arXiv research

Non-injectivity proven for trace map on character varieties.

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.

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

Study shows infinite volumes of moduli spaces for certain groups.

We use Bonahon-Wong's trace map to study character varieties of the once-punctured torus and of the 4-punctured sphere. We clarify a relationship with cluster algebra associated with ideal triangulations of surfaces, and we show that the Goldman Poisson algebra of loops on surfaces is recovered from the Poisson structu…

Study of measured laminations on surfaces using Newton polytopes and Poisson brackets.

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 …

For a compact oriented surface $Σ$ of genus $g$ with $n+1$ boundary components, the space $\mathfrak{g}(Σ)$ spanned by free homotopy classes of loops in $Σ$ carries the structure of a Lie bialgebra equipped with a natural decreasing filtration, whose structure morphisms are called the Goldman bracket and the (framed) T…

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…

Quantum trace map defined for 3-manifolds with torus boundaries.

The paper constructs a noncommutative bracket on surface groups and proves it's Hamiltonian.

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

The definition of the grafting operation for quasifuchsian groups is extended by Bromberg to all $b$-groups. Although the grafting maps are not necessarily continuous at boundary groups, in this paper, we show that the grafting maps take every "standard" convergent sequence to a convergent sequence. As a consequence of…

Goldman bracket distinguishes surface homeomorphisms.

3D quantum trace map connects 3-manifold quantizations.

Study on random representations of surface groups into SU(n), focusing on asymptotic expansions.

Propose a new 3d quantum trace map that agrees with Garoufalidis and Yu's construction and extends to certain manifolds with ideal triangulated boundaries.

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

The paper shows compatibility between two quantum maps for surfaces and 3-manifolds.

In this paper we consider the action of the mapping class group of a surface on the space of homomorphisms from the fundamental group of a surface into PSL(2,R). Goldman conjectured that when the surface is closed and of genus bigger than one, the action on non-Teichmuller connected components of the associated moduli …

Introduces a new model for mapping matrices to matrices, subsuming linear regression.

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.…

Study shows mapping class group action is ergodic on specific representations.

The author connects Poincaré embeddings to Reidemeister traces and diagonal maps.

New Lie algebras from knot homology.

In this paper by using Teichmuller theory of a sphere with four holes/orbifold points, we obtain a system of flat coordinates on the general affine cubic surface having a D_4 singularity at the origin. We show that the Goldman bracket on the geodesic functions on the four-holed/orbifold sphere coincides with the Etingo…

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…

The study finds all trace field degrees for Torelli group mappings.

Study 2-loop part of Johnson cokernel using trace map.

Study on heat trace expansion for thermoelastic Dirichlet-to-Neumann map.

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.

Quantum trace map defines invariants for knots and links, confirming a length conjecture.

Research characterizes traces of Sobolev mappings into manifolds, identifying analytical obstructions and proving surjectivity under certain conditions.

Quantum traces embed into quantum tori for surface skein algebras.

Defines a map connecting 3d-index and skein module.

Deroin and Tholozan's representations are mapped to complex projective space via action-angle coordinates.

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 provide some language for algebraic study of the mapping class groups for surfaces with non-connected boundary. As applications, we generalize our previous results on Dehn twists to any compact connected oriented surfaces with non-empty boundary. Moreover we embed the `smallest' Torelli group in the sense of Putman …

The purpose of this paper is to study the action of the mapping class group on the moduli space of representations of the fundamental group of a non-orientable surface into SU(2). The action is shown to be ergodic with respect to a natural measure. This measure is defined using the push-forward measure associated to a …

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.