Isotopic tiling theory for hyperbolic surfaces (Q2025653)

An equivariant tiling is a pair consisting of a tiling and its (maybe not full) symmetry group. A Delaney-Dress symbol is a certain colored graph together with some functions that encodes a tiling equivariant type. The Delaney-Dress tiling theory allows to obtain all possible equivariant types of tile-\(k\)-transitive (for any reasonable \(k \in \mathbb N\)) tilings by disks of all the three two-dimensional spaces of constant curvature: Euclidean plane, sphere and hyperbolic plane. Some algorithms were developed and implemented on computer by D.~H.~Huson and O.~Delgado Friedrichs. The paper under review generalizes the Delaney-Dress tiling theory to classify equivariant tilings of any hyperbolic surface of finite genus, possibly non-orientable, with boundary, and punctured (with cusps). The notion of orbifold and mapping class groups are essentially used. The Conway notations for orbifolds are extended by hyperbolic transformations corresponding to non-mirror boundary components and parabolic transformations corresponding to punctures. In addition to non-Euclidean crystallographic (NEC) groups by A.~M.~Macbeath the authors consider groups which have unbounded fundamental domain of finite area. Mapping class groups are extended to orbifolds. The results lay basis for enumerating isotopically distinct tilings of hyperbolic surfaces. Remark that in the study of tilings with transitivity properties the authors use different terminology. In the classical monograph on tilings of the Euclidean plane, B.~Grünbaum and G.~C.~Shephard say of homeomeric tilings. E.~Zamorzaeva uses the concept of Delone class when enumerating isohedral tilings of the hyperbolic plane for translation group of genus two as well as when enumerating isohedral tilings on some Riemann surfaces of genus two. Both terms mean the same as equivariantly equivalent tilings.