On cohomological dimension of group homomorphisms

Abstract

The (co)homological dimension of homomorphism $\varphi :G\to H$ 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 $\varphi :G\to H$ of a geometrically finite group $G$ the homological dimension of $\varphi$ equals the cohomological dimension, $hd(\varphi )=cd(\varphi )$. THEOREM 2. For every homomorphism $\varphi :G\to H$ of geometrically finite groups $cd(\varphi \times \varphi )=2cd(\varphi )$.