PUB - Publikationen an der Universität Bielefeld

The main achievement of this thesis 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_{PZ_n}}$$PZ_n$($\\Sigma_\\Gamma$($L$))$\\xrightarrow{\\pi_{PZ_n}}$$PZ_{n-1}$($\\Sigma_\\Gamma$($L$))$\\rightarrow$1.

In particular, we observe that the kernel $K$ of $\iota_{PZ_n}$ 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 $K$.

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.