arXiv research

Unified proof of Nielsen-Thurston classification via Teichmüller's theorem.

The paper extends Thurston's method to new variants of Mather-Thurston theorem.

Article explores Thurston's circle packing theorem in 3-manifold geometry.

Proves Thurston's bounded image theorem for Haken manifolds.

The paper extends Menelaus' and Ceva's theorems to translation triangles in various Thurston geometries.

Proof of Thurston's earthquake theorem using Anti-de Sitter geometry.

Modernizes Thurston's proof of entropy theorem for traintrack maps.

Revises a theorem by Thurston, finding a counter-example and a weaker version.

Developed theory for Thurston maps with a small set of essential singularities.

Teichmüller space rigidity proven for Thurston metric.

Study of infinity in Teichmüller space using Thurston boundary.

Two flexible, degenerate constructions related to Thurston's theorem.

3-manifolds explained through geometry, proving Thurston's conjecture.

Survey of recent Kleinian representation convergence results.

Historical notes on Thurston's 3-manifold geometry.

We establish a form of the h-principle for the existence of foliations quasi-complementary to a given one; the same methods also provide a proof of the classical Mather-Thurston theorem.

Survey of combination theorems in geometry and dynamics.

We prove a version the local Reeb-Thurston stability theorem for symplectic foliations.

In this note, we provide a description of the structure of homomorphisms from a finitely generated group to any torsion-free (3-dimensional) Kleinian group with uniformly bounded finite covolume. This is analogous to the Jorgensen-Thurston Theorem in hyperbolic geometry.

Paper extends circle pattern theory to obtuse angles.

This is an expository paper. We prove the Cannon-Thurston property for bounded geometry surface groups with or without punctures. We prove three theorems, due to Cannon-Thurston, Minsky and Bowditch. The proofs are culled out of earlier work of the author.

We define analogue of theta-functions on the Kodaira--Thurston manifold which is a compact 4-dimensional symplectic manifold and use them to construct canonical symplectic embedding of the Kodaira--Thurston manifold into the complex projective space (analogue of the Lefshetz theorem).

We provide analogues for non-orientable surfaces with or without boundary or punctures of several basic theorems in the setting of the Thurston theory of surfaces which were developed so far only in the case of orientable surfaces. Namely, we provide natural analogues for non-orientable surfaces of the Fenchel-Nielsen …

Paper confirms conjecture for PL foliations of codimension 2.

In 1976, Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that conversely, any integral second cohomology class with norm equal to one is the Euler class of a taut foliation. This is the first from a series of two papers that together give a neg…

We give a complete proof of Thurston's celebrated hyperbolic Dehn filling theorem, following the ideal triangulation approach of Thurston and Neumann-Zagier. We avoid to assume that a genuine ideal triangulation always exists, using only a partially flat one, obtained by subdividing an Epstein-Penner decomposition. Thi…

We present a new, completely three-dimensional proof of the fact, due to Gabai-Eliashberg-Thurston, that every closed, oriented, irreducible 3-manifold with nonzero second homology carries a universally tight contact structure.

We give a proof of the Neilsen-Thurston classification theorem of a homeomorphism f of a standard surface of finite type as either periodic, pseudo-Anosov, or reducible. In the periodic case, we show that there exists an integer n>0 such that f is isotopic to h with h^n isotopic to the identity. This is the weaker vers…

Geometrization Theorem solves complex geometry problems.

Study of Teichmüller space geometry using infinitesimal and global methods.

This paper contains a generalization of the convex ideal case of the Thurston-Andreev theorem when the genus is greater than 1. The heart of the paper concerns taking formal angle data on a surface and ``conformally flowing'' this formal angle data to uniquely associated uniform angle data. This flow turns out to be th…

Study of homeomorphisms on infinite type surfaces with a classification theorem.

Generalized Thurston's characterization for branched coverings of the 2-sphere.

The article extends Thurston's Grafting Theorem to signed spaces and defines a framed monodromy map.

Thurston obtained a classification of individual surface homeomorphisms via the dynamics of the corresponding mapping class elements on Teichmüller space. In this paper we present certain extended versions of this, first, to random products of homeomorphisms and second, to holomorphic self-maps of Teichmüller spaces.

New finding links hyperbolic manifold systolic volume to triangulation complexity.

Proofs non-realizability of mapping class group via homeomorphisms, resolves Thurston's conjecture.

We prove that the Calabi-Yau equation can be solved on the Kodaira-Thurston manifold for all given $T^2$-invariant volume forms. This provides support for Donaldson's conjecture that Yau's theorem has an extension to symplectic four-manifolds with compatible but non-integrable almost complex structures.

The paper studies pseudo-Anosov maps from typical Thurston constructions.

Mahan Mitra (Mj) proved Cannon--Thurston maps exist for normal hyperbolic subgroups of a hyperbolic group. We prove that Cannon--Thurston maps do not exist for infinite normal hyperbolic subgroups of non-hyperbolic CAT(0) groups with isolated flats with respect to the visual boundaries. We also show Cannon--Thurston ma…

Thurston related $\mathbb{C}{\rm P}^1$-structures (complex projective structures) and equivariant pleated surfaces in the hyperbolic-three space $\mathbb{H}^3$, in order to give a parameterization of the deformation space of $\mathbb{C}{\rm P}^1$-structures. In this note, we summarize Thurston's parametrization of $\ma…

We prove that the twisted Reidemeister torsion of a 3-manifold corresponding to a fibered class is monic and we show that it gives lower bounds on the Thurston norm. The former fixes a flawed proof in [FV10], the latter gives a quick alternative argument for the main theorem of [FK06].

We show that the analog of Hamilton's Ricci flow in the combinatorial setting produces solutions which converge exponentially fast to Thurston's circle packing on surfaces. As a consequence, a new proof of Thurston's existence of circle packing theorem is obtained. As another consequence, Ricci flow suggests a new algo…

Study of strongly invertible Legendrian links in contact 3-space.

Extends Thurston's combinatorial characterization to all branched coverings of the 2-sphere.

This article has been withdrawn due to a mistake which is explained in version 2.

We study the geometry of the Thurston metric on the Teichmüller space $\mathcal{T}(S)$ of hyperbolic structures on a surface $S$. Some of our results on the coarse geometry of this metric apply to arbitrary surfaces $S$ of finite type; however, we focus particular attention on the case where the surface is a once-punct…

The notion of i-bounded geometry generalises simultaneously bounded geometry and the geometry of punctured torus Kleinian groups. We show that the limit set of a surface Kleinian group of i-bounded geometry is locally connected by constructing a natural Cannon-Thurston map. This is an exposition of a special case of th…