstract

Let G be a countable group and k a positive integer, we show that the L^{2} -Betti numbers of the group G vanish up to degree k provided that there is some infinite index subgroup H with finite k th L^{2} -Betti number containing a normal subgroup of G whose L^{2} -Betti numbers are all zero below degree k . This generalizes previous criteria of both Sauer and Thom, and Peterson and Thom. In addition, we exhibit a purely algebraic proof of a well-known theorem of Gaboriau concerning the first L^{2} -Betti number which was requested by Bourdon, Martin and Valette. Finally, we provide evidence of a positive answer for a question posted by Hillman that wonders whether the above statement holds for k=1 and H containing a subnormal subgroup instead.