On cohomological dimension of group homomorphisms

The (co)homological dimension of homomorphism $\phi :G\to H$ is the maximal number $k$ such that the induced homomorphism is nonzero for some $H$-module. The following theorems are proven: THEOREM 1. For every homomorphism $\phi :G\to H$ of a geometrically finite group $G$ the homological dimension of $\phi$ equals the cohomological dimension, $hd(\phi )=cd(\phi )$. THEOREM 2. For every homomorphism $\phi :G\to H$ of geometrically finite groups $cd(\phi \times \phi )=2cd(\phi )$.