Braid groups and mapping class groups for 2-orbifolds

The main achievement of this thesis is that pure orbifold braid groups fit into an exact sequence 1$\\rightarrow$ $K$$\\rightarrow$\π₁^orb(\\Sigma_\\Gamma($n-1+L$))\\xrightarrow{\\iota_PZn}PZ_n(\\Sigma_\\Gamma($L$))\\xrightarrow{\π_PZn}PZ_n-1(\\Sigma_\\Gamma($L$))$\\rightarrow$1. <br /><br /> In particular, we observe that the kernel $K$ of \iota_PZn is non-trivial. This corrects Theorem 2.14 in [42]. Using the relation pictured in the figure on the title page, we construct non-trivial elements in the kernel. Moreover, we determine $K$. For this purpose, we introduce orbifold mapping class groups (with marked points) and establish a Birman exact sequence for them. Comparing the 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. In particular, this yields a presentation of the pure orbifold braid group which allows us to determine the kernel $K$. <br /><br /> Furthermore, we generalize a result of Allcock [2] about Artin groups contained in orbifold braid groups and we analyze the connectivity of bipartite matching complexes of $\Gamma$-arcs. This allows us to deduce highly generating families of subgroups in Map_n^id,orb(\\Sigma_\\Gamma($L$)). For Z_n(\\Sigma_\\Gamma($L$)) and the contained Artin groups, we also obtain a highly generating family of subgroups.

Level thesis:degree_level

thesis.doctoral

Grantor dc:publisher

Universität Bielefeld

Year

2023

Author dc:creator

• Flechsig, Jonas

Repository record source_url

https://pub.uni-bielefeld.de/record/2979933

OAI identifier oai:identifier

oai:pub.uni-bielefeld.de:2979933