Mild Pro-$p$ Groups and Ordered Monoids

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 Gärtner. 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.