Orbifold braid groups

TL;DR

This paper investigates the algebraic properties of orbifold braid groups, demonstrating injectivity of certain homomorphisms and triviality of their centers, advancing understanding of their structure and implications for related Artin groups.

Contribution

It applies previous results to establish injectivity of inclusion-induced homomorphisms and trivial centers in orbifold braid groups, deepening their structural understanding.

Findings

Inclusion maps induce injective homomorphisms

Most orbifold braid groups have trivial centers

Results support the study of Artin groups via orbifold braid groups

Abstract

The orbifold braid groups of two dimensional orbifolds were defined in [1] (arXiv:math/9907194) to understand certain Artin groups as subgroups of some suitable orbifold braid groups. We studied orbifold braid groups in some more detail in [17] (arXiv:2006.07106) and [18] (arXiv:2106.08110), to prove the Farrell-Jones Isomorphism conjecture for orbifold braid groups and as a consequence for some Artin groups. In this article we apply the results from [17] and [18], to study two aspects of the orbifold braid groups. First we show that the homomorphisms induced on the orbifold braid groups by the inclusion maps of a generic class of sub-orbifolds of an orbifold are injective. Then, we prove that the centers of most of the orbifold braid groups are trivial.