Related papers: Strongly quasisymmetirc homeomorphisms being compa…

In this paper, we first generalize a result of Bishop and Steger [Representation theoretic rigidity in PSL(2, R). Acta Math., 170, (1993), 121-149] by proving that for a Fuchsian group $G$ of divergence type and non-lattice, if $h$ is a…

This article analyzes sublinearly quasisymmetric homeo-morphisms (generalized quasisymmetric mappings), and draws applications to the sublinear large-scale geometry of negatively curved groups and spaces. It is proven that those…

Suppose $\mu$ be a Beltrami coefficient on the unit disk, which is compatible with a convex co-compact Fuchsian group $G$ of the second kind. In this paper we show that if $\displaystyle\frac{|\mu|^{2}}{1-|z|^{2}}dxdy $ satisfies the…

Let $\Omega$ be a Carleson-Denjoy domain and $G$ be its covering group. Let $\mu$ be a Beltrami coefficient on the unit disk which is compatible with the group $G$. In this paper we show that if $\frac{|\mu|^{2}}{1-|z|^{2}}dxdy$ satisfies…

We show that if $f:X\to Y$ is a quasisymmetric mapping between Ahlfors regular spaces, then $\dim_H f(E)\leq\dim_H E$ for "almost every" bounded Ahlfors regular set $E\subseteq X$. If additionally, $X$ and $Y$ are Loewner spaces then…

An almost-Fuchsian group is a quasi-Fuchsian group such that the quotient hyperbolic manifold contains a closed incompressible minimal surface with principal curvatures contained in (-1,1). We show that the domain of discontinuity of an…

We extend the classical Carath\'eodory extension theorem to quasiconformal Jordan domains $( Y, d_{Y} )$. We say that a metric space $( Y, d_{Y} )$ is a quasiconformal Jordan domain if the completion $\overline{Y}$ of $( Y, d_{Y} )$ has…

A homemorphism between domains in $\mathbb R^n$, $n\ge 2$ is quasiconformal, with its intricate analytic and geometric consequences, if the (pointwise) linear dilatation -- a purely metric quantity -- is uniformly bounded. Gehring proved…

Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least 2, that $D\subset E$ and $D'\subset E'$ are domains, and that $f: D\to D'$ is a homeomorphism. In this paper, we prove that if there exists some constant $M>1$…

Uniformly quasiconformally homogeneous domains in $\mathbb{R}^n$ carry a transitive collection of $K$-quasiconformal maps for a fixed $K\geq 1.$ In this paper, we study two questions in this setting. The first is to show that…

Suppose that $X$ and $Y$ are quasiconvex and complete metric spaces, that $G\subset X$ and $G'\subset Y$ are domains, and that $f: G\to G'$ is a homeomorphism. Our main result is the following subinvariance property of the class of uniform…

For a finitely generated group $G$ and collection of subgroups $\mathcal{P}$ we prove that the relative Dehn function of a pair $(G,\mathcal{P})$ is invariant under quasi-isometry of pairs. Along the way we show quasi-isometries of pairs…

The aim of this work is to link the quasiconformal geometry of a Euclidean domain $U$ to the spectral properties of its Dirichlet integral $\D$, through the algebra of multipliers $\M(H^{1,2}(U))$ of the Sobolev space. In the main result we…

Suppose that $X$ and $Y$ are quasiconvex and complete metric spaces, that $G\subset X$ and $G'\subset Y$ are domains, and that $f: G\to G'$ is a homeomorphism. In this paper, we first give some basic properties of short arcs, and then we…

We investigate strongly symmetric homeomorphisms of the real line which appear in harmonic analysis aspects of quasiconformal Teichm\"uller theory. An element in this class can be characterized by a property that it can be extended…

We provide a general sufficient condition for extendability of quasimorphisms on subgroups. This condition recovers the result of Hull--Osin on quasimorphisms on hyperbolically embedded subgroups, and the proof given in this paper is much…

A subgroup $H\leq G$ is said to be almost normal if every conjugate of $H$ is commensurable to $H$. If $H$ is almost normal, there is a well-defined quotient space $G/H$. We show that if a group $G$ has type $F_{n+1}$ and contains an almost…

We prove that the hitting measure is singular with respect to Lebesgue measure for any random walk on a cocompact Fuchsian group generated by translations joining opposite sides of a symmetric hyperbolic polygon. Moreover, the Hausdorff…

In this paper, we prove that if a homeomorphism is quasisymmetric relative to the boundary of the domain then it maps a length John domain to a diameter John domain. Moreover, we prove a necessary and sufficient condition for a diameter…

We continue investigations started by Lakeland on Fuchsian and Kleinian groups which have a Dirichlet fundamental domain that also is a Ford domain in the upper half-space model of hyperbolic $2$- and $3$-space, or which have a Dirichlet…