# A Hyperbolic Filling for Ultrametric Spaces

A Hyperbolic Filling for Ultrametric Spaces 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 $$h_\mathcal B$$ h B 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 ) , d B ) 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

# A Hyperbolic Filling for Ultrametric Spaces

, Volume 14 (3) – Feb 19, 2014
15 pages

/lp/springer-journals/a-hyperbolic-filling-for-ultrametric-spaces-DkQ6OVRNtG
Publisher
Springer Journals
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-014-0050-6
Publisher site
See Article on Publisher Site

### Abstract

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 $$h_\mathcal B$$ h B 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 ) , d B ) 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.

### Journal

Computational Methods and Function TheorySpringer Journals

Published: Feb 19, 2014

### References

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