A Hyperbolic Filling for Ultrametric Spaces
A Hyperbolic Filling for Ultrametric Spaces
复制标题
超度量空间的双曲填充
DOI：
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发表时间：
2014
期刊：
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通讯作者：
Z. Ibragimov
中科院分区：
文献类型：
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作者：
Z. Ibragimov
所谓超度量空间的双曲填充，我们指的是一个格罗莫夫0000-双曲空间，它的无穷远处的边界可以通过莫比乌斯映射与该空间等同。众所周知，一个配备了标准视觉度量的Gromov $$0$$0-双曲空间在无穷远处的边界是一个完备的有界超度量空间，并且Gromov $$0$$0-双曲空间之间在无穷远处的等距扩展到它们在无穷远处边界之间的莫比乌斯映射。本文构造了完备超度量空间的一个标准双曲填充。更精确地说，给定这样一个空间$$X $$X，我们在$$X $$X中所有非退化球的集合$$mathcal B（X）$$B（X）上引入度量$$h_mathcal B $$h B。我们证明了空间$$（mathcal B（X），d_mathcal B）$$（B（X），dB）是Gromov $$0$0-双曲的，并且它在无穷远处的边界，配备了一个规范的视觉度量，可以通过一个莫比乌斯映射和在有界的情况下，通过一个相似性用$$X $$X的度量完备化来标识.
By a hyperbolic filling of an ultrametric space we mean a Gromov $$0$$0-hyperbolic space whose boundary at infinity can be identified with the space via a Möbius map. It is well known that the boundary at infinity of a Gromov $$0$$0-hyperbolic space, equipped with a canonical visual metric, is a complete bounded ultrametric space, and that the isometries at infinity between Gromov $$0$$0-hyperbolic spaces extend to Möbius maps between their boundaries at infinity. In this paper we construct a canonical hyperbolic filling for perfect ultrametric spaces. More precisely, given such a space $$X$$X, we introduce a metric $$h_mathcal B$$hB on the collection $$mathcal B(X)$$B(X) of all non-degenerate balls in $$X$$X. We show that the space $$(mathcal B(X), d_mathcal B)$$(B(X),dB) is Gromov $$0$$0-hyperbolic and that its boundary at infinity, equipped with a canonical visual metric, can be identified with the metric completion of $$X$$X via a Möbius map and, in the bounded case, via a similarity.