{"id":{"repo_id":"bielefeld","oai_identifier":"oai:pub.uni-bielefeld.de:2979933"},"canonical_url":"https://search.dev.ndltd.org/etd/bielefeld/oai:pub.uni-bielefeld.de:2979933","repository":{"repo_id":"bielefeld","name":"Universität Bielefeld","base_url":"https://pub.uni-bielefeld.de/oai"},"display":{"title":"Braid groups and mapping class groups for 2-orbifolds","abstract":"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. <br /><br /> 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 $K$. <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.","abstract_html":"The main achievement of this thesis is that pure orbifold braid groups fit into an exact sequence 1$\\\\rightarrow$ $K$$\\\\rightarrow$<span class=\"etd-inline-math\">\\&pi;<sub>1</sub><sup><sup>orb</sup></sup></span>(<span class=\"etd-inline-math\">\\\\Sigma<sub>\\</sub>\\Gamma</span>($n-1+L$))<span class=\"etd-inline-math\">\\\\xrightarrow{\\\\iota<sub>PZ<sub>n</sub></sub>}</span><span class=\"etd-inline-math\">PZ<sub>n</sub></span>(<span class=\"etd-inline-math\">\\\\Sigma<sub>\\</sub>\\Gamma</span>($L$))<span class=\"etd-inline-math\">\\\\xrightarrow{\\&pi;<sub>PZ<sub>n</sub></sub>}</span><span class=\"etd-inline-math\">PZ<sub>n-1</sub></span>(<span class=\"etd-inline-math\">\\\\Sigma<sub>\\</sub>\\Gamma</span>($L$))$\\\\rightarrow$1. &lt;br /&gt;&lt;br /&gt; In particular, we observe that the kernel $K$ of <span class=\"etd-inline-math\">\\iota<sub>PZ<sub>n</sub></sub></span> 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$. &lt;br /&gt;&lt;br /&gt; 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<span class=\"etd-inline-math\"><sub>n</sub><sup>id,orb</sup></span>(<span class=\"etd-inline-math\">\\\\Sigma<sub>\\</sub>\\Gamma</span>($L$)). For <span class=\"etd-inline-math\">Z<sub>n</sub></span>(<span class=\"etd-inline-math\">\\\\Sigma<sub>\\</sub>\\Gamma</span>($L$)) and the contained Artin groups, we also obtain a highly generating family of subgroups.","abstract_has_math":true,"creators":["Flechsig, Jonas"],"institution":"Universität Bielefeld","degree_name":null,"degree_level":"thesis.doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-06-05","date_published":"2023-06-05","updated_at":"2026-07-27T18:50:07Z","subjects":[],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://pub.uni-bielefeld.de/record/2979933","outbound_label":"Repository record","outbound_source":"source_url"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Flechsig, Jonas"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:publisher","label":"Institution","values":["Universitätsbibliothek Bielefeld"]},{"key":"dc:type","label":"Dc Type","values":["doctoralThesis"]},{"key":"thesis:degree_level","label":"Degree Level","values":["thesis.doctoral"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Universität Bielefeld"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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}}$$PZ_{n-1}$($\\\\Sigma_\\\\Gamma$($L$))$\\\\rightarrow$1. <br /><br /> In particular, we observe that the kernel $K$ of $\\iota_{PZ_n}$ is non-trivial. This corrects Theorem 2.14 in [42]. Using the relation pictured in the figure on the title page, we construct non-trivial elements in the kernel. Moreover, we determine $K$. For this purpose, we introduce orbifold mapping class groups (with marked points) and establish a Birman exact sequence for them. Comparing the orbifold mapping class groups with the orbifold braid groups, reveals a surprising behavior: in contrast to the classical case, the orbifold braid group is a proper quotient of the orbifold mapping class group. In particular, this yields a presentation of the pure orbifold braid group which allows us to determine the kernel $K$. <br /><br /> Furthermore, we generalize a result of Allcock [2] about Artin groups contained in orbifold braid groups and we analyze the connectivity of bipartite matching complexes of $\\Gamma$-arcs. This allows us to deduce highly generating families of subgroups in Map$_n^{id,orb}$($\\\\Sigma_\\\\Gamma$($L$)). For $Z_n$($\\\\Sigma_\\\\Gamma$($L$)) and the contained Artin groups, we also obtain a highly generating family of subgroups."]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Braid groups and mapping class groups for 2-orbifolds"]}]}],"canonical_facts":{"dc:creator":["Flechsig, Jonas"],"dc:description.abstract":["The main achievement of this thesis is that pure orbifold braid groups fit into an exact sequence 1$\\\\rightarrow$ $K$$\\\\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. <br /><br /> 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 $K$. <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."],"dc:format.medium":["application/pdf"],"dc:publisher":["Universitätsbibliothek Bielefeld"],"dc:title":["Braid groups and mapping class groups for 2-orbifolds"],"dc:type":["doctoralThesis"],"thesis:degree_level":["thesis.doctoral"],"thesis:institution_name":["Universität Bielefeld"]},"updated_at":"2026-07-27T18:50:07Z"}