Orbit Braid Action on a Finite Generated Group

TL;DR

This paper extends Artin's ideas to establish a correspondence between orbit braid groups on punctured planes and certain homeomorphism groups, providing a faithful representation in a finitely generated group.

Contribution

It introduces a faithful representation of orbit braid groups on punctured planes into finitely generated groups, generalizing classical braid theory concepts.

Findings

Established a one-to-one correspondence between orbit braid groups and homeomorphism quotients.

Constructed a faithful representation of orbit braid groups in finitely generated groups.

Characterized orbit braid representations to support the main correspondence.

Abstract

This paper aims to generalize Artin's ideas to establish an one-to-one correspondence between the orbit braid group $B^{orb}_n(\mathbb{C},\mathbb{Z}_p)$ and a quotient of a group formed by some particular homeomorphisms of a punctured plane. First, we find a faithful representation of $B^{orb}_n(\mathbb{C},\mathbb{Z}_p)$ in a finite generated group whose generators are corresponding to generators of fundamental group of the punctured plane, by examining the representation from $B^{orb}_n(\mathbb{C}^{\times},\mathbb{Z}_p)$ to the fundamental group is faithful. Then we investigate some characterizations of orbit braid representation to come to our conclusion.