Goldman symplectic form and complex structure compatible on SL(3,R) Hitchin component.
problem Compatibility of Goldman's symplectic form with complex structure on SL(3,R) Hitchin component. method Proof of compatibility between Goldman's symplectic form and Labourie-Loftin complex structure.
result Goldman symplectic form and complex structure determine a pseudo-Kähler structure on SL(3,R) Hitchin component. Shearing deformations in Hitchin representations are computed for a symplectic form.
problem Computing symplectic form pairings for Hitchin representations.
method Shearing deformations of Hitchin representations.
result Pairings of shearing deformations computed for the Atiyah-Bott-Goldman symplectic form.
Study shows infinite volumes of moduli spaces for certain groups.
problem Infinite volumes of Hitchin-Riemann moduli spaces for specific groups.
method Employed Goldman flows to find infinite disjoint subsets of identical volume.
result Proved infinite Atiyah-Bott-Goldman covolume for mapping class group actions.
We prove a generalization of Kawai theorem for the case of orbifold Riemann surface. The computation is based on a formula for the differential of a holomorphic map from the cotangent bundle of the Teichmüller space to the PSL(2,C)-character variety, which allows to evaluate explicitly the pullback …
New proof of Wolpert's Magic Formula using de Rham cohomology.
problem Proving Wolpert's Magic Formula for Teichmüller space.
method Using de Rham cohomology and Goldman's symplectic form.
result Derive a new proof of Wolpert's Magic Formula.
Symplectic coordinates found on a Hitchin component for a hyperbolic surface.
problem Parametrizing the PSL3(R)-Hitchin component with canonical coordinates. method Proved global Darboux coordinates with half canonical Goldman coordinates.
result Global Darboux coordinates exist for the PSL3(R)-Hitchin component. Symplectic forms match on circle pattern space.
problem Matching symplectic forms on circle pattern space.
method Pullback of symplectic forms to circle pattern space.
result Symplectic forms on circle pattern space coincide.
This article is the second of a pair of articles about the Goldman symplectic form on the PGL(V)-Hitchin component of a closed, connected, oriented, hyperbolic surface S. We show that any ideal triangulation on S and any compatible bridge system determine a symplectic trivialization of the tangent bundle to the PGL(V)-…
Paper describes a pseudo-Kähler structure on a specific Hitchin component.
problem Existence and description of a pseudo-Kähler structure on the SL(3,R)-Hitchin component.
method Explicit construction of a pseudo-Riemannian metric and symplectic form compatible with complex structure.
result Existence of a pseudo-Kähler structure on a neighborhood of the Fuchsian locus.
The paper studies the center of the Goldman Lie algebra and its properties.
problem Identifying the center of the Goldman Lie algebra and its properties.
method Analyzing the Goldman Lie algebra as a Z_2-graded Lie algebra and using properties of the even part.
result The center of the even part of the Goldman Lie algebra is generated by specific classes of loops.
Goldman bracket distinguishes surface homeomorphisms.
problem Characterizing homeomorphisms between non-compact surfaces.
method Using the Goldman bracket to distinguish homeomorphisms.
result A homotopy equivalence is a homeomorphism if it preserves the Goldman bracket.
Criteria for loop separability on surfaces using Goldman bracket.
problem Determining when two loops on an oriented surface can have disjoint representatives.
method Algebraic criteria in terms of the Goldman bracket, extended using hyperbolic geometry.
result Explicit algebraic conditions for loop separability on surfaces.
The Goldman-Parker Conjecture classifies the complex hyperbolic C-reflection ideal triangle groups up to discreteness. We proved the Goldman-Parker Conjecture in [Ann. of Math. 153 (2001) 533--598] using a rigorous computer-assisted proof. In this paper we give a new and improved proof of the Goldman-Parker Conjecture.…
Deroin and Tholozan's representations are mapped to complex projective space via action-angle coordinates.
problem Mapping representations of a punctured sphere into PSL(2,R) to a simpler geometric space. method Polygonal model and chains of triangles to extract action-angle coordinates.
result Action-angle coordinates give an explicit isomorphism and almost global Darboux coordinates.
