A Hyperbolic Filling for Ultrametric Spaces

作者：Ibragimov Zair^*

来源：Computational Methods and Function Theory, 2014, 14(2-3): 315-329.

DOI：10.1007/s40315-014-0050-6

• Ultrametric space
• Hyperbolic filling
• Mobius maps

By a hyperbolic filling of an ultrametric space we mean a Gromov -hyperbolic space whose boundary at infinity can be identified with the space via a Mobius map. It is well known that the boundary at infinity of a Gromov -hyperbolic space, equipped with a canonical visual metric, is a complete bounded ultrametric space, and that the isometries at infinity between Gromov -hyperbolic spaces extend to Mobius 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 , we introduce a metric on the collection of all non-degenerate balls in . We show that the space is Gromov -hyperbolic and that its boundary at infinity, equipped with a canonical visual metric, can be identified with the metric completion of via a Mobius map and, in the bounded case, via a similarity.

• 出版日期2014-10