Weak commutativity for pro-$p$ groups

TL;DR

This paper introduces a pro-$p$ analogue of Sidki's weak commutativity construction, exploring its properties and implications for finitely presented and analytic pro-$p$ groups, including tensor products and finiteness conditions.

Contribution

It defines the pro-$p$ weak commutativity construction and investigates its properties, extending known results to pro-$p$ groups and related structures.

Findings

Pro-$p$ weak commutativity construction preserves finite presentability.

The construction maintains analyticity in pro-$p$ groups.

Established a pro-$p$ version of the $(n-1)-n-(n+1)$ Theorem.

Abstract

We define and study a pro-$p$ version of Sidki's weak commutativity construction. This is the pro-$p$ group $\mathfrak{X}_p(G)$ generated by two copies $G$ and $G^{\psi}$ of a pro-$p$ group, subject to the defining relators $[g,g^{\psi}]$ for all $g \in G$. We show for instance that if $G$ is finitely presented or analytic pro-$p$, then $\mathfrak{X}_p(G)$ has the same property. Furthermore we study properties of the non-abelian tensor product and the pro-$p$ version of Rocco's construction $\nu(H)$. We also study finiteness properties of subdirect products of pro-$p$ groups. In particular we prove a pro-$p$ version of the $(n-1)-n-(n+1)$ Theorem.