A Hyperbolic Filling for Ultrametric Spaces

Abstract

By a hyperbolic filling of an ultrametric space we mean a Gromov $$0$$ -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 $$0$$ -hyperbolic space, equipped with a canonical visual metric, is a complete bounded ultrametric space, and that the isometries at infinity between Gromov $$0$$ -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 $$X$$ , we introduce a metric $$h_\mathcal B$$ on the collection $$\mathcal B(X)$$ of all non-degenerate balls in $$X$$ . We show that the space $$(\mathcal B(X), d_\mathcal B)$$ is Gromov $$0$$ -hyperbolic and that its boundary at infinity, equipped with a canonical visual metric, can be identified with the metric completion of $$X$$ via a Mobius map and, in the bounded case, via a similarity.