Relative extensions and cohomology of profinite groups

TL;DR

This paper develops a correspondence between relative cohomology groups of profinite groups and classes of relative extensions, linking cohomological properties with algebraic extensions and exploring their relation to free products and projective pairs.

Contribution

It introduces a new correspondence between relative cohomology and relative extensions for profinite groups, extending classical results to the profinite setting.

Findings

Established a correspondence for both discrete and profinite groups.

Connected cohomological dimension one groups with free products.

Provided insights into the structure of projective group pairs.

Abstract

We construct a correspondence between the cohomology groups of a group $G$ relative to a family of subgroups $\famS$ and the classes of `relative extensions' of $G$ by abelian groups, modulo a certain equivalence relation. We establish this correspondence for both discrete and profinite group pairs. We go on to discuss the relationships of profinite group pairs of cohomological dimension one with free products and projective group pairs.