The groups $G_{k+1}^{k}$ and fundamental groups of configuration spaces

TL;DR

This paper shows that for n=k+1, the group G_{k+1}^{k} is isomorphic to the fundamental group of a certain configuration space, providing new insights and solutions for related algebraic problems.

Contribution

It establishes an isomorphism between G_{k+1}^{k} and a fundamental group of a configuration space for the case n=k+1, advancing understanding of these groups.

Findings

G_{k+1}^{k} is isomorphic to a fundamental group of a configuration space for n=k+1.

This leads to solutions for word and conjugacy problems in G_{4}^{3}.

Provides insights into the structure of G_{k+1}^{k} for higher k.

Abstract

In \cite{HigherGnk}, the author has constructed natural maps from fundamental groups of topological spaces (restricted configuration spaces) to the groups $G_{n}^{k}$. In the present paper, we show that in the case of $n=k+1$, the group $G_{k+1}^{k}$ is isomorphic to the fundamental group of some (quotient space of) some configuration space. In particular, this leads to the solution of word and conjugacy problems in $G_{4}^{3}$ and sheds light on $G_{k+1}^{k}$ for higher $k$.