Using classical results of infinite-dimensional geometry, we show that the isometry group of the Urysohn space, endowed with its usual Polish group topology, is homeomorphic to the separable Hilbert space. The proof is basedon a lemma about extensions of metric spaces by finite metric spaces, which wealso use to investigate (answering a question of I. Goldbring) the relationship, when A,B are finite subsets of the Urysohn space, between the group of isometries fixing A pointwise, the group of isometries fixing B pointwise, and the group of isometries fixing the intersection of A and B pointwise.
The Urysohn space is a separable complete metric space with two fundamental properties: (a) universality: every separable metric space can be isometrically embedded in it; (b) ultrahomogeneity: every finite isometry between two finite subspaces can be extended to an auto-isometry of the whole space. The Urysohn space is uniquely determined up to isometry within separable metric spaces by these two properties. We introduce an analogue of the Urysohn space for diversities, a recently developed variant of the concept of a metric space. In a diversity any finite set of points is assigned a non-neg…
In this paper, we give new constructions of Urysohn universal ultrametric spaces. We first characterize a Urysohn universal ultrametric subspace of the space of all continuous functions whose images contain the zero, from a zero-dimensional compact Hausdorff space without isolated points into the space of non-negative real numbers equipped with the nearly discrete topology. As a consequence, the whole function space is Urysohn universal, which can be considered as a non-Archimedean analog of Banach--Mazur theorem. As a more application, we prove that the space of all continuous pseudo-ultramet…
In this paper we define a notion of S-extension for a metric space and study minimality and coherence of S-extensions. We show that every S-extension can be identified with an algebraic object. We use this algebraic representation to give a complete characterization of all finite minimal S-extensions of a given finite metric space and a complete characterization of all minimal coherent S-extensions. We also define a notion of ultraextensive metric spaces and show that every countable metric space can be extended to a countable ultraextensive metric space. %As an application, we show that every…
We first prove a version of Tietze-Urysohn's theorem for proper functions taking values in non-negative real numbers defined on $σ$-compact locally compact Hausdorff spaces. As its application, we prove an extension theorem of proper metrics, which states that if $X$ is a $σ$-compact locally compact space, $A$ is a closed subset of $X$, and $d$ is a proper metric on $A$ that generates the same topology of $A$, then there exists a proper metric on $X$ such that $D$ generates the same topology of $X$ and $D|_{A^{2}}=d$. Moreover, if $A$ is a proper retraction, we can choose $D$ so that $(A, d)$ …
The 'contracting boundary' of a proper geodesic metric space consists of equivalence classes of geodesic rays that behave like rays in a hyperbolic space. We introduce a geometrically relevant, quasi-isometry invariant topology on the contracting boundary. When the space is the Cayley graph of a finitely generated group we show that our new topology is metrizable.
The course was given at Peking University, Fall 2019. We discuss the following subjects: (1) Introduction to general topology, hyperspaces, metric and pseudometric spaces, graph theory. (2) Graphs in metric spaces, minimum spanning tree, Steiner minimal tree, Gromov minimal filling. (3) Hausdorff distance, Vietoris topology, Limits theory, inheritance of completeness, total boundedness, compactness by hyperspaces. (4) Gromov-Hausdorff distance, triangle inequality, positive definiteness for isometry classes of compact spaces, counterexample for boundedly compact spaces. (5) Gromov-Hausdorff di…
We consider Lie groups equipped with arbitrary distances. We only assume that the distance is left-invariant and induces the manifold topology. For brevity, we call such object metric Lie groups. Apart from Riemannian Lie groups, distinguished examples are sub-Riemannian Lie groups and, in particular, Carnot groups equipped with Carnot-Carathéodory distances. We study the regularity of isometries, i.e., distance-preserving homeomorphisms. Our first result is the analyticity of such maps between metric Lie groups. The second result is that if two metric Lie groups are connected and nilpotent th…
