Related papers: Quasiconformal maps on non-rigid Carnot groups

We show that if $M$ is a sub-Riemannian manifold and $N$ is a Carnot group such that the nilpotentization of $M$ at almost every point is isomorphic to $N$, then there are subsets of $N$ of positive measure that embed into $M$ by…

Let $f : G \to H$ be a Lipschitz map between two Carnot groups. We show that if $B$ is ball of $G$, then there exists a subset $Z \subset B$, whose image in $H$ under $f$ has small Hausdorff content, such that $B \backslash Z$ can be…

The paper deals with quasigroups having a trivial group of automorphisms and a trivial group of autotopisms. Examples of such quasigroups and methods of their verification are given.

We demonstrate the existence of quasiconformal mappings on closed manifolds that cannot be decomposed as a composition of mappings with arbitrarily small conformal distortion.

We provide a new geometric proof of Reimann's theorem characterizing quasiconformal mappings as the ones preserving functions of bounded mean oscillation. While our proof is new already in the Euclidean spaces, it is applicable in…

This is the first in a series of papers on geometric mapping theory in Carnot groups -- and more generally equiregular manifolds -- in which we prove a number of new structural results for Sobolev (in particular quasisymmetric) mappings,…

We consider the existence problem of lift F of a map f between Carnot group with different smoothness, where we use central extension to define lifting. Our main result is the existence of the contact lifts of Lipschitz and Sobolev maps and…

The conformal deformations are contained in two classes of mappings: quasiconformal and harmonic mappings. In this paper we consider the intersection of these classes. We show that, every $K$ quasiconformal harmonic mapping between…

Quasiconformal maps are homeomorphisms with useful local distortion inequalities; infinitesimally, they map balls to ellipsoids with bounded eccentricity. This leads to a number of useful regularity properties, including quantitative…

In this paper we prove some approximation results for biLipschitz maps in the Heisenberg group. Namely, we show that a biLipschitz map with biLipschitz constant close to one can be pointwise approximated, quantitatively on any fixed ball,…

We study the mean radius growth function for quasiconformal mappings. We give a new sub-class of quasiconformal mappings in $\mathbb{R}^n$, for $n\geq 2$, called bounded integrable parameterization mappings, or BIP maps for short. These…

Two groups are virtually isomorphic if they can be obtained one from the other via a finite number of steps, where each step consists in taking a finite extension or a finite index subgroup (or viceversa). Virtually isomorphic groups are…

The triangular ratio metric is studied in subdomains of the complex plane and Euclidean $n$-space. Various inequalities are proven for it. The main results deal with the behavior of this metric under quasiconformal maps. We also study the…

We give an example of a totally disconnected set E in R^3 which is not removable for quasiconformal homeomorphisms, i.e., there is a homeomorphism f of R^3 to itself which is quasiconformal off E, but not quasiconformal on all of R^3. The…

We introduce a notion of rectifiability modeled on Carnot groups. Precisely, we say that a subset E of a Carnot group M and N is a subgroup of M, we say E is N-rectifiable if it is the Lipschitz image of a positive measure subset of N.…

We give lower bound estimates for the macroscopic scale of coarse differentiability of Lipschitz maps from a Carnot group with the Carnot-Carath\'{e}odory metric $(G,\dcc)$ to a few different classes of metric spaces. Using this result, we…

We present a brief overview of the Kor\'anyi-Reimann theory of quasiconformal mappings on the Heisenberg group stressing on the analogies as well as on the differences between the Heisenberg group case and the classical two-dimensional…

We study metric contraction properties for metric spaces associated with left-invariant sub-Riemannian metrics on Carnot groups. We show that ideal sub-Riemannian structures on Carnot groups satisfy such properties and give a lower bound of…

We prove that the boundary of an almost minimizer of the intrinsic perimeter in a plentiful group can be approximated by intrinsic Lipschitz graphs. Plentiful groups are Carnot groups of step~$2$ whose center of the Lie algebra is generated…

We show that for some negatively curved solvable Lie groups, all self quasiisometries are almost isometries. We prove this by showing that all self quasisymmetric maps of the ideal boundary (of the solvable Lie groups) are bilipschitz with…