arXiv research

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 that an Ahlfors 2-regular, linearly locally contractible metric 2-sphere is quasisymmetrically homeomorphic to the standar…

Study on mappings between nonrigid Carnot groups, proving quasisymmetric rigidity.

We give upper bounds on the principal curvatures of a maximal surface of nonpositive curvature in three-dimensional Anti-de Sitter space, which only depend on the width of the convex hull of the surface. Moreover, given a quasisymmetric homeomorphism $φ$, we study the relation between the width of the convex hull of th…

We give parameterizations of homeomorphisms, quasisymmetric maps and symmetric maps of the unit circle in terms of shear coordinates for the Farey tesselation.

The paper constructs thin Loewner carpets and their embeddings in $S^2$.

We give a parametrization to the asymptotic Teichmuller space of the open unit disk through equivalent classes of shear functions induced by quasisymmetric homeomorphisms on the Farey tesselation of the unit disk. Then using the parametrization, we define a new metric on the asymptotic Teichmuller space. Two other rela…

Suppose G is a Gromov hyperbolic group, and the boundary at infinity of G is quasisymmetrically homeomorphic to an Ahlfors Q-regular metric 2-sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and isometrically on hyperbolic 3-space.

We prove that any weakly acausal curve $Γ$ in the boundary of Anti-de Sitter (2+1)-space is the asymptotic boundary of two spacelike $K$-surfaces, one of which is past-convex and the other future-convex, for every $K\in(-\infty,-1)$. The curve $Γ$ is the graph of a quasisymmetric homeomorphism of the circle if and only…

Survey on uniformization of metric surfaces, including fractal and topological manifolds.

The paper sketches a recent progress and formulates several open problems in studying equivariant quasiconformal and quasisymmetric homeomorphisms in negatively curved spaces as well as geometry and topology of noncompact geometrically finite negatively curved manifolds and their boundaries at infinity having Carnot--C…

Study quasisymmetric maps on hyperbolic plane boundaries.

A new proof shows how to characterize maps using simple geometry.

We study a new class of square Sierpiński carpets $F_{n,p}$ ($5\leq n, 1\leq p<\frac{n}{2}-1$) on $\mathbb{S}^2$, which are not quasisymmetrically equivalent to the standard Sierpiński carpets. We prove that the group of quasisymmetric self-maps of each $F_{n,p}$ is the Euclidean isometry group. We also establish that …

Given a quasisymmetric homeomorphism $\varphi$ of the circle, Bonsante and Schlenker proved the existence and uniqueness of the minimal Lagrangian extension $f_\varphi:\mathbb{H}^2\to\mathbb{H}^2$ to the hyperbolic plane. By previous work of the author, its maximal dilatation satisfies $\log K(f_\varphi)\leq C||\varphi…

The study characterizes quasiperiodic surfaces in pseudo-hyperbolic spaces with curvature conditions.

Study compares hyperbolic and quasihyperbolic metrics in plane domains.

We consider decomposition spaces $\R^3/G$ that are manifold factors and admit defining sequences consisting of cubes-with-handles. Metrics on $\R^3/G$ constructed via modular embeddings into Euclidean spaces promote the controlled topology to a controlled geometry. The quasisymmetric parametrizability of the metric spa…

Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a generalization for unbounded convex subsets, we consider convex regions of hyperbolic three-space bounded by two properly embedd…

In the first part of the paper we describe the complex geometry of the universal Teichmüller space $\mathcal T$, which may be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The quotient $\mathcal S$ of the diffeomorphism group of the circle modulo Möbius …

Study groups admitting unbounded quasimorphisms to R with coarsely-connected quasikernel.

Maps and embeddings between hyperbolic spaces and their boundaries studied.

New fractal spaces not quasisymmetric to Loewner spaces discovered.

For a given $ε>0$, we show that there exist two finite index subgroups of $PSL_2(\mathbb{Z})$ which are $(1+ε)$-quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any $ε>0$ there are two finite regular covers of the Modular once punctured torus $T_0$ (or just the Mod…

We discuss a variation of Gromov's notion of asymptotic dimension that was introduced and named Nagata dimension by Assouad. The Nagata dimension turns out to be a quasisymmetry invariant of metric spaces. The class of metric spaces with finite Nagata dimension includes in particular all doubling spaces, metric trees, …

Uniformly branching trees are equivalent to certain metric spaces.

We give a short proof of the fact that bounded earthquakes of the unit disk induce quasisymmetric maps of the unit circle. By a similar method, we show that symmetric maps are induced by bounded earthquakes with asymptotically trivial measures.

A variant of Gromov's H{ö}lder-equivalence problem, motivated by a pinching problem in Riemannian geometry, is discussed. A partial result is given. The main tool is a general coarea inequality satisfied by packing energies of maps.

We prove that the linearly controlled asymptotic dimension of the fundamental group of any 3-dimensional graph-manifold does not exceed 7. As applications we obtain that the universal cover of such a graph-manifold is an absolute Lipschitz retract and it admits a quasisymmetric embedding into the product of 8 metric tr…

Groups with cusped spaces are quasi-isometric to symmetric spaces.

Study on mappings in Carnot groups, proving rigidity results.

Using a flow first introduced by J.P. Anderson, we obtain some existence theorems for harmonic maps from a noncompact complete Riemannian manifold into a complete Riemannian manifold. In particular, we prove as a corollary a recent result of Hardt and Wolf stating that any quasisymmetric map of the sphere that is suffi…

Unified approach to conformal and modular invariants on surfaces.

We study the asymmetry of the Lipschitz metric d on Outer space. We introduce an (asymmetric) Finsler norm that induces d. There is an Out(F_n)-invariant potential Ψon Outer space such that when the Lipschitz norm is corrected by the derivative of Ψ, the resulting norm is quasisymmetric. As an application, we give new …

We describe relations between hyperbolic geometry and codimension two knots or, more exactly, between varieties of conjugacy classes of discrete faithful representations of the fundamental groups of hyperbolic n-manifolds M into $\operatorname{SO}^{\circ} (n+2,1)$ and (n-1)-dimensional knots in the (n+1)-sphere. This a…

The elliptic Hall algebra governs torus link homology.

Extends Palais' result on diffeomorphisms to homeomorphisms and bi-Lipschitz mappings.

Study on homeomorphism groups of manifolds using set theory.

New proof associates partitions to isotopic pseudo-Anosov homeomorphisms.

Uniform interpretation of group theory in manifold homeomorphisms.

Proves existence of sentences to identify homeomorphic manifolds.

New homeomorphism found in Klein bottle group.

Study weak conjugacy in surface homeomorphisms.

Paper explains dynamics of homeomorphisms to mapping tori geometry.

Unified framework recovers and improves classical Brouwer homeomorphism results.

Every homeomorphism of Euclidean space is a commutator of two homeomorphisms.

Paper improves proof of theorem on convex 2-disk homeomorphisms.

The study of periodic subgroups in homeomorphism groups of manifolds.

Adapts pivoting technique to circle homeomorphisms for proofs.