arXiv research

Explains Thurston geometries and visualization techniques.

This paper uses combinatorial Ricci flow to tackle Thurston's triangulation conjecture.

Historical notes on Thurston's 3-manifold geometry.

The Euler class conjecture links geometric structures to integral points on the Thurston norm ball.

We show that for a strongly convergent sequence of geometrically finite Kleinian groups with geometrically finite limit, the Cannon-Thurston maps of limit sets converge uniformly. If however the algebraic and geometric limits differ, as in the well known examples due to Kerckhoff and Thurston, then provided the geometr…

The paper solves a geometric P=W conjecture for SL(2,C) using Thurston's compactification.

We establish basic geometric and topological properties of Thurston's Master Teapot and the Thurston set for superattracting unimodal self-maps of intervals. In particular, the Master Teapot is connected, contains the unit cylinder, and its intersection with a set $\mathbb{D} \times \{c\}$ grows monotonically with $c$.…

This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings-Jaco theorem which established a similar result for the Poincare Conjecture. The paper also gives two other algebraic conjectures; one is equi…

Solves a 45-year-old Poincaré Conjecture using geometrization of 3-manifolds.

Survey of combination theorems in geometry and dynamics.

Geometrization Theorem solves complex geometry problems.

The paper shows Thurston geometries don't support Anosov diffeomorphisms.

Study continuity of limit sets in symmetric spaces.

Study geometric manifolds in arbitrary dimensions, focusing on maps and diffeomorphisms.

The paper shows how profinite completions can reveal 4D geometries except for specific cases.

The seven non euclidean geometries of the Thurston's geometrization program are proved to originate naturally from singularization morphisms and versal deformations on euclidean 3-manifolds generated in the frame of the Langlands global program. The Poincare conjecture for a 3-manifold appears as a particular case of t…

This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. The main goal is to describe Thurston's geometrisation of three-manifolds, proved by Perelman in 2002. The book is divided into three parts: the first is devoted to hyperbolic geometry, the second to surfaces,…

The paper describes geometric properties of Teichmüller space metrics.

The paper proves inequalities linking geometric norms and Thurston norms in hyperbolic 3-manifolds.

Study geometric structures on LVM threefolds, focusing on resonant structures.

Study of Teichmüller space geometry using infinitesimal and global methods.

We give an overview of the theory of Cannon-Thurston maps which forms one of the links between the complex analytic and hyperbolic geometric study of Kleinian groups. We also briefly sketch connections to hyperbolic subgroups of hyperbolic groups and end with some open questions.

We study canonical decompositions of postcritically finite branched coverings of the 2-sphere, as defined by K. Pilgrim. We show that every hyperbolic cycle in the decomposition does not have a Thurston obstruction. It is thus Thurston equivalent to a rational map.

This paper determines the flexible exponent for non-geometric 3-manifolds.

We obtain an ordering of closed aspherical 4-manifolds that carry a non-hyperbolic Thurston geometry. As application, we derive that the Kodaira dimension of geometric 4-manifolds is monotone with respect to the existence of maps of non-zero degree.

We apply an equivariant version of Perelman's Ricci flow with surgery to study smooth actions by finite groups on closed 3-manifolds. Our main result is that such actions on elliptic and hyperbolic 3-manifolds are conjugate to isometric actions. Combining our results with results by Meeks and Scott [17], it follows tha…

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

We describe recent links between two topics: geometric structures on manifolds in the sense of Ehresmann and Thurston, and dynamics "at infinity" for representations of discrete groups into Lie groups.

Given a sub-hyperbolic semi-rational branched covering which is not CLH-equivalent a rational map, it must have the non-empty canonical Thurston obstruction. By using this canonical Thurston obstruction, we decompose this dynamical system in this paper into several sub-dynamical systems. Each of these sub-dynamical sys…

The Cannon-Thurston map's measures become singular with respect to sphere measures.

Algorithm determines Thurston equivalence of topological polynomials.

We show that for a strongly convergent sequence of purely loxodromic finitely generated Kleinian groups with incompressible ends, Cannon-Thurston maps, viewed as maps from a fixed base limit set to the Riemann sphere, converge uniformly. For algebraically convergent sequences we show that there exist examples where eve…

The paper classifies hypersurfaces in a specific 4D geometry.

In this paper we extend Thurston's hyperbolic Dehn surgery theorem to a class of geometrically infinite hyperbolic 3-manifolds. As an application we prove a modest density theorem for Kleinian groups. We also discuss hyperbolic Dehn surgery on geometrically finite hypebolic cone-manifolds.

New quasi space forms solve Thurston's geometrical space form problem.

Let N^h be a hyperbolic 3-manifold of bounded geometry corresponding to a hyperbolic structure on a pared manifold (M,P). Further, suppose that (\partial{M} - P) is incompressible, i.e. the boundary of M is incompressible away from cusps. Further, suppose that M_{gf} is a geometrically finite hyperbolic structure on (M…

This note summarizes in an informal way some geometric properties of Anosov representations into the symplectic group, which were presented in a talk at the conference What is Next. The mathematical legacy of Bill Thurston, held in June 2014 in Cornell.

We prove that twisted correction terms in Heegaard Floer homology provide lower bounds on the Thurston norm of certain cohomology classes determined by the strong concordance class of a 2-component link $L$ in $S^3$. We then specialise this procedure to knots in $S^2\times S^1$, and obtain a lower bound on their geomet…

We shall show that for a given homeomorphism type and a set of end invariants (including the parabolic locus) with necessary topological conditions which a topologically tame Kleinian group with that homeomorphism type must satisfy, there is an algebraic limit of minimally parabolic, geometrically finite Kleinian group…

Unique geodesics selected by energy minimization in Teichmüller space.

New metrics derived from Hölder distortion on Hitchin components.

As an example of the transitions between some of the eight geometries of Thurston, investigated before, we study the geometries supported by the cone-manifolds obtained by surgery on the trefoil knot with singular set the core of the surgery. The geometric structures are explicitly constructed. The most interesting phe…

We give a brief survey of Hamilton's program for 3-manifolds as an approach toward Thurston's Geometrization Conjectre.

Surjectivity of Cannon-Thurston map proven for metric graph bundles.

In the genus one case, we make explicit some constructions of Veech on flat surfaces and generalize some geometric results of Thurston about moduli spaces of flat spheres as well as some equivalent ones but of an analytico-cohomological nature of Deligne-Mostow, which concern the monodromy of Appell-Lauricella hypergeo…

The density conjecture of Bers, Sullivan and Thurston predicts that each complete hyperbolic 3-manifold M with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We prove that the conjecture obtains for each complete hyperbolic 3-manifold with no cusps and incompr…

Study of geometric structures on projective space complement without Schwarz conditions.

Let $S$ be a closed orientable surface with genus $g\geq 2$. For a sequence $\s_i$ in the Teichmüller space of $S$, which converges to a projective measured lamination $[\lam]$ in the Thurston boundary, we obtain a relation between $\lam$ and the geometric limit of pants decompositions whose lengths are uniformly bound…