Survey on Thurston metric on Teichmüller space, focusing on extremal maps.
problem Constructing extremal Lipschitz maps between hyperbolic surfaces.
method Review of constructions including Thurston's original work.
result Coarse geometry and isometry rigidity of Thurston metric discussed.
Unified proof of Nielsen-Thurston classification via Teichmüller's theorem.
problem Classifying mapping class groups and rational maps.
method Unified proof following Bers' approach.
result Unified proof of Nielsen-Thurston classification.
The paper extends Thurston's method to new variants of Mather-Thurston theorem.
problem Proving new variants of Mather-Thurston theorem for PL homeomorphisms and contactomorphisms.
method Generalizing Thurston's technique to prove new variants of Mather-Thurston theorem for PL homeomorphisms and contactomorphisms.
result The paper answers questions posed by Gelfand-Fuks and Greenberg on PL foliations and Rybicki on contactomorphisms.
Developed theory for Thurston maps with a small set of essential singularities.
problem Characterizing Thurston maps with essential singularities.
method Analyzed pullback maps on Teichmüller space to characterize Thurston maps.
result Established a characterization theorem for Thurston maps with four postsingular values.
Article explores Thurston's circle packing theorem in 3-manifold geometry.
problem Understanding Thurston's circle packing theorem in 3-manifold geometry.
method Analyzes the Koebe-Andre'ev-Thurston Theorem and its relation to Thurston's circle packing theorem.
result Illustrates the significance of Thurston's circle packing theorem in 3-manifold geometry.
Overview of Thurston's work in math.
problem None explicitly stated, focuses on Thurston's contributions.
method Presentation of significant results.
result Impact of Thurston's work on mathematics.
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 D×{c} grows monotonically with c.…
Study shows genus two Heegaard diagrams for all Thurston geometries.
problem Finding Heegaard diagrams for Thurston geometries.
method Collecting genus two Heegaard diagrams for all eight Thurston geometries.
result Each Thurston geometry has a genus two Heegaard example.
Uniformly finite Cannon--Thurston fibers in most hyperbolic settings.
problem Existence and finiteness of Cannon--Thurston maps.
method Analysis of proper maps between hyperbolic metric spaces.
result Uniform finiteness of Cannon--Thurston fibers in most known settings.
In earlier work, we had shown that Cannon-Thurston maps exist for Kleinian surface groups. In this paper we prove that pre-images of points are precisely end-points of leaves of the ending lamination whenever the Cannon-Thurston map is not one-to-one. In particular, the Cannon-Thurston map is finite-to-one. This comple…
We show that Cannon-Thurston maps exist for degenerate free groups without parabolics, i.e. for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon-Thurston maps for surface groups, we show that Cannon-Thurston maps exist for arbitrary finitely generated Kleinian groups witho…
Thurston's jiggling lemma simplifies triangulations.
problem Simplifying triangulations into a general position.
method Alternative, conceptual proof and generalization to manifolds.
result A more straightforward proof of Thurston's jiggling lemma.
Historical notes on Thurston's 3-manifold geometry.
problem Exploring Thurston's notes on 3-manifold geometry.
method Historical commentary and references.
result Discussion of Lobachevsky, Andreev, and Milnor's works.
Proves Vaisman solvmanifolds are finite quotients of Kodaira-Thurston manifolds.
problem Characterizing Vaisman solvmanifolds and their properties.
method Analyzing fundamental groups and quotient structures.
result Every Vaisman solvmanifold is a finite quotient of a Kodaira-Thurston manifold.
Veering triangulations link Thurston norm and isotopy of surfaces.
problem Understanding the relationship between Thurston norm and isotopy of surfaces.
method Analyzing veering triangulations and their relation to Thurston norm and isotopy.
result Veering triangulations specify faces of Thurston norm balls and link isotopy of surfaces.
