标题： 法雷藓 显示英文标题

标题： Farey Bryophylla

摘要： 双曲平面上法雷镶嵌的构造始于一个理想三角形的对称性有限生成群，即 一个所有顶点都在边界上的三角形。 它在双曲平面的边界上产生了一个显著的分形结构，通过与镶嵌结构相关的连分数编码每个元素。 将这种构造推广到高维双曲空间的问题已经悬而未决多年。 在本文中，我们迈出了在三维情况下进行推广的第一步。 我们引入了共形苔藓，即双曲3空间边界的一类子集，它们具有类似于法雷镶嵌的分形特性。 我们对所有共形苔藓进行了分类，并研究了它们极限集的性质。 显示英文摘要

摘要： The construction of the Farey tessellation in the hyperbolic plane starts with a finitely generated group of symmetries of an ideal triangle, i.e. a triangle with all vertices on the boundary. It induces a remarkable fractal structure on the boundary of the hyperbolic plane, encoding every element by the continued fraction related to the structure of the tessellation. The problem of finding a generalisation of this construction to the higher dimensional hyperbolic spaces has remained open for many years. In this paper we make the first steps towards a generalisation in the three-dimensional case. We introduce conformal bryophylla, a class of subsets of the boundary of the hyperbolic 3-space which possess fractal properties similar to the Farey tessellation. We classify all conformal bryophylla and study the properties of their limiting sets.