arXiv subjects

Debattam Das

Publications and source records attributed to Debattam Das.

7 recordsLinked to original sources

arXiv2026

Reversibility and its asymptotic counting in Picard group

We investigate reversible elements in the Picard modular group $\mathrm{PSL}(2,\mathbb{Z}[i])$. We show that reversibility coincides with strong reversibility for Kleinian groups, in particular for the Picard group. We classify reversible elements in the Picard group and characterize loxodromic reversible elements up to conjugacy. We prove that each such conjugacy class contains exactly eight special representatives. We also obtain asymptotic estimates for the number of reversible conjugacy classes with bounded trace.

math.GR↗

arXiv2026

Minimal Filling pair of non orientable surfaces

For $g\ge 3$, let $N_g$ denote the non-orientable surface of genus $g$. In this article, we establish the existence of filling pairs on $N_g$ that intersect minimally by construction using the theory of fat graphs. The mapping class group $\mathrm{Mod}(N_g)$ acts on the set of all such filling pairs. We count $\mathrm{Mod}(N_g)$-orbits of this action by providing both lower and upper bounds. Furthermore, we show that both bounds grow super-exponentially with $g$ using graph cohomology. Also, we investigate the lengths of minimally intersecting filling pairs on hyperbolic non-orientable surfaces $X$ in moduli space $\mathcal{M}_g$ of $N_g$. We define a function $\mathcal{F}_g:\mathcal{M}_g\to\mathbb{R}_{>0}$, where for $X\in \mathcal{M}_g$, the function $\mathcal{F}_g(X)$ is the shortest total length of a minimally intersecting filling pair on $X$. We determine its minimum $m_g$ and show that the set of minimizers is in bijection with the $\mathrm{Mod}(N_g)$-orbits of minimally intersecting filling pairs. We further extend $\mathcal{F}_g$ to $\mathcal{Y}_g$, defined by minimizing the length over all filling pairs, and show that $\mathcal{Y}_g$ attains the same minimum value as $\mathcal{F}_g$.

math.GT↗

arXiv2026

Combinatorial Structure of Inert and Ambiguous Classes in Modular Group

We study inert, and ambiguous conjugacy classes in the modular group $\mathrm{PSL}(2,\mathbb{Z})$ from a purely combinatorial perspective. Using word length in the free product representation $\mathbb{Z}_2 * \mathbb{Z}_3$ of the modular group, we obtain exact counting formulas and asymptotic growth rates for inert and ambiguous classes. Our results provide the first counting formulas for inert classes obtained independently of Sarnak's analytic trace-based methods, while also establishing a combinatorial framework for ambiguous classes.

math.GT↗

arXiv2025

Asymptotic growth of the number of Reciprocal Classes in the Hecke Groups

We estimate the asymptotic growth of reciprocal conjugacy classes in Hecke groups using their free product structure and word lengths of reciprocal elements. Our approach is different from other works in this direction and uses tools from basic probability theory.

math.GR↗

arXiv2024

Spherical growth of reciprocal classes in the Hecke Groups

Let $\Gamma_p$ denote the Hecke group where $p=2r$, $r>0$. Let $\mathcal{N}_l$ denote the set of conjugacy classes of reciprocal elements of word length $l$ in $\Gamma_p$. We prove that for $l \to \infty$, $$|\mathcal{N}_l| = \mathcal{O}\left(\left\lfloor \tfrac{l+1}{2} \right\rfloor^{s-1} \rho^{\left\lfloor \tfrac{l+1}{2} \right\rfloor} \right), $$ where $\mathcal O$ is the `big O', $\rho \in [\sqrt{2}, 2]$ is the unique positive real root of $$ p(x) = x^{r+1} - 2\sum_{j=1}^{r-1} x^{r-j} - 1, $$ and $s$ is the maximal multiplicity among the roots of $p(x)$. Our method relies on the free product structure of the Hecke group $\Gamma_p$, a combinatorial counting function, and recurrence relations derived from cyclically reduced representatives. We also derive that the growth rate of the primitive reciprocal classes of word length $l$ is in agreement with that of $\mathcal{N}_l$. This work generalizes previous results for odd $p$ and provides an explicit asymptotic bound for all Hecke groups.

math.GR↗

arXiv2024

Reversible and other generalised torsion elements in Seifert-fibered spaces

An element $a$ in a group $\Gamma$ is called \emph{reversible} if there exists $g \in \Gamma$ such that $gag^{-1}=a^{-1}$. The reversible elements are also known as `real elements' or `reciprocal elements' in literature. In this paper, we classify the reversible elements in Fuchsian groups, and use this classification to find all reversible elements in a Seifert-fibered group. We then apply the classification to the braid groups, particularly to the braid group on $3$ strands. We further study generalised 3-torsion elements in PSL(2,Z), and use this to analyse the existence of generalised 3-torsion elements in Seifert-fibered spaces in general, and braid groups on 3 strands in particular.

math.GT↗

arXiv2023

Reciprocity in the Hecke Groups

An element $g$ in a group $G$ is called \emph{reciprocal} if there exists $h \in G$ such that $g^{-1}=hgh^{-1}$. The reciprocal elements are also known as `real elements' or `reversible elements' in the literature. We classify the reciprocal elements and parametrize the reciprocal classes in the Hecke groups $\Gamma_p$ for $p\geq 3$. This generalizes a result by Sarnak for reciprocal elements in the modular group.

math.GR↗