arXiv research

Goldman symplectic form and complex structure compatible on $\mathrm{SL}(3,\mathbb R)$ Hitchin component.

Study describes moduli of quaternionic hyperbolic triples of points.

Let M be a compact surface of negative Euler characteristic and let C(M) be the deformation space of convex real projective structures on M. For every choice of pants decomposition for M, there is a well known parameterization of C(M) known as the Goldman parameterization. In this paper, we study how some geometric pro…

The paper studies the center of the Goldman Lie algebra and its properties.

Goldman bracket distinguishes surface homeomorphisms.

Criteria for loop separability on surfaces using Goldman bracket.

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.…

Let $Γ$ be a lattice in a connected semisimple Lie group $G$ with trivial center and no compact factors. We introduce a volume invariant for representations of $Γ$ into $G$, which generalizes the volume invariant for representations of uniform lattices introduced by Goldman. Then, we show that the maximality of this vo…

New Lie algebras from knot homology.

We construct a new Riemannian metric on Goldman space $\mathcal{B}(S)$, 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 Teichm$\ddot{u}$ller space, embe…

Classifies invariant measures on specific character varieties.

In the eighties Goldman discovered a Lie algebra structure on the vector space generated by the free homotopy classes of oriented curves on an oriented surface. The Lie bracket [a,b] is defined as the signed sum over the intersection points of a and b of the loop product of at the intersection points. If one of the cla…

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.

We show the set of faithful representations of a closed orientable hyperbolic surface group is dense in both irreducible components of the PSL(2,K) representation variety, where K is the field of real or complex numbers, answering a question of W. Goldman. We also prove the existence of faithful representations into PU…

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.

Decomposes Goldman-Turaev Lie bialgebra via cutting a 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 $\mathrm{PSL}_n(\mathbb{R})$-Hitchin-Riemann moduli space.

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.

Paper describes a pseudo-Kähler structure on a specific Hitchin component.

Study shows infinite volumes of moduli spaces for certain groups.

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.

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.

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 $\mathrm{PSL}(2,\mathbb{C})$-character variety, which allows to evaluate explicitly the pullback …

Geometric proof of curve characterization using loop-bundles.

Goldman parametrizes the $\mathrm{PSL}_3(\mathbb{R})$-Hitchin component of a closed oriented hyperbolic surface of genus $g$ by $16g-16$ parameters. Among them, $10g-10$ coordinates are canonical. We prove that the $\mathrm{PSL}_3(\mathbb{R})$-Hitchin component equipped with the Atiyah-Bott-Goldman symplectic form admi…

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…

We explore the dynamics of the action of the mapping class group in genus 2 on the PSL(2,R)-character variety. We prove that this action is ergodic on the connected components of Euler class 1 and -1, as it was conjectured by Goldman. In the connected component of Euler class 0 there are two invariant open subsets, on …

Defines Fenchel-Nielsen coordinates for SL(3,C) representations.

We study the action of the mapping class group Mod(S) on the boundary dQ of quasifuchsian space Q. Among other results, Mod(S) is shown to be topologically transitive on the subset C in dQ of manifolds without a conformally compact end. We also prove that any open subset of the character variety X(pi_1(S),SL(2,C)) inte…

Develops a method to define and characterize geodesics on hyperbolic surfaces.

Study examines Hilbert area of inscribed polygons in projective geometry.

Signed seminorms linked to real tropical spaces and matroids.

Let $Σ_{\infty, 1}$ be the inductive limit of compact oriented surfaces with one boundary component. We prove the center of the Goldman Lie algebra of the surface $Σ_{\infty,1}$ is spanned by the constant loop. A similar statement for a closed oriented surface was conjectured by Chas and Sullivan, and proved by Etingof…

For the moduli space of unmarked convex $\mathbb{RP}^2$ structures on the surface $S_{g,m}$ with negative Euler characteristic, we investigate the subsets of the moduli space defined by the notions like boundedness of projective invariants, area, Gromov hyperbolicity constant, quasisymmetricity constant etc. These subs…

New method computes automorphisms of surface groups using skein algebras.

In this paper, we describe a surprising link between the theory of the Goldman-Turaev Lie bialgebra on surfaces of genus zero and the Kashiwara-Vergne (KV) problem in Lie theory. Let $Σ$ be an oriented 2-dimensional manifold with non-empty boundary and $\mathbb{K}$ a field of characteristic zero. The Goldman-Turaev Lie…

The Hitchin component Hit_n(S) of a closed surface S is a preferred component of the character variety X_PSL_n(R)(S) consisting of homomorphisms from the fundamental group pi_1(S) to the Lie group PSL_n(R)(S), whose elements enjoy remarkable geometric and dynamical properties. We consider a certain type of deformations…

The Margulis invariant is a function defined on a group of Lorentzian transformations $G$ acting on Minkowski space $\R^{2,1}$, that contains no elliptic elements. The spectrum of $G$ is the sequence of values of the Margulis invariant for all its elements. If the underlying linear group of $G$ is fixed, Drumm and Gold…