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Isotopic tiling theory for hyperbolic surfaces

Benedikt Kolbe 1 Myfanwy Evans 2
1 GAMBLE - Geometric Algorithms and Models Beyond the Linear and Euclidean realm
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonori-entable, 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 un-ambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces.
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Submitted on : Tuesday, October 27, 2020 - 5:23:46 PM
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Benedikt Kolbe, Myfanwy Evans. Isotopic tiling theory for hyperbolic surfaces. Geometriae Dedicata, Springer Verlag, 2020, ⟨10.1007/s10711-020-00554-2⟩. ⟨hal-02981001⟩

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