arXiv research

Survey of recent Kleinian representation convergence results.

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).

This paper shows similarities in deformation spaces of Kleinian groups and anti-holomorphic maps.

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

We prove that a 3--dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by its Gauss image. Furthermore, any spherical metric on the torus with cone singularities of negative curvature and all closed contractible geodesics of length greater than $2π$ is the metric of the Gauss image of som…

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

Compact leaves with amenable groups are stable under small perturbations.

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

Proves Thurston's bounded image theorem for Haken manifolds.

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

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

Thurston's hyperbolization theorem for Haken manifolds and normal surface theory yield an algorithm to determine whether or not a compact orientable 3-manifold with nonempty boundary consisting of tori admits a complete finite-volume hyperbolic metric on its interior. A conjecture of Gabai, Meyerhoff, and Milley reduce…

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

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

In 1970, E. M. Andreev published a classification of all three-dimensional compact hyperbolic polyhedra having non-obtuse dihedral angles. Given a combinatorial description of a polyhedron, $C$, Andreev's Theorem provides five classes of linear inequalities, depending on $C$, for the dihedral angles, which are necessar…

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.

We consider a compact orientable hyperbolic 3-manifold with a compressible boundary. Suppose that we are given a sequence of geometrically finite hyperbolic metrics whose conformal boundary structures at infinity diverge to a projective lamination. We prove that if this limit projective lamination is doubly incompressi…

We give a generalization of Thurston's Bounded Image Theorem for skinning maps, which applies to pared 3-manifolds with incompressible boundary that are not necessarily acylindrical. Along the way we study properties of divergent sequences in the deformation space of such a manifold, establishing the existence of compa…

The study proves properties of 4D projective manifolds and builds non-hyperbolic examples.

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

The study provides a structure theorem for a new class of noncompact 3-manifolds.

Bonahon's method for compactifying Teichmüller space extended to non-compact surfaces.

Study shows genus two Heegaard diagrams for all Thurston geometries.

Historical notes on Thurston's 3-manifold geometry.

The purpose of this article is to give a proof of the Orbifold Theorem announced by Thurston in late 1981: If $O$ is a compact, connected, orientable, irreducible and topologically atoroidal 3-orbifold with non-empty ramification locus, then $O$ is geometric. As a corollary, any smooth orientation preserving non-free f…

In the context of Thurstons geometrisation program we address the question which compact aspherical 3-manifolds admit Riemannian metrics of nonpositive curvature. We show that non-geometric Haken manifolds generically, but not always, admit such metrics. More precisely, we prove that a Haken manifold with, possibly emp…

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.

Third paper in series defines monopole Floer homology and gluing theorem for 3-manifolds.

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.

For every positive, continuous and homogeneous function $f$ on the space of currents on a compact surface $\overlineΣ$, and for every compactly supported filling current $α$, we compute as $L \to \infty$, the number of mapping classes $φ$ so that $f(φ(α))\leq L$. As an application, when the surface in question is close…

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 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.

Schmutz Schaller and Thurston's approaches are dual.

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…

Proves Vaisman solvmanifolds are finite quotients of Kodaira-Thurston manifolds.

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.

Let S be a compact, connected, orientable surface of positive genus. Let HT(S) be the Hatcher-Thurston complex of S. We prove that Aut(HT(S)) is isomorphic to the extended mapping class group of S modulo its center.