arXiv subjects

Kashyap Rajeevsarathy

Publications and source records attributed to Kashyap Rajeevsarathy.

At least 19 recordsLinked to original sources

arXiv2026

General primitivity in the mapping class group

For $g\geq 2$, let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g$. In this paper, we obtain necessary and sufficient conditions under which a given pseudo-periodic mapping class can be a root of another up to conjugacy. Using this characterization, the canonical decomposition of (non-periodic) mapping classes, and some known algorithms, we give an algorithm for determining the conjugacy classes of roots of arbitrary mapping classes. Furthermore, we derive realizable bounds on the degrees of roots of pseudo-periodic mapping classes in $\mathrm{Mod}(S_g)$, the Torelli group, the level-$m$ subgroup of $\mathrm{Mod}(S_g)$, and the commutator subgroup of $\mathrm{Mod}(S_2)$. In particular, we show that the highest possible (realizable) degree of a