On a relation between certain cohomological invariants

Let GG be a group, spliZG the supremum of the projective lengths of the injective ZGZG-modules and silpZG the supremum of the injective lengths of the projective ZGZG-modules. The invariants spliZG and silpZG were studied in [T.V. Gedrich, K.W. Gruenberg, Complete cohomological functors on groups, Topology Appl. 25 (1987) 203–223] in connection with the existence of complete cohomological functors. If spliZG is finite then silpZG=spliZG=findimZG [T.V. Gedrich, K.W. Gruenberg, Complete cohomological functors on groups, Topology Appl. 25 (1987) 203–223] and cd¯ZG≤spliZG≤cd¯ZG+1, where cd¯ZG is the generalized cohomological dimension of GG [B.M. Ikenaga, Homological dimension and Farrell cohomology, J. Algebra 87 (1984) 422–457]. Note that cd¯ZG=vcdG if GG is of finite virtual cohomological dimension. It has been conjectured in [O. Talelli, On groups of type ΦΦ, Arch. Math. 89 (1) (2007) 24–32] that if spliZG is finite then GG admits a finite dimensional model for E¯G, the classifying space for proper actions.We conjecture that spliZG=cd¯ZG+1 for any group GG and we prove the conjecture for duality groups, fundamental groups of graphs of finite groups and fundamental groups of certain finite graphs of groups of type FP∞.