We study the Thurston-Bennequin number of complete and complete bipartite Legendrian graphs. We define a new invariant called the total Thurston-Bennequin number of the graph. We show that this invariant is determined by the Thurston-Bennequin numbers of 3-cycles for complete graphs and by the Thurston-Bennequin number…
Develop criteria to distinguish Gromov-Thurston manifolds using algebraic Dehn fillings.
problem Distinguishing homotopy types of Gromov-Thurston manifolds
method Using virtual Dehn fillings of relatively hyperbolic groups
result Criteria to distinguish homotopy types
We show that link Floer homology detects the Thurston norm of a link complement. As an application, we show that the Thurston polytope of an alternating link is dual to the Newton polytope of its multi-variable Alexander polynomial. To illustrate these techniques, we also compute the Thurston polytopes of several speci…
The paper extends Menelaus' and Ceva's theorems to translation triangles in various Thurston geometries.
problem Extending classical theorems to non-Euclidean geometries.
method Using projective models of Thurston geometries and defining a ``surface of a translation-like triangle".
result Generalization of Menelaus' and Ceva's theorems to non-constant curvature Thurston geometries.
Article provides polytopes as dual unit balls of Thurston norms on 3-manifolds.
problem Understanding the dual unit ball shape of Thurston norms.
method Introduced a family of polytopes in Z^2g that can be dual unit balls of Thurston norms on 3-manifolds.
result Polytopes with mod 2 congruent vertices can be realized as dual unit balls of Thurston norms.
Paper shows mapping class group-equivariant Teichmüller space deformation to Thurston spine.
problem Mapping Teichmüller space to Thurston spine.
method Equivariant deformation retraction of Teichmüller space onto a cell complex.
result Thurston spine contains points corresponding to hyperbolic surfaces with shortest geodesics forming polygons.
Modernizes Thurston's proof of entropy theorem for traintrack maps.
problem Proving the entropy theorem for traintrack maps using Thurston's methods.
method Modernizes Thurston's original proof, fills gaps, and proves ergodicity.
result A cohesive proof of the traintrack theorem, including ergodicity.
Survey of Thurston norm properties and connections to 3-manifold invariants.
problem Understanding Thurston norm in 3-manifolds.
method Review and analysis of existing literature.
result Relationships between Thurston norm and various topological invariants.
Constructs Teichmüller curve to study Thurston spine structure.
problem Understanding the structure of Thurston spine in Teichmüller space.
method Constructs a Teichmüller curve and characterizes its intersection with Thurston spine.
result Characterizes Thurston spine as a trivalent tree and equivariant deformation retract of Teichmüller curve.
Proves Thurston's bounded image theorem for Haken manifolds.
problem Proving Thurston's bounded image theorem for Haken manifolds.
method Using recent developments in Kleinian group theory.
result A proof of Thurston's original bounded image theorem.
The paper studies pseudo-Anosov maps from typical Thurston constructions.
problem Estimating the entropy of pseudo-Anosov maps from Thurston's constructions.
method Developed a method to extract information about random walks associated with Thurston's construction.
result Random walks eventually become pseudo-Anosov under certain conditions.
New method calculates Thurston norm for 3-manifolds with toroidal boundaries.
problem Computing the Thurston norm for 3-manifolds with toroidal boundaries.
method maw dual graph construction and sutured manifold hierarchies.
result Explicit procedure to compute Thurston norm from hierarchies.
Numerical study confirms Brennan's conjecture for a counterexample to Thurston's K=2 conjecture.
problem Thurston's K=2 conjecture and Brennan's conjecture in planar domains. method Numerical analysis of a specific counterexample to Thurston's conjecture.
result The counterexample does not contradict Brennan's conjecture.
Explains Thurston geometries and visualization techniques.
problem Classifying Thurston geometries and their visualization.
method Survey of history and recent techniques.
result Discussion of recent immersive visualization techniques.
Teichmüller space rigidity proven for Thurston metric.
problem Understanding isometries in Teichmüller space with Thurston metric.
method Analyzing R-linear surjective isometries between cotangent spaces. result Every isometry between hyperbolic surfaces induces an isometry in Teichmüller space.
The Thurston spine's properties are studied in relation to Morse-Smale complexes.
problem Understanding the Thurston spine's local properties and their global implications.
method Analyzes the Thurston spine as a subset of Teichmüller space and studies its local properties in relation to the systole function.
result The Thurston spine satisfies properties analogous to Morse-Smale complexes, demonstrating its topological significance.
We show that solutions of Thurston equation on triangulated 3-manifolds in a commutative ring carry topological information. We also introduce a homogeneous Thurston equation and a commutative ring associated to triangulated 3-manifolds.
