Mild Pro-$p$ Groups and Ordered Monoids

TL;DR

This paper establishes a criterion for the mildness of finitely presented pro-$p$ groups, generalizing previous cohomological and circuit criteria, and explores connections with right-angled Artin groups.

Contribution

It introduces a new criterion for pro-$p$ group mildness that unifies and extends existing criteria, including cohomological and circuit-based approaches.

Findings

Proves a general mildness criterion for finitely presented pro-$p$ groups.

Generalizes cohomological mildness criteria via Massey products.

Connects mildness conditions with properties of right-angled Artin groups.

Abstract

We prove a criterion for the mildness of a finitely presented pro-$p$ group $G$. It implies as a special case a cohomological mildness criterion via Massey products, generalizing results due to Schmidt and G\"artner. It subsumes Labute's non-singular circuit criterion. We further show connections with the triangle condition for the mildness of pro-$p$ right-angled Artin groups, due to Quadrelli, Snopce and Vannacci.