Searcharxiv⌕ Search

arXiv subjects

Arpan Kabiraj

Publications and source records attributed to Arpan Kabiraj.

12 recordsLinked to original sources
arXiv2026

Poisson bracket of trace functions on the Fuchsian locus and Wolpert's formulas

We develop a systematic method for computing traces of products of Möbius transformations associated with oriented geodesics on a hyperbolic surface. The method is based on a normalization of matrices in ${SL}_2(\mathbb R)$ which expresses trace identities in terms of hyperbolic lengths, intersection angles, and signed distances along geodesics. Using these trace computations together with Goldman's description of the Atiyah-Bott-Goldman symplectic form on the character variety, we derive explicit geometric formulas for the Poisson brackets of trace functions associated with closed geodesics. More precisely, we express the Poisson bracket of two trace functions and the iterated Poisson bracket of three trace functions in terms of the hyperbolic lengths of the corresponding geodesics, their intersection angles, and the signed distances between intersection points. The results naturally lead to a unified perspective for revisiting Wolpert's cosine and sine formulas and deriving new proofs of them.

math.GT↗
arXiv2026

Simple lift of non-simple closed curves

Given a compact, oriented surface $S$ of finite genus and finitely many boundary components, we provide examples of finite covers $\tilde{S}$ of $S$ and non-simple closed curves $γ$ on $S$ which lifts to simple closed curves on $\tilde{S}$. In particular, given any positive integer $n\geq 2$, we construct explicit non-simple closed curves on $S$ which has a simple lift to a degree $n$ cover of $S$.

math.GT↗
arXiv2026

Non-injectivity of the trace map for character varieties

Given a closed oriented surface $Σ$ of genus at least two, the Goldman trace map defines a function from the vector space generated by the free homotopy classes of oriented closed curves to the Poisson algebra of regular functions on the $G$-character variety where $G$ is a reductive (real or complex) linear Lie group. In this article, we prove that this map is never injective. For each $n$, we construct an explicit nonzero element of the vector space whose associated trace function vanishes on every homomorphism from $π_1(Σ)$ to $GL_n$. The construction is based on the Amitsur-Levitzki identity, together with a choice of words in a free subgroup of $π_1(Σ)$, ensuring that no cancellation occurs at the level of free homotopy classes. This gives a uniform family of explicit kernel elements, proving Goldman's predicted non-injectivity of the trace map in arbitrary rank.

math.GT↗
arXiv2026

Lifting closed curves to finite covers of free groups

In this article, we show that given any integer $l\geq 2$, every closed curve $γ$ on the bouquet of $n$-circles $Γ$, admits a lift to a finite $l$-sheeted normal covering of $Γ$. Equivalently, identifying the free group $F_n$ of $n$ generators with the fundamental group of $Γ$, this statement asserts that $F_n$ is a union of $ l$-index normal subgroups for any $l\geq 2.$ The proof proceeds by explicitly constructing families of $l$-sheeted normal coverings of $Γ$, together with a characterization, in terms of necessary and sufficient conditions, of when a closed curve $γ$ on $Γ$ lifts to these covers.

math.GT↗
arXiv2023

Explicit Lie bracket of closed geodesics on a hyperbolic surface with applications

In this note we develop a tool box of non-Euclidean plane geometry methods that yield a constructive way to define in terms of closed geodesics the Goldman bracket on deformation classes of closed, directed curves. We use this construction to algebraically characterize closed geodesics without self-intersection on hyperbolic surfaces.

math.GT↗
arXiv2022

Center of Poisson and skein algebras associated to loops on surfaces

We discuss and develop a systematic method to compute the Poisson center (Casimir) of various Poisson algebras associated to loops on orientable surfaces (possibly with boundary and punctures) introduced by Goldman and Wolpert in 80's while studying Thurston's earthquakes deformations. Our computation extends a result of Etingof to all finite type hyperbolic surfaces. We use these methods to compute the center of various skein algebras introduced by Turaev for the quantization of these Poisson algebras. As another application of our results we compute the center of homotopy skein algebra introduced by Hoste and Przytycki.