Study on Vaisman metrics on Kodaira-Thurston surface using pluriclosed flow.
problem Characterizing and preserving Vaisman metrics on Kodaira-Thurston surface.
method Characterization of T2-invariant Vaisman metrics, analysis of pluriclosed flow behavior. result Pluriclosed flow preserves Vaisman condition on Kodaira-Thurston surface, including non-constant scalar curvature.
The Euler class conjecture links geometric structures to integral points on the Thurston norm ball.
problem Determining if integral points on the Thurston norm dual ball correspond to geometric structures.
method Examining various geometric, topological, and dynamical structures on 3-manifolds.
result Integral points on the Thurston norm dual ball correspond to the Euler class of taut foliations and other structures.
Graph manifolds' Thurston norms are sums of linear functionals, and every such norm can be realized.
problem Understanding Thurston norms of graph manifolds and their realizability.
method Analyzing the structure of Thurston norms as sums of linear functionals and showing realizability.
result Every Thurston norm of a graph manifold can be expressed as a sum of absolute values of linear functionals with rational coefficients.
We consider Gromov-Thurston examples of negatively curved n-manifolds which do not admit metrics of constant sectional curvature. We show that for each n some of the Gromov-Thurston manifolds admit strictly convex real-projective structures.
We simplify Thurston norm computation for 2-bridge link complements.
problem Understanding the complexity of Thurston norm unit balls in 3-manifolds.
method Utilized Floyd and Hatcher's surface description and integral class minimization.
result Thurston norm unit balls of 2-bridge link complements have at most 8 faces.
Researchers determine the Thurston unit ball for a family of n-chained links and find conditions for fibered faces.
problem Determining the Thurston unit ball and conditions for fibered faces in a family of n-chained links. method Analyzing the family of n-chained links C(n,p), proving the Thurston unit ball is an n-dimensional cocube for p>0, and finding conditions for fibered faces. result The Thurston unit ball for C(n,p) is an n-dimensional cocube for p>0 and provides at least one fibered face for any p. The Thurston norm of a 3-manifold measures the complexity of surfaces representing two-dimensional homology classes. We study the possible unit balls of Thurston norms of 3-manifolds M with b1(M)=2, and whose fundamental groups admit presentations with two generators and one relator. We show that even among this…
Constructs a universal Cannon-Thurston map for a new curve complex.
problem Mapping class groups and their boundaries.
method Using Birman exact sequence, proves hyperbolicity, constructs map.
result Universal Cannon-Thurston map to surviving curve complex boundary.
The paper studies rigidity of sphere packings on 3D manifolds with boundary.
problem Rigidity of sphere packings on 3D manifolds with boundary.
method Introduced generalized Thurston's sphere packings and proved their rigidity properties.
result Generalized Thurston's sphere packings are locally determined by combinatorial scalar curvatures and cannot be deformed while keeping combinatorial Ricci curvatures fixed.
Study of Teichmüller space geometry using infinitesimal and global methods.
problem Understanding the geometry of Teichmüller space and its tangent/cotangent spheres.
method Systematic study of Thurston metric's infinitesimal and global properties.
result Rigidity statements for the Thurston metric analogous to Royden theorem.
The paper classifies homogeneous hypersurfaces in three 4D Thurston geometries.
problem Classifying homogeneous hypersurfaces in specific 4D geometries.
method Analyzing isometry groups and applying classification techniques.
result Homogeneous hypersurfaces identified in Sol14, Solm,n4 and Nil4. 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).
New metrics for Anosov representations defined from Thurston's asymmetric metrics.
problem Defining metrics for Anosov representations.
method Generalizing Thurston's asymmetric metric to Anosov representations.
result Provides a (possibly asymmetric) Finsler distance in some cases.
Unique geodesics selected by energy minimization in Teichmüller space.
problem Finding a unique geodesic between points in Teichmüller space.
method Energy minimization of harmonic map rays, extending Thurston boundary.
result Selection of a unique Thurston geodesic through points in Teichmüller space.
A Thurston map is a branched covering map from §2 to §2 with a finite postcritical set. We associate a natural Gromov hyperbolic graph $\G=\G(f,\mathcal C)$ with an expanding Thurston map f and a Jordan curve C on §2 containing $\post(f)$. The boundary at infinity of $\G$ with associated visual me…