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.
We describe a method to classify crystallographic tilings of the Euclidean and hyperbolic planes by tiles whose stabiliser group contains translation isometries or whose topology is not that of a closed disk. We tackle this problem from two different viewpoints, one with constructive techniques to enumerate such tilings and the other from a viewpoint of classification. The methods are purely topological and generalise Delaney-Dress combinatorial tiling theory. The classification is up to equivariant equivalence and is achieved by viewing tilings as decorations of orbifolds.
We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This provides a generalization of the enumeration of Delaney-Dress combinatorial tiling theory on the basis of isotopic tiling theory. To accomplish this, we derive representations of the mapping class group of the orbifold associated to the symmetry group in the group of outer automorphisms of the symmetry group of a tiling. We explicitly give descriptions of certain subgroups of mapping class groups and of tilings as decorations on orbifolds, namely those that are commensurate with the Primitive, Diamond and Gyroid triply-periodic minimal surfaces. We use this explicit description to give an array of examples of isotopically distinct tilings of the hyperbolic plane with symmetries generated by rotations, outlining how the approach yields an unambiguous enumeration
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