arXiv subjects

Shiv Parsad

Publications and source records attributed to Shiv Parsad.

9 recordsLinked to original sources

arXiv2026

Minima of geodeics length functions for non-uniform filling

Kerckhoff proved that the geodesic length function $\ell_\Omega$ of a filling $\Omega$ on $S_g$ attains a unique minimum in Teichm\"uller space. Recent work of Ernesto Girondo et al. computed these minima for uniform fillings using the algebraic machinery of dessins d'enfants and Grothendieck-Belyi surfaces. We present an elementary optimization approach for $4$-regular topological uniform fillings, bypassing this framework. Furthermore, we analyze two special classes of non-uniform $4$-regular fillings using fat graphs and optimization techniques. We explicitly compute their minima and prove that in both classes, the minimum of these length functions is attained at a triangle surface.

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arXiv2025

Some analogues of isoperimetric inequality

The discrete isoperimetric inequality states that among all n -gons with a fixed area, the regular n -gon has the least perimeter. We prove analogues of the discrete isoperimetric inequality (involving circumradius or inradius) for cyclic and tangential polygons in hyperbolic geometry, considering both single and multiple polygons. Furthermore, we establish two versions of the isoperimetric inequality for multiple polygons in hyperbolic geometry with some restriction on their area or perimeter.

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arXiv2025

Filling systems of maximum size

Let $S_g$ be a closed orientable surface of genus $g\geq 2$. A collection $\Omega = \{ \gamma_1, \dots, \gamma_s\}$ of pairwise non-homotopic simple closed curves on $S_g$ such that $\gamma_i$ and $\gamma_j$ are in minimal position, is called a \emph{filling system} or a \emph{filling} of $S_g$ if the complement $S_g\setminus \Omega$ is a disjoint union of $b$ topological discs for some $b\geq 1$. The \emph{size} of a filling system is defined as the number of its elements. We prove that the maximum size of a filling system on $S_g$ with $ 1 \leq b \leq 2g-2$ boundary components is $2g+b-1$. Furthermore, we give a lower bound on mapping class group orbits of filling systems of maximum size with $ 1 \leq b \leq g-2$ boundary components.

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arXiv2024

Angle Parametrization of Teichmüller space and hyperelliptic surfaces

Let $S_g$ be a closed orientable surface of genus $g \geq 2$, and let $\mathcal{T}_g$ be the Teichmüller space of $S_g$. Let $\mathcal{H}_g$ denotes the space of all hyperelliptic surfaces of genus $g$. For $g\geq 3$, we have proved that $\mathcal{T}_g$ can be parametrized by $6g-5$ angle parameters. We also prove that for $g\geq 2$, $\mathcal{H}_g$ can be parametrized by $4g-2$ angle parameters.

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arXiv2021

Geometric realizations of cyclic actions on surfaces -- II

Let $\mathrm{Mod}(S_g)$ denote the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$. Given a finite subgroup $H$ of $\mathrm{Mod}(S_g)$, let $\mathrm{Fix}(H)$ denote the set of fixed points induced by the action of $H$ on the Teichmüller space $\mathrm{Teich}(S_g)$. When $H$ is cyclic with $|H| \geq 3$, we show that $\mathrm{Fix}(H)$ admits a decomposition as a product of two-dimensional strips at least one of which is of bounded width. For an arbitrary $H$ with at least one generator of order $\geq 3$, we derive a computable optimal upper bound for the restriction $\mathrm{sys} : \mathrm{Fix}(H) \to \mathbb{R}^+$ of the systole function. Furthermore, we show that in such a case, $\mathrm{Fix}(H)$ is not symplectomorphic to the Euclidean space of the same dimension. Finally, we apply our theory to recover three well-known results, namely: (a) Harvey's result giving the dimension of $\mathrm{Fix}(H)$, (b) Gilman's result that $H$ is irreducible if and only if the corresponding orbifold is a sphere with three cone points, and (c) the Nielsen realization theorem for cyclic groups.

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arXiv2018

Conjugation Orbits of Loxodromic Pairs in SU(n,1)

Let ${\bf H}_{\mathbb C}^n$ be the $n$-dimensional complex hyperbolic space and ${\rm SU}(n,1)$ be the (holomorphic) isometry group. An element $g$ in ${\rm SU}(n,1)$ is called loxodromic or hyperbolic if it has exactly two fixed points on the boundary $\partial {\bf H}_{\mathbb C}^n$. We classify ${\rm SU}(n,1)$ conjugation orbits of pairs of loxodromic elements in ${\rm SU}(n,1)$.

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arXiv2018

Filling systems on surfaces

Let $F_g$ be a closed orientable surface of genus $g$. A set $Ω= \{ γ_1, \dots, γ_s\}$ of pairwise non-homotopic simple closed curves on $F_g$ is called a \emph{filling system} or simply a \emph{filling} of $F_g$, if $F_g\setminus Ω$ is a union of $b$ topological discs for some $b\geq 1$. A filling system is called \emph{minimal}, if $b=1$. The \emph{size} of a filling is defined as the number of its elements. We prove that the maximum size of a filling of $F_g$ with $b$ complementary discs is $2g+b-1$. Next, we show that for $g\geq 2, b\geq 1\text{ with }(g,b)\neq (2,1)$ (resp. $(g,b)=(2,1)$) and for each $2\leq s\leq 2g+b-1$ (resp. $3\leq s\leq 2g+b-1$), there exists a filling of $F_g$ of size $s$ with $b$ complementary discs. Furthermore, we study geometric intersection number of curves in a minimal filling. For $g\geq 2$, we show that for a minimal filling $Ω$ of size $s$, the \emph{geometric intersection numbers} satisfy $\max \left\lbrace i(γ_i, γ_j)| i\neq j\right\rbrace\leq 2g-s+1$, and for each such $s$ there exists a minimal filling $Ω=\left\lbrace γ_1, \dots, γ_s \right\rbrace$ such that $\max\left\lbrace i(γ_i, γ_j) | i\neq j\right\rbrace = 2g-s+1$.

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arXiv2017

Geometric realizations of cyclic actions on surfaces

Let $ \text{Mod}(S_g)$ denote the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$, and let $f\in \text{Mod}(S_g)$ be of finite order. We give an inductive procedure to construct an explicit hyperbolic structure on $S_g$ that realizes $f$ as an isometry. In other words, this procedure yields an explicit solution to the Nielsen realization problem for cyclic subgroups of $ \text{Mod}(S_g)$. Furthermore, we give a purely combinatorial perspective by showing how certain finite order mapping classes can be viewed as fat graph automorphisms. As an application of our realizations, we determine the sizes of maximal reduction systems for certain finite order mapping classes. Moreover, we describe a method to compute the image of finite order mapping classes and the roots of Dehn twists, under the symplectic representation $Ψ: \text{Mod}(S_g) \to \text{Sp}(2g; \mathbb{Z})$.

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arXiv2017

On Fenchel-Nielsen Coordinates of Surface Group Representations into SU(3,1)

Let $Σ_g$ be a compact, connected, orientable surface of genus $g \geq 2$. We ask for a parametrization of the discrete, faithful, totally loxodromic representations in the deformation space ${\rm Hom}(π_1(Σ_g), {\rm SU}(3,1))/{\rm SU}(3,1)$. We show that such a representation, under some hypothesis, can be determined by $30g-30$ real parameters.

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