Related papers: Some isometry groups of Urysohn space
WWe define the notion of a random metric space and prove that with probability one such a space is isometricto the Urysohn universal metric space. The main technique is the study of universal and random distance matrices; we relate the…
We view ultrametric spaces as two-sorted structures consisting of a set of points and of a linearly ordered set of distances. We call the appropriate notion of embeddings distance-carrying (dc for short). Those are obtained by combining…
In this thesis, we study the existence of universal objets of two differents types in the theory of topological groups and theirs actions on compacts spaces. In the first part, we contribute to the problem of existence of test spaces for…
We investigate the structure of subspaces of a Hilbert space that are invariant under unitary representations of a discrete group. We work with square integrable representations, and we show that they are those for which we can construct an…
In a paper published posthumously, P.S. Urysohn constructed a complete, separable metric space that contains an isometric copy of every complete separable metric space, nowadays referred to as the Urysohn universal space. Here we study…
Given a locally compact Polish space X, a necessary and sufficient condition for a group G of homeomorphisms of X to be the full isometry group of (X,d) for some proper metric d on X is given. It is shown that every locally compact Polish…
We compute the full isometry group of any left invariant metric on a simply connected, non-unimodular Lie group of dimension three. As an application, we determine the index of symmetry of such metrics and prove that the singularities of…
We show that infinite cyclic subgroups of groups acting uniformly properly on injective metric spaces are uniformly undistorted. In the special case of hierarchically hyperbolic groups, we use this to study translation lengths for actions…
Answering a question of Gao and Kechris, we show that, given any polish group G, there exists a closed subset F of Urysohn's universal metric space U such that G is (topologically) isomorphic to the subgroup of isometries of U which map F…
By a recent result of Juh\'{a}sz and van Mill, a locally compact topological group whose dense subspaces are all separable is metrizable. In this note we investigate the following question: is every locally compact group having all dense…
Prompted by a recent question of G. Hjorth as to whether a bounded Urysohn space is indivisible, that is to say has the property that any partition into finitely many pieces has one piece which contains an isometric copy of the space, we…
A thorough classification of the topologies of compact homogeneous universes is given except for the hyperbolic spaces, and their global degrees of freedom are completely worked out. To obtain compact universes, spatial points are…
Recently, much interest was devoted to the Urysohn universal metric space U and its isometries; this paper is a contribution to this field of research. In particular, we study some properties of isometries of U, and prove the following…
We prove that for any compact manifold of dimension greater than $1$, the set of pseudo-Riemannian metrics having a trivial isometry group contains an open and dense subset of the space of metrics.
Given a finitely generated group $\Gamma$, we study the space ${\rm Isom}(\Gamma,{\mathbb Q\mathbb U})$ of all actions of $\Gamma$ by isometries of the rational Urysohn metric space ${\mathbb Q\mathbb U}$, where ${\rm Isom}(\Gamma,{\mathbb…
For an Alexandrov space (with curvature bounded below), we determine the maximal dimension of its isometry group and show that the space is isometric to a Riemannian manifold, provided the dimension of its isometry group is maximal. We also…
We introduce an universum of the Polish (=complete separable metric) space - the convex cone of distance matrices and study its geometry. It happened that the generic Polish spaces in this sense of this universum is so called Urysohn spaces…
We prove that for any constant $K>0$ there exists a separable group equipped with a complete bi-invariant metric bounded by $K$, isometric to the Urysohn sphere of diameter $K$, that is of `almost-universal disposition'. It is thus an…
We classify all group topologies coarser than the topology of stabilizers of finite sets in the case of automorphism groups of countable free-homogeneous structures, Urysohn space and Urysohn sphere, among other related results.
A metric space is said to be all-set-homogeneous if any of its partial isometries can be extended to a genuine isometry. We give a classification of a certain subclass of all-set-homogeneous length spaces.