arXiv subjects

Deblina Das

Publications and source records attributed to Deblina Das.

6 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 M\"obius 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

A structure theorem for centralizers of dilations in $QI(\mathbb{R}_{+})$

We study centralizers of dilations in the quasi-isometry group of the positive real line. We introduce an asymptotic invariant defined via coarsely dense sequences at infinity and establish a rigidity theorem for quasi-isometries that coarsely commute with a dilation. As an application, we identify the subgroup of the centralizer consisting of elements with non-empty asymptotic invariant and prove that it is naturally isomorphic to the multiplicative group of positive real numbers.

math.GR↗

arXiv2026

Non-injectivity of the trace map for character varieties

Given a closed oriented surface $\Sigma$ 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 $\pi_1(\Sigma)$ to $GL_n$. The construction is based on the Amitsur-Levitzki identity, together with a choice of words in a free subgroup of $\pi_1(\Sigma)$, 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↗

arXiv2025

Orderability and Asymptotic Structure of $\mathrm{QI}(\mathbb{R}^n)$

In this article, we study the algebraic and dynamical structure of certain normal subgroups of the quasi-isometry group of Euclidean spaces. We first consider the normal subgroup consisting of quasi-isometries that are asymptotically equal to the identity, and introduce a nested family of normal subgroups that distinguish different orders of sublinear deviation from the identity. We show that the centers of the resulting quotient groups are trivial. We further prove that these quotient groups are neither left-orderable nor locally indicable. We also introduce an asymptotic topology on the quasi-isometry group, yielding a natural metric structure on the quotient and providing a framework for studying large-scale invariants.

math.GT↗

arXiv2024

Lifting closed curves to finite covers of free groups

In this article, we show that given any integer $l\geq 2$, every closed curve $\gamma$ on the bouquet of $n$-circles $\Gamma$, admits a lift to a finite $l$-sheeted normal covering of $\Gamma$. Equivalently, identifying the free group $F_n$ of $n$ generators with the fundamental group of $\Gamma$, 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 $\Gamma$, together with a characterization, in terms of necessary and sufficient conditions, of when a closed curve $\gamma$ on $\Gamma$ lifts to these covers.

math.GT↗

arXiv2024

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 $\gamma$ 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↗