Quasirational relation modules and p-adic Malcev completions

TL;DR

This paper introduces quasirational relation modules for pro-p groups, showing their connection to p-adic Malcev completions and providing new tools for understanding group presentations in a p-adic context.

Contribution

It defines quasirational relation modules and demonstrates their relation to p-adic Malcev completions, expanding the understanding of pro-p group presentations.

Findings

Quasirational relation modules include CA-presentations.

All single-relation pro-p-group presentations are quasirational.

p-adic rationalizations are isomorphic to abelianized p-adic Malcev completions.

Abstract

We introduce the concept of quasirational relation modules for discrete (pro- p) presentations of discrete (pro-p) groups. It is shown, that this class of presentations for discrete groups contains CA-presentations and their subpresentations. For pro-p-groups we see that all presentations of pro-p-groups with a single defining relation are quasirational. We offer definitions of p- adic G(p)-completion and p-adic rationalization of relation modules which are adjusted to quasirational pro-p-presentations. p-adic rationalizations of quasirational relation modules of pro-p-groups are isomorphic to abelianized p-adic Malcev completions.