On cohomological dimension of group homomorphisms

TL;DR

This paper investigates the cohomological and homological dimensions of group homomorphisms, establishing their equality for geometrically finite groups and analyzing the behavior of these dimensions under group products.

Contribution

It proves that for geometrically finite groups, the homological and cohomological dimensions of a homomorphism are equal, and explores how these dimensions behave under group products.

Findings

Homological and cohomological dimensions are equal for geometrically finite groups.

The cohomological dimension of a product of a homomorphism with itself doubles.

Theorems provide a deeper understanding of the structure of group homomorphisms in geometric group theory.

Abstract

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 $cd(\phi\times\phi)=2cd(\phi)$.