On the Burde--de Rham theorem for finitely presented pro-$p$ groups

TL;DR

This paper explores the Burde--de Rham theorem in the context of finitely presented pro-$p$ groups, providing new examples and cohomological insights under specific algebraic conditions.

Contribution

It extends the Burde--de Rham theorem to certain pro-$p$ groups with relators of total degree zero and offers concrete examples and cohomological interpretations.

Findings

The theorem holds for groups with relators of total degree zero.

Provides higher-dimensional examples under Iwasawa conditions.

Offers cohomological perspectives on the theorem.

Abstract

We consider the Burde--de Rham theorem for finitely presented pro-$p$ groups under the assumption that the total degrees of all relators are $0$. We also give some concrete examples including higher-dimensional cases under Iwasawa theoretic conditions, and consider some cohomological interpretations.