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/lp/springer-journals/a-hyperbolic-filling-for-ultrametric-spaces-DkQ6OVRNtG
- Publisher
- Springer Journals
- Copyright
- Copyright © 2014 by Springer-Verlag Berlin Heidelberg
- 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 Theory – Springer Journals
Published: Feb 19, 2014