Isometric embeddings of Polish ultrametric spaces (Q714745)

Given a countable set \(R \subseteq \mathbb{R}^{>0}\), let \(\mathcal{U}_R\) be the collection of all Polish ultrametric spaces with nonzero distances contained in \(R\), and let \(X_R \in \mathcal{U}_R\) be the Polish ultrametric Urysohn space for such a class, i.e.\ the unique (up to isometry) Polish ultrametric space which is ultrahomogeneous and universal for elements of \(\mathcal{U}_R\). In [\textit{S. Gao} and \textit{C. Shao}, Topology Appl. 158, No. 3, 492--508 (2011; Zbl 1220.54012)] it was shown that every isometric embedding \(e : X \to X_R\) of a compact \(X \in \mathcal{U}_R\) into \(X_R\) is \textit{extensive}, i.e.\ it has the property that there is a topological group embedding \(\Phi : \mathrm{Iso}(X) \to \mathrm{Iso}(X_R)\) between the isometry groups of the two spaces such that \(\Phi(\phi) \restriction e[X] = e \circ \phi \circ e^{-1}\) for every \(\phi \in \mathrm{Iso}(X)\). In this paper, the author generalizes the mentioned Gao-Shao result by giving a full characterization of the collection of those \(X \in \mathcal{U}_R\) such that every isometric embedding \(e : X \to X_R\) is extensive: a space \(X \in \mathcal{U}_R\) belongs to such a collection if and only if for every `closed' ball \(b\) in \(X\), either \(b\) is set-wise fixed by every \(\phi \in \mathrm{Iso}(X)\), or else it is spherically complete (i.e.\ \(\bigcap_n b_n \neq \emptyset\) for every decreasing sequence \(\{ b_n \}_{n \in \mathbb{N}}\) of `closed' balls in \(b\)) and it does not contain an infinite polygon (i.e.\ an infinite set \(P\) such that \(d_X(x,y) = d_X(y,z)\) for all distinct \(x,y,z \in P\)).