Related papers: Quasihyperbolic metric and M\"obius transformation…
Let $D$ be a nonempty open set in a metric space $(X,d)$ with $\partial D\neq \emptyset$. Define \begin{equation*} h_{D,c}(x,y)=\log\left(1+c\frac{d(x,y)}{\sqrt{d_D(x)d_D(y)}}\right), \end{equation*} where $d_D(x)=d(x,\partial D)$ is the…
Due to the invariance properties of cross-ratio, M\"obius transformations are often used to map a set of points or other geometric object into a symmetric position to simplify a problem studied. However, when the points are mapped under a…
In this paper, we introduce a new metric $\tilde{c}$ which is associated with the domain boundary for a Ptolemy space $(X,d)$. Moreover, we study the inclusion relation of the $\tilde{c}$ metric balls and some related hyperbolic type metric…
We introduce a new intrinsic metric in subdomains of a metric space and give upper and lower bounds for it in terms of well-known metrics. We also prove distortion results for this metric under quasiregular maps.
The purpose of this paper is to study the notion of the quasihyperbolic volume and to find growth estimates for the quasihyperbolic volume of balls in a domain $G \subset {\mathbb R}^n\,$ in terms of the radii.
The regular fractional transformations of the extended quaternionic space have been recently introduced as variants of the classical linear fractional transformations. These variants have the advantage of being included in the class of…
We introduce a new quasi-isometry invariant of metric spaces called the hyperbolic dimension, hypdim, which is a version of the Gromov's asymptotic dimension, asdim. The hyperbolic dimension is at most the asymptotic dimension, however,…
We survey the recent developments in the theory of quasireg- ular mappings in metric spaces. In particular, we study the geometric porosity of the branch set of quasiregular mappings in general metric measure spaces, and then, introduce the…
Let G be a group which is hyperbolic relative to a collection of subgroups A, and it is also hyperbolic relative to a collection of subgroups B. Suppose that the collection A contains B. We characterize, for subgroups of G, when…
We investigate the relationship between quasisymmetric and convergence groups acting on the circle. We show that the M\"obius transformations of the circle form a maximal convergence group. This completes the characterization of the…
In this note, we present examples of non-quasi-geodesic metric spaces which are hyperbolic (i.e., satisfying the Gromov's $4$-point condition) while the intersection of any two metric balls therein does not either "look like" a ball or has…
We develop the foundations of the theory of quasi-visual approximations of bounded metric spaces. Roughly speaking, these are sequences of covers of a given space for which the diameters of the sets in the covers shrink to zero and for…
We study local convexity properties of the triangular ratio metric balls in proper subdomains of the real coordinate space. We also study inclusion properties of the visual angle metric balls and related hyperbolic type metric balls in the…
We introduce a relaxed version of the metric definition of quasiconformality that is natural also for mappings of low regularity, including $W_{\mathrm{loc}}^{1,1}(\mathbb{R}^n;\mathbb{R}^n)$-mappings. Then we show on the plane that this…
Let $M$ be a compact oriented 3-manifold with non-empty boundary consisting of surfaces of genii $>1$ such that the interior of $M$ is hyperbolizable. We show that for each spherical cone-metric $d$ on $\partial M$ such that all cone-angles…
We survey several methods of extending quasisymmetric homeomorphisms of the real line to bi-Lipschitz diffeomorphisms of the upper half-plane with respect to the hyperbolic metric.
Suppose n>2, let M,M' be n-dimensional connected complete finite-volume hyperbolic manifolds with non-empty geodesic boundary, and suppose that the fundamental group of M is quasi-isometric to the fundamental group of M' (with respect to…
For a domain $G$ in the one-point compactification $\overline{\mathbb{R}}^n = \mathbb{R}^n \cup \{ \infty\}$ of $\mathbb{R}^n, n \ge 2$, we characterize the completeness of the modulus metric $\mu_G$ in terms of a potential-theoretic…
We give several characterizations of parabolic (quasisuper)- minimizers in a metric measure space equipped with a doubling measure and supporting a Poincar\'e inequality. We also prove a version of comparison principle for super- and…
In this paper, we introduce the concept of quasihyperbolically visible spaces. As a tool, we study the connection between the Gromov boundary and the metric boundary.