Abstract
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.
Highlights
The enumerative approaches of Delaney–Dress tiling theory [16] in the two-dimensional hyperbolic plane have facilitated a novel investigation of three-dimensional Euclidean networks, where hyperbolic tilings of triply-periodic minimal surfaces (TPMS) are used for an enumeration of crystallographic nets in R3 [9,40,51,57,61,62]
The connection of isotopic tiling theory and mapping class groups is novel, there is a well-known connection between the Teichmüller space of Riemann surfaces of genus g and certain tilings of the hyperbolic plane with 4g sided geodesical polygons that we will use as inspiration [23]
We introduce the mapping class group (MCG) of orbifolds and prove fundamental results facilitating its applications to tiling theory
Summary
The enumerative approaches of Delaney–Dress tiling theory [16] in the two-dimensional hyperbolic plane have facilitated a novel investigation of three-dimensional Euclidean networks, where hyperbolic tilings of triply-periodic minimal surfaces (TPMS) are used for an enumeration of crystallographic nets in R3 [9,40,51,57,61,62]. Computer implementations of algorithms based on Delaney–Dress tiling theory can exhaustively enumerate the combinatorial types of equivariant tilings in connected spaces of constant sectional curvature [32]. This gives us a description of all combinatorially distinct tilings of an orbifold. The connection of isotopic tiling theory and mapping class groups is novel, there is a well-known connection between the Teichmüller space of Riemann surfaces of genus g and certain tilings of the hyperbolic plane with 4g sided geodesical polygons that we will use as inspiration [23]. We make an effort to make the results more accessible by explaining the intuition behind the main ideas
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