Braid groups and mapping class groups for 2-orbifolds

TL;DR

This paper studies the structure of pure orbifold braid groups and their relation to orbifold mapping class groups, revealing new exact sequences and correcting previous theorems, with implications for understanding orbifold group behaviors.

Contribution

It introduces a new exact sequence for pure orbifold braid groups, corrects a prior theorem, and compares orbifold braid groups with mapping class groups, highlighting their differences.

Findings

The kernel of the exact sequence is non-trivial.

Orbifold braid groups are proper quotients of orbifold mapping class groups.

A new presentation of pure orbifold braid groups is provided.

Abstract

The main result of this article is that pure orbifold braid groups fit into an exact sequence $1\rightarrow K\rightarrow\pi_1^{orb}(\Sigma_\Gamma(n-1+L))\xrightarrow{\iota_{\textrm{PZ}_n}}\textrm{PZ}_n(\Sigma_\Gamma(L))\xrightarrow{\pi_{\textrm{PZ}_n}}\textrm{PZ}_{n-1}(\Sigma_\Gamma(L))\rightarrow1.$ In particular, we observe that the kernel $K$ of $\iota_{\textrm{PZ}_n}$ is non-trivial. This corrects Theorem 2.14 in [12](arXiv:2006.07106). Moreover, we use the presentation of the pure orbifold mapping class group $\textrm{PMap}^{\textrm{id},orb}_n(\Sigma_\Gamma(L))$ from [8] to determine $K$. Comparing these orbifold mapping class groups with the orbifold braid groups, reveals a surprising behavior: in contrast to the classical case, the orbifold braid group is a proper quotient of the orbifold mapping class group. This yields a presentation of the pure orbifold braid group which allows us to read off the kernel $K$.