Filtrations and cohomology on graph products

TL;DR

This paper investigates the algebraic and cohomological properties of graph products of groups, especially fundamental groups of surfaces, by analyzing filtrations, graded Lie algebras, and cohomology over finite fields, providing new examples and resolving existing questions.

Contribution

It introduces methods to compute filtrations, graded Lie algebras, and cohomology of graph products, extending previous work and providing explicit examples with torsion-free properties.

Findings

Computed cohomology over _p for graph products of surface groups.

Determined graded Lie algebras associated with filtrations of these groups.

Provided explicit examples satisfying torsion-freeness conditions.

Abstract

Let $p$ be a prime. We resolve a question posed by Min\'a\v{c}-Rogelstad-T\^an. 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 Min\'a\v{c}-Rogelstad-T\^an.