The paper defines and calculates Reidemeister torsion for a specific class of representations.
problem Defining and calculating Reidemeister torsion for G-Anosov representations.
method Symplectic chain complex method to establish a novel formula for R-torsion.
result Reidemeister torsion is well-defined and calculated for G-Anosov representations.
New Lie algebras from knot homology.
problem Defining Lie algebras from knot homology.
method Using group homology, analogous to Goldman Lie algebra.
result Relations among new Lie algebras discussed.
Researchers create coordinates for hyperbolic surfaces, proving a magic formula.
problem Constructing coordinates for hyperbolic structures on genus-2 surfaces.
method Developed Fenchel-Nielsen coordinates and Wolpert's magic formula analogues.
result Found Darboux charts for the Goldman symplectic form on branched hyperbolic structures.
We construct a new Riemannian metric on Goldman space B(S), the space of the equivalence classes of convex projective structures on the surface S, and then prove the new metric, as well as the metric of Darvishzadeh and Goldman, restricts to be the Weil-Petersson metric on Teichmu¨ller space, embe…
A new metric model for quasi-Fuchsian space defined by Bers metrics.
problem Understanding the quasi-Fuchsian space of a surface.
method Introducing Bers metrics and studying their properties to model QF(S).
result New integral representations of the Goldman symplectic form and holomorphic extension of the Weil-Petersson metric.
We survey a geometric approach to the Johnson homomorphisms using the Goldman-Turaev Lie bialgebra.
We determine the second homology group of the homological Goldman Lie algebra for an oriented surface.
We determine all the ideals of the homological Goldman Lie algebra, which reflects the structure of an oriented surface.
Decomposes Goldman-Turaev Lie bialgebra via cutting a surface.
problem Decomposing the Goldman-Turaev Lie bialgebra of a surface.
method Algebraic construction of double Lie bimodules and their combination.
result Decomposes the Goldman-Turaev Lie bialgebra along a simple separating curve.
We determine the minimal number of generators of the homological Goldman Lie algebra of a surface consisting of elements of the first homology group of the surface.
We show that a homotopy equivalence between compact, connected, oriented surfaces with non-empty boundary is homotopic to a homeomorphism if and only if it commutes with the Goldman bracket.
We study the geometric properties of the terms of the Goldman bracket between two free homotopy classes of oriented closed curves in a hyperbolic surface. We provide an obstruction for the equality of two terms in the Goldman bracket, namely if two terms in the Goldman bracket are equal to each other then for every hyp…
In the mid eighties Goldman proved an embedded curve could be isotoped to not intersect a closed geodesic if and only if their Lie bracket (as defined in that work) vanished. Goldman asked for a topological proof and about extensions of the conclusion to curves with self-intersection. Turaev, in the late eighties, aske…
Infinite volume found in the thick part of PSLn(R)-Hitchin-Riemann moduli space.
problem Proving infinite volume in the thick part of PSLn(R)-Hitchin-Riemann moduli space. method Employing Goldman flows and internal sequences to find an infinite series of subsets of identical volume.
result Infinite total Atiyah--Bott--Goldman volume for n>2. We show that the Goldman flows preserve the holomorphic structure on the moduli space of homomorphisms of the fundamental group of a Riemann surface into U(1), in other words the Jacobian.
New combinatorial approach to Goldman-Turaev Lie bialgebra using cyclic word partitions.
problem Defining the Goldman bracket and Turaev cobracket combinatorially.
method Focus on partitions of cyclic words to define the bracket and cobracket.
result Combinatorial definition of the bracket and cobracket.
For closed and oriented hyperbolic surfaces, a formula of Witten establishes an equality between two volume forms on the space of representations of the surface in a semisimple Lie group. One of the forms is a Reidemeister torsion, the other one is the power of the Atiyah-Bott-Goldman symplectic form. We introduce an h…
We prove a quasi-Poisson bracket formula for the space of representations of the fundamental groupoid of a surface with boundary, which generalizes Goldman's Poisson bracket formula. We also deduce a similar formula for quasi-Poisson cross-sections.
Study formalities on closed surfaces using connections.
problem Formalities of Goldman-Turaev Lie bialgebra on closed surfaces.
method Reformulated Kashiwara-Vergne groups and associators in higher genera using non-commutative connections.
result Determined pro-unipotent automorphism group of associated graded.
