Related papers: The Metric Completion of Outer Space

We introduce the notion of (almost isometric) local retracts in metric space as a natural non-linear version of the concepts of locally complemented and almost isometric ideals from Banach spaces. We prove that given two metric spaces…

Let $\mathrm{Sym}_q(m)$ be the space of symmetric matrices in $\mathbb{F}_q^{m\times m}$. A subspace of $\mathrm{Sym}_q(m)$ equipped with the rank distance is called a symmetric rank-metric code. In this paper we study the covering…

In this work, a metric is presented on the set of boundedly-compact pointed metric spaces that generates the Gromov-Hausdorff topology. A similar metric is defined for measured metric spaces that generates the Gromov-Hausdorff-Prokhorov…

For metric spaces, the doubling property, the uniform disconnectedness, and the uniform perfectness are known as quasi-symmetric invariant properties. The David-Semmes uniformization theorem states that if a compact metric space satisfies…

We define $\partial$-biLipschitz homeomorphisms between uniform metric spaces and show that these maps are always quasim\"obius. We also show that a homeomorphism being $\partial$-biLipschitz is equivalent to the map biLipschitz in the…

We solve the following three questions concerning surjective linear isometries between spaces of Lipschitz functions $\mathrm{Lip}(X,E)$ and $\mathrm{Lip}(Y,F)$, for strictly convex normed spaces $E$ and $F$ and metric spaces $X$ and $Y$:…

We show the existence of Lipschitz-free spaces verifying the Point of Continuity Property with arbitrarily high weak-fragmentability index. For this purpose, we use a generalized construction of the countably branching diamond graphs. As a…

We investigate the space of Hermitian metrics on a fixed complex vector bundle. This infinite-dimensional space has appeared in the study of Hermitian-Einstein structures, where a special L2-type Riemannian metric is introduced. We compute…

We study the limits of sequences of spheres and complex projective spaces with unbounded dimensions. A sequence of spheres (resp. complex projective spaces) either is a Levy family, infinitely dissipates, or converges to (resp. the Hopf…

We study Lipschitz-free spaces over compact and uniformly discrete metric spaces enjoying certain high regularity properties - having group structure with left-invariant metric. Using methods of harmonic analysis we show that, given a…

A labeled metric space is intuitively speaking a metric space together with a special set of points to be understood as the geometric boundary of the space. We study basic properties of a recently introduced labeled Gromov-Hausdorff…

Chapter 1 is a short history of non-Euclidean geometry, which synthesises my readings of mostly secondary sources. Chapter 2 presents each of the main models of hyperbolic geometry, and describes the tesselation of the upper half-plane…

We study metric spheres Z obtained by gluing two hemispheres of the Euclidean sphere along an orientation-preserving homeomorphism mapping the equator onto itself, where the distance on Z is the canonical distance that is locally isometric…

A metric space $X$ is {\em injective} if every non-expanding map $f:B\to X$ defined on a subspace $B$ of a metric space $A$ can be extended to a non-expanding map $\bar f:A\to X$. We prove that a metric space $X$ is a Lipschitz image of an…

We prove that any length metric space homeomorphic to a 2-manifold with boundary, also called a length surface, is the Gromov-Hausdorff limit of polyhedral surfaces with controlled geometry. As an application, using the classical…

In this manuscript, we claim that the newly introduced $\mathcal{F}$-metric space \cite[\, M.~Jleli and B.~Samet, On a new generalization of metric spaces, J. Fixed Point Theory Appl, 20(3) 2018]{JS1} is metrizable. Also, we deduce that the…

We discuss the asymptotic symmetry algebra of the Schrodinger-invariant metrics in d+3 dimensions and its realization on finite temperature solutions of gravity coupled to matter fields. These solutions have been proposed as gravity…

We define metrics in space that are natural counterparts of the hyperbolic metric in plane domains, using the characterization of the hyperbolic metric due to Beardon and Pommerenke. We obtain inequalities for these metrics under…

Fix an integer $r\geq 3$. We consider metric spaces on $n$ points such that the distance between any two points lies in $\{1,..., r\}$. Our main result describes their approximate structure for large $n$. As a consequence, we show that the…

Geometric characteristics of metric spaces that appear in formulas of the Gromov--Hausdorff distances from these spaces to so-called simplexes, i.e., to the metric spaces, all whose non-zero distances are the same are studied. The…