On a relation between certain cohomological invariants

Let G be a group, spli Z G the supremum of the projective lengths of the injective Z G -modules and silp Z G the supremum of the injective lengths of the projective Z G -modules. The invariants spli Z G and silp Z G 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 spli Z G is finite then silp Z G = spli Z G = findim Z G [T.V. Gedrich, K.W. Gruenberg, Complete cohomological functors on groups, Topology Appl. 25 (1987) 203–223] and cd ¯ Z G ≤ spli Z G ≤ cd ¯ Z G + 1 , where cd ¯ Z G is the generalized cohomological dimension of G [B.M. Ikenaga, Homological dimension and Farrell cohomology, J. Algebra 87 (1984) 422–457]. Note that cd ¯ Z G = vcd G if G is of finite virtual cohomological dimension. It has been conjectured in [O. Talelli, On groups of type Φ , Arch. Math. 89 (1) (2007) 24–32] that if spli Z G is finite then G admits a finite dimensional model for E ¯ G , the classifying space for proper actions. We conjecture that spli Z G = cd ¯ Z G + 1 for any group G 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 ∞ .