math.GT↗
arXiv2020

The Lie bracket of undirected curves on a surface

A Lie bracket defined on the linear span of the free homotopy classes of undirected closed curves was discovered in stages passing through Thurston's earthquake deformations, Wolpert's corresponding calculations with Hamiltonian vector fields and Goldman's algebraic treatment of the latter leading to a Lie bracket on the span of directed closed curves. The purpose of this work is to deepen the understanding of the former Lie bracket which will be referred to as the Thurston-Wolpert-Goldman Lie bracket or, briefly, the TWG bracket. We give a local direct geometric definition of the TWG bracket and use this geometric point of view to prove three results: firstly, the center of the TWG-bracket is the Lie sub algebra generated by the class of the trivial loop and the classes of loops parallel to boundary components or punctures; secondly the analogous result hold for the centers of the universal enveloping algebra and of the symmetric algebra determined by the TWG Lie algebra; and thirdly, in terms of the natural basis, the TWG bracket of two non-central curves is always a linear combination of non-central curves. We also give a brief and more illuminating proof of a known result, namely, the TWG bracket counts intersection. We conclude by discussing substantial computer evidence suggesting an unexpected and strong conjectural statement relating the intersection structure of curves and the TWG bracket, namely, if the TWG bracket of two distinct undirected curves is zero then these curve classes have disjoint representatives. The main tools are basic hyperbolic geometry and Thurston's earthquake theory.

math.GR↗
arXiv2018

Elementary equivalence in Artin groups of finite type

Irreducible Artin groups of finite type can be parametrized via their associated Coxeter diagrams into six sporadic examples and four infinite families, each of which is further parametrized by the natural numbers. Within each of these four infinite families, we investigate the relationship between elementary equivalence and isomorphism. For three out of the four families, we show that two groups in the same family are equivalent if and only if they are isomorphic; a positive, but weaker, result is also attained for the fourth family. In particular, we show that two braid groups are elementarily equivalent if and only if they are isomorphic. As a consequence of our work, we prove that there are infinitely many elementary equivalence classes of irreducible Artin groups of finite type. We also show that mapping class groups of closed surfaces - a geometric analogue of braid groups - are elementarily equivalent if and only if they are isomorphic.

math.GR↗
arXiv2018

Equal angles of intersecting geodesics for every hyperbolic metric

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 hyperbolic metric, the angles corresponding to the intersection points are equal to each other. As a consequence, we obtain an alternative proof of a theorem of Chas, i.e. if one of the free homotopy classes contains a simple representative then the geometric intersection number and the number of terms (counted with multiplicity) in the Goldman bracket are the same.

math.GT↗
arXiv2017

Goldman bracket and length equivalent filling curves

A pair of distinct free homotopy classes of closed curves in an orientable surface $F$ with negative Euler characteristic is said to be length equivalent if for any hyperbolic structure on $F$, the length of the geodesic representative of one class is equal to the length of the geodesic representative of the other class. Suppose $α$ and $β$ are two intersecting oriented closed curves on $F$ and $P$ and $Q$ are any two intersection points between them. If the two terms $\langleα*_Pβ\rangle$ and $\langleα*_Qβ\rangle$ in $[\langleα\rangle,\langleβ\rangle]$, the Goldman bracket between them, are the same, then we construct infinitely many pairs of length equivalent curves in $F.$ These pairs correspond to the terms of the Goldman bracket between a power of $α$ and $β$. As a special case, our construction shows that given a self-intersecting geodesic $α$ of $F$ and any self-intersection point $P$ of $α$, we get a sequence of such pairs. Furthermore if $α$ is a filling curve then these pairs are also filling.

math.GT↗
arXiv2015

Center Of the Goldman algebra

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.

math.GT↗