Deficiency and the geometric invariants of a group (with an appendix by Pascal Schweitzer)

It is known that the geometric invariants of a group G (which contain information on finiteness properties of certain submonoids and normal subgroups of G) have a description in terms of the vanishing of group homology of G with Novikov-ring-coefficients [see J.-C. Sikorav, Homologie de Novikov associée à une classe de cohomologie réelle de degré un, Thèse Orsay, 1987; R. Bieri, The geometric invariants of a group, in: G.A. Niblo, M.A. Roller (Eds.), Geometric Group Theory, in: London Math. Soc. Lecture Notes Series, vol. 181, Cambridge University Press, Cambridge, 1993; R. Bieri, R. Strebel, Geometric invariants for discrete groups, manuscript-preprint of a monograph (in preparation)], and [R. Bieri, R. Geoghegan, Kernels of actions on non-positively curved spaces, in: P.H. Kropholler, G. Niblo, R. Stöhr (Eds.), Geometry and Cohomology in Group Theory, in: London Math. Soc. Lecture Notes Series, vol. 252, Cambridge University Press, Cambridge, 1998, pp. 24–38]. In a recent paper Kochloukova [D. Kochloukova, Some Novikov rings that are von Neumann finite and knot-like groups (submitted for publication)] uses this to prove a conjecture of E. Rapaport-Strasser on knot-like groups. We extend her approach to establish a rather general relationship between deficiency and the geometric invariants of a group.