发表机构
Hong Kong University of Science and Technology（香港科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究闭曲面由全等最弯曲多边形构成的铺砌，通过平面图和顶点类型建立存在性条件，刻画最小铺砌，并给出双曲情形下的极大无挠子群。
AI 中文摘要
我们研究闭曲面由全等最弯曲多边形构成的铺砌。通过将此类铺砌与由平面图和顶点类型确定的分支覆盖联系起来，我们建立了给定闭拓扑曲面承认常曲率度量并支持具有指定组合数据和瓦片数量的铺砌的充分必要条件。我们还刻画了最小铺砌——即不通过非平凡无分支覆盖下降为更少瓦片的铺砌——用相关置换的公共块分解来表征。显式置换构造产生了具有指定顶点类型的无限族最小铺砌，即使平面图和顶点类型固定时也是如此。在双曲情形下，这些结果还给出了通用铺砌对称群的有限指数子群，这些子群在无挠子群中是极大的，指数等于瓦片数量。
英文摘要
We study tilings of closed surfaces by congruent most curvilinear polygons. By relating such tilings to branched coverings determined by planar diagrams and vertex types, we establish necessary and sufficient conditions for a given closed topological surface to admit a metric of constant curvature supporting a tiling with prescribed combinatorial data and number of tiles. We also characterize minimal tilings, which do not descend through nontrivial unbranched coverings to tilings with fewer tiles, in terms of common block decompositions of the associated permutations. Explicit permutation constructions yield infinite families of minimal tilings with prescribed vertex types, even when the planar diagram and vertex types are fixed. In the hyperbolic setting, these results also give finite-index subgroups of the universal tiling symmetry groups that are maximal among torsion-free subgroups, with the index equal to the number of tiles.
Comments38 pages, 13 figures
