Embeddings of trees in the hyperbolic disk
Embeddings of trees in the hyperbolic disk
复制标题
双曲盘中树的嵌入
DOI：
10.1080/17476939408814723
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发表时间：
1994
期刊：
影响因子：
--
通讯作者：
Flavia Colonna
中科院分区：
文献类型：
--
作者：
Joel M. Cohen;Flavia Colonna
我们考虑偶数度齐次树（即自由群的凯莱图）可以嵌入双曲盘中的条件，使得由群元素的旋转和平移引起的自同构可以由盘的自同构来表示。我们专注于研究特定的最佳嵌入。我们表明，在这种情况下，磁盘上的有界解析函数是由它们在嵌入树的顶点处的值确定的。使用 的构造，我们可以定义齐次树上的调和性，使得盘上的有界函数限制为树上的调和函数，当且仅当它们是调和的时。我们还展示了这棵树的简单而优雅的结构。
We consider the conditions under which a homogeneous tree of even degree (i.e. the Cayley graph of a free group) may be embedded in the hyperbolic disk in such a way that the automorphisms induced by rotation and by translation by group elements may be represented by automorphisms of the disk. We concentrate on the study of a particular optimal such embedding. We show that in this case bounded analytic functions on the disk are determined by their values at the vertices of the embedded tree. Using a construction of we can define harmonicity on a homogeneous tree in such a way that bounded functions on the disk restrict to harmonic functions on the tree if and only if they are harmonic. We also show a simple and elegant construction of this tree.