On homogeneous ultrametric spaces

TL;DR

This paper characterizes homogeneous ultrametric spaces, showing that isometries on singletons extend to the whole space, and explores their properties, constructions, and embeddability into universal homogeneous ultrametric spaces.

Contribution

It proves that ultrametric spaces are homogeneous if isometries on points extend, and constructs universal and countable homogeneous ultrametric spaces with various properties.

Findings

The group of isometries of an ultrametric space has arity at most 2.

The Cauchy completion of a homogeneous ultrametric space is homogeneous.

Every ultrametric space embeds into a homogeneous ultrametric space with the same set of distances.

Abstract

A metric space M is homogeneous if every isometry between finite subsets extends to a surjective isometry defined on the whole space. We show that if M is an ultrametric space, it suffices that isometries defined on singletons extend, i.e that the group of isometries of M acts transitively. We derive this fact from a result expressing that the arity of the group of isometries of an ultrametric space is at most 2. An illustration of this result with the notion of spectral homogeneity is given. With this, we show that the Cauchy completion of a homogeneous ultrametric space is homogeneous. We present several constructions of homogeneous ultrametric spaces, particularly the countable homogeneous ultrametric space, universal for rational distances, and its Cauchy completion. From a general embeddability result, we prove that every ultrametric space is embeddable into a homogeneous ultrametric space with the same set of distances values and we also derive three embeddability results due respectively to F. Delon, A. Lemin and V. Lemin, and V. Fevinberg. Looking at ultrametric spaces as 2-structures, we observe that the nerve of an ultrametric space is the tree of its robust modules.