The Groups $G_{n}^{k}$ and Fundamental Groups of Configuration Spaces

TL;DR

This paper explores the relationship between fundamental groups of plane configuration complements and the algebraic groups $G_{n}^{k}$, providing new insights into their connections and geometric realizations.

Contribution

It introduces a map linking fundamental groups to $G_{n}^{k}$ groups and discusses their interrelations and geometric interpretations.

Findings

Established a map from fundamental groups to $G_{n}^{k}$ groups.

Analyzed connections between different $G_{n}^{k}$ groups.

Explored geometric realizations of these groups.

Abstract

We construct a map from fundamental groups of complements to some plane configurations to the groups $G_{n}^{k}$ for large $k$. We discuss connection between the groups $G_{n}^{k}$ for different $G_{n}^{k}$ and their geometric realization.