Filtrations and cohomology on graph products
Oussama Hamza
TL;DR
This work resolves a key question of Mináč–Rogelstad–Tân by linking the Zassenhaus and lower central filtrations under a torsion-freeness hypothesis for pro-$p$ groups, and extends these filtrations and cohomology computations to graph products. It establishes that, under suitable conditions, the Lie algebra of a graph product decomposes as the graph product of the Lie algebras, and proves torsion-freeness alongside rank preservation in both field and ring settings. The paper then computes Golod–Shafarevich gocha series and shows Koszulity transfers to graph products, with cohomology $H^ullet( ext{ΓG})$ identified as the quadratic dual of the graph-product algebra, yielding explicit formulas for h^n(ΓG) and gocha(ΓG,t). Applications include cohomology computations for graph products of surface-group pro-$p$ groups and the construction of new torsion-free examples, with parallel results extended to abstract groups and their pro-$p$ completions, including p-cohomological completeness.
Abstract
Let $p$ be a prime. We resolve a question posed by Mináč-Rogelstad-Tân. We relate the Zassenhaus and the lower central series of pro-$p$ groups under a torsion-freeness condition. We also study graph products of (pro-$p$) groups under natural assumptions. In particular, we compute their graded Lie algebras associated with the previous filtrations, as well as their cohomology over $\mathbb{F}_p$. Our approach relies on various filtrations of amalgamated products, as studied in Leoni's PhD thesis. Explicit examples are provided using the Koszul property. As a concrete application, we compute the cohomology over $\mathbb{F}_p$ and the graded Lie algebras associated with the filtrations of graph products of fundamental groups of surfaces. These groups furnish new examples satisfying the torsion-freeness condition, which arises in the question of Mináč-Rogelstad-Tân.