Isotopic tiling theory for hyperbolic surfaces
- In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to orbifolds, in turn using this to study tilings of covering spaces of orbifolds. Moreover, we study finite subgroups of these mapping class groups. Our results can be used to extend the Delaney-Dress combinatorial encoding of a tiling to yield a finite symbol encoding the complexity of an isotopy class of tilings. The results of this paper provide the basis for a complete and unambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces.
| Author details: | Benedikt Maximilian KolbeGND, Myfanwy E. EvansORCiD |
|---|---|
| DOI: | https://doi.org/10.1007/s10711-020-00554-2 |
| ISSN: | 0046-5755 |
| ISSN: | 1572-9168 |
| Title of parent work (English): | Geometriae dedicata |
| Publisher: | Springer |
| Place of publishing: | Dordrecht |
| Publication type: | Article |
| Language: | English |
| Date of first publication: | 2020/07/25 |
| Publication year: | 2020 |
| Release date: | 2023/03/22 |
| Tag: | delaney-dress tiling theory; groups; hyperbolic tilings; isotopic tiling theory; mapping class; maps on surfaces; orbifolds |
| Volume: | 212 |
| Issue: | 1 |
| Number of pages: | 28 |
| First page: | 177 |
| Last Page: | 204 |
| Funding institution: | Projekt DEAL - Emmy Noether Programme of the Deutsche; ForschungsgemeinschaftGerman Research Foundation (DFG); Deutscher; Akademischer AustauschdienstDeutscher Akademischer Austausch Dienst; (DAAD) |
| Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
| DDC classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
| Peer review: | Referiert |
| Grantor: | Publikationsfonds der Universität Potsdam |
| Publishing method: | Open Access / Hybrid Open-Access |
| License (German): |  CC-BY - Namensnennung 4.0 International |
| External remark: | Zweitveröffentlichung in der Schriftenreihe Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe ; 1347 |