We show that the center of the Goldman algebra associated to a closed oriented hyperbolic surface is trivial. For a hyperbolic surface of finite type with nonempty boundary, the center consists of closed curves which are homotopic to boundary components or punctures.
Cataclysm deformations study Anosov representations and their convergence.
problem Understanding convergence of Anosov representations under deformation.
method Cataclysm deformation of Anosov representations using twisted transverse cocycles.
result Uniform convergence of cataclysm deformations on compact sets.
Study compares two pseudo-Kähler structures on a specific mathematical component.
problem Comparing two pseudo-Kähler structures on the SL(3,R)-Hitchin component. method Examined Rungi-Tamburelli's ωf and Goldman's ωG forms, and aligned Killing forms. result Rungi-Tamburelli's semi-pseudo-Kähler structure is non-degenerate and matches another structure after normalization.
Cataclysm deformations study Anosov representations, leading to new formulas and non-open sets.
problem Understanding Anosov representations and their deformations.
method Cataclysm deformations based on twisted transverse cocycles.
result Uniform convergence of cataclysm deformations on compact sets.
Geometric proof of curve characterization using loop-bundles.
problem Characterization of simple curves in terms of the Goldman-Turaev bracket.
method Combining combinatorial approach with geometric proof.
result Geometric proof of curve characterization conjecture.
New proof connects Kashiwara-Vergne equations to Goldman-Turaev Lie bialgebra.
problem Proving Kashiwara-Vergne equations from isomorphism of Lie bialgebras.
method Novel characterization of conjugacy classes in free Lie algebra via cyclic words.
result Automorphisms inducing isomorphisms in Goldman-Turaev Lie bialgebra satisfy Kashiwara-Vergne equations.
In this paper we show that, after completion in the I-adic topology, the Goldman bracket on the space spanned by homotopy classes of loops on a smooth, complex algebraic curve is a morphism of mixed Hodge structure. We prove similar statements for the natural action (defined by Kawazumi and Kuno) of the loops in X on p…
By introducing a refinement of the Goldman-Turaev Lie bialgebra, we interpret the divergence cocycle in the Kashiwara-Vergne problem and the Enomoto-Satoh obstructions for the surjectivity of the Johnson homomorphisms as some part of a regular homotopy version of the Turaev cobracket.
One of the more memorable moments of last summer's credit crunch came when the CFO of Goldman Sachs, David Viniar, announced in August that Goldman's flagship GEO hedge fund had lost 27% of its value since the start of the year. As Mr. Viniar explained, "We were seeing things that were 25-standard deviation moves, seve…
the main theorem gives a sufficient condition for a n elements of SL(2,R) to generate a free group.The idea behind it is to use a nonorientable version of the Dehn-Wolpert-Goldman twist and to sew it with the original representation of a free group to get representation of the closed surfase group and then to apply Gol…
Explicit computation of symplectic form for PGLn(R)-Hitchin component.
problem Symplectic structure of PGLn(R)-Hitchin component. method Atiyah-Bott-Goldman symplectic form and global coordinates.
result Coefficients of the symplectic form are constant.
Defines Fenchel-Nielsen coordinates for SL(3,C) representations.
problem No specific problem stated; coordinates defined for a new context.
method Introduced Fenchel-Nielsen coordinates for mSL(3,C) representations. result Relates to classical and generalized Fenchel-Nielsen coordinates.
The paper constructs a noncommutative bracket on surface groups and proves it's Hamiltonian.
problem Noncommutative Hamiltonian structures on surface groups.
method Double quasi Poisson bracket construction and noncommutative r-matrix formalism. result Noncommutative Hamiltonian structures on cyclic spaces of unbased loops.
Develops a method to define and characterize geodesics on hyperbolic surfaces.
problem Characterizing closed geodesics on hyperbolic surfaces without self-intersection.
method Constructive definition of the Goldman bracket using closed geodesics.
result Algebraic characterization of geodesics on hyperbolic surfaces.
Signed seminorms linked to real tropical spaces and matroids.
problem Understanding signed seminorms and their real tropicalizations.
method Introducing signed Goldman-Iwahori space, identifying it as inverse limit of real tropicalizations, and giving matroid-theoretic description.
result Signed seminorms identified as inverse limit of real tropicalizations of projective space.