On Deficiency Gradient of Groups

Deficiency gradient is a higher dimensional analog of rank gradient. In this paper, we give a combinatorial proof that the fundamental group of a simply connected complex of amenable groups has deficiency gradient zero. We apply this to establish the vanishing of deficiency gradient in special linear groups over polynomial rings and number fields, and in Artin groups for which the nerve of the Coxeter matrix is simply connected. This implies that the first and second l2-Betti numbers vanish for these Artin groups without recourse to the K(π,1) conjecture. We propose a conjecture about the stabilization of deficiency gradient, which characterizes groups with 2-dimensional classifying spaces.Communicated by Marc Burger