On cohomological dimension of homomorphisms

The (co)homological dimension of a homomorphism ϕ : G → H φ :G→ H is the maximal number k k such that the induced homomorphism in k k -th (co)homology groups is nonzero for coefficients in some H H -module. It is known that for geometrically finite groups G , c d ( G ) = h d ( G ) G, cd(G)=hd(G) and c d ( G × G ) = 2 c d ( G ) cd(G× G)=2cd(G) . We prove analogous theorems for homomorphisms of geometrically finite groups. The analogy stops working on the Eilenberg-Ganea equality c d ( G ) = g d ( G ) cd(G)=gd(G) where c d ( G ) > 2 cd(G)>2 and g d ( G ) gd(G) is the geometric dimension of G G . We show that for every k > 2 k>2 there is a group homomorphism ϕ k : π k → Z k φ _k:π _k→ Z^k with c d ( ϕ k ) > k cd(φ _k)>k and g d ( ϕ k ) = k gd(φ _k)=k where π k π _k is the fundamental group of a closed aspherical ( k + 1 ) (k+1) -dimensional manifold.