Toric arrangements and Bloch-Kato pro-$p$ groups

TL;DR

This paper establishes combinatorial obstructions for the Bloch-Kato property in fundamental groups of toric arrangement complements, with implications for pure braid and mapping class groups' pro-p completions.

Contribution

It introduces new combinatorial criteria to determine the Bloch-Kato property for groups arising from toric arrangements, extending to pure braid and mapping class groups.

Findings

Pro-p completion of pure braid group on 3 or fewer strands has Bloch-Kato property.

Pro-p completion of pure mapping class group on 4 or fewer punctures has Bloch-Kato property.

Provides combinatorial obstructions for cohomology generation in supersolvable arrangements.

Abstract

We prove a purely combinatorial obstruction for the Bloch-Kato property within the class of fundamental groups of complement manifolds of toric arrangements (i.e., arrangements of hypersurfaces in the complex torus). As a stepping stone we obtain a combinatorial obstruction for the cohomology of a supersolvable arrangement to be generated in degree 1. Our result allows us to prove that - for all prime numbers $p$, the pro-$p$ completion of the pure braid group on $k$ strands has the Bloch-Kato property if and only if $k\leq 3$; - for all prime numbers $p$, the pro-$p$ completion of the pure mapping class group of the sphere $S^2$ with $k$ punctures has the Bloch-Kato property if and only if $k\leq 4$.