# Geometry and quasisymmetric parametrization of Semmes spaces

@article{Pankka2011GeometryAQ, title={Geometry and quasisymmetric parametrization of Semmes spaces}, author={Pekka Pankka and Jang-Mei Gloria Wu}, journal={Revista Matematica Iberoamericana}, year={2011}, volume={30}, pages={893-960} }

We consider decomposition spaces R/G that are manifold factors and admit defining sequences consisting of cubes-with-handles. Metrics on R/G constructed via modular embeddings of R/G into Euclidean spaces promote the controlled topology to a controlled geometry. The quasisymmetric parametrizability of the metric space R/G×R by R for any m ≥ 0 imposes quantitative topological constraints, in terms of the circulation and the growth of the cubes-with-handles, to the defining sequences for R/G. We…

## 15 Citations

### Uniformization of two-dimensional metric surfaces

- Mathematics
- 2014

We establish uniformization results for metric spaces that are homeomorphic to the Euclidean plane or sphere and have locally finite Hausdorff 2-measure. Applying the geometric definition of…

### Quasisymmetric spheres constructed over quasidisks

- Mathematics
- 2014

In this paper we provide some concrete examples of quasispheres and quasisymmetric spheres. We present two different constructions of surfaces in R over a quasidisk Ω ⊂ R. In the first construction,…

### Geometric function theory: the art of pullback factorization

- Mathematics
- 2016

In this paper, we develop the foundations of the theory of quasiregular mappings in general metric measure spaces. In particular, nine definitions of quasiregularity for a discrete open mapping with…

### Remarks on conformal modulus in metric spaces

- Mathematics
- 2022

. We give an example of an Ahlfors 3 -regular, linearly locally connected metric space homeomorphic to R 3 containing a nondegenerate continuum E with zero capacity, in the sense that the conformal…

### Quasispheres and metric doubling measures

- Mathematics
- 2017

Applying the Bonk-Kleiner characterization of Ahlfors 2-regular quasispheres, we show that a metric two-sphere $X$ is a quasisphere if and only if $X$ is linearly locally connected and carries a weak…

### Deformation and quasiregular extension of cubical Alexander maps

- Mathematics
- 2019

In this article we prove that, for an oriented PL $n$-manifold $M$ with $m$ boundary components and $d_0\in \mathbb N$, there exist mutually disjoint closed Euclidean balls and a $\mathsf…

### Bi-Lipschitz embedding of the generalized Grushin plane in Euclidean spaces

- Mathematics
- 2015

We show that, for all $\alpha\geq 0$, the generalized Grushin plane $\mathbb{G}_{\alpha}$ is bi-Lipschitz homeomorphic to a $2$-dimensional quasiplane in the Euclidean space $\mathbb{R}^{[\alpha…

### Quasiregular Ellipticity of Open and Generalized Manifolds

- Mathematics
- 2013

We study the existence of geometrically controlled branched covering maps from $$\mathbb R^3$$R3 to open $$3$$3-manifolds or to decomposition spaces $$\mathbb {S}^3/G$$S3/G, and from $$\mathbb…

### Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients

- Mathematics
- 2015

Preface 1. Introduction 2. Review of basic functional analysis 3. Lebesgue theory of Banach space-valued functions 4. Lipschitz functions and embeddings 5. Path integrals and modulus 6. Upper…

### Quasisymmetric spheres over Jordan domains

- Mathematics
- 2014

Let $\Omega$ be a planar Jordan domain. We consider double-dome-like surfaces $\Sigma$ defined by graphs of functions of $dist( \cdot ,\partial \Omega)$ over $\Omega$. The goal is to find the right…

## References

SHOWING 1-10 OF 37 REFERENCES

### Quasiconformality and quasisymmetry in metric measure spaces.

- Mathematics
- 1998

A homeomorphism f : X → Y between metric spaces is called quasisymmetric if it satisfies the three-point condition of Tukia and Vaisala. It has been known since the 1960’s that when X = Y = R (n ≥…

### Good metric spaces without good parameterizations

- Mathematics
- 1996

A classical problem in geometric topology is to recognize when a topological space is a topological manifold. This paper addresses the question of when a metric space admits a quasisymmetric…

### Quasiconformal maps in metric spaces with controlled geometry

- Mathematics
- 1998

This paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry. The principal message is that such a theory is both…

### Quasisymmetric nonparametrization and spaces associated with the Whitehead continuum

- Mathematics
- 2010

The decomposition space R 3 =Wh associated with the Whitehead continuum Wh is not a manifold, but the product .R 3 =Wh/ R m is homeomorphic to R 3Cm for any m 1 (known since the 1960’s). We study the…

### Quasisymmetric parametrizations of two-dimensional metric spheres

- Mathematics
- 2002

We study metric spaces homeomorphic to the 2-sphere, and find conditions under which they are quasisymmetrically homeomorphic to the standard 2-sphere. As an application of our main theorem we show…

### Strange actions of groups on spheres

- Mathematics
- 1987

In [FS] we investigated certain topological analogs of Schottky groups, called admissible actions, and their compatibility with various structures on spheres. We constructed an action ϕ : F 2 × S 3→…

### Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities

- Mathematics
- 1996

In many metric spaces one can connect an arbitrary pair of points with a curve of finite length, but in Euclidean spaces one can connect a pair of points with a lot of rectifiable curves, curves that…

### Cell-like closed-0-dimensional decompositions of ³ are ⁴ factors

- Mathematics
- 1976

It is proved that the product of a cell-like closed-0dimensional upper semicontinuous decomposition of R3 with a line is R4. This establishes at once this feature for all the various dogbone-inspired…

### Quasisymmetric parametrizations of two‐dimensional metric planes

- Mathematics
- 2008

The classical uniformization theorem states that any simply connected Riemann surface is conformally equivalent to the disk, the plane, or the sphere, each equipped with a standard conformal…