ON n-DEFICIENCY OF GROUPS

Abstract

Let G be a group of type Fn with n ≥ 2, i.e. G admits a finite n-dimensional (n - 1)-aspherical CW-complex X such that π1(X) is isomorphic to G. We introduce the notion of the n-deficiency dn(G) which is a generalized notion of deficiency of G and present various results about the relation between dn(G) and the L2-Betti numbers of G. We also give some partial results about the Eilenberg–Ganea conjecture and the Whitehead conjecture. These results correspond to a generalization of several earlier results [Quart. J. Math. Oxford (2)38 (1987) 35–44; J. Pure Appl. Algebra7 (1976) 241–248; Quart. J. Math. Oxford (2)32 (1981) 45–55; 2-Knots and Their Groups (Cambridge University Press, 1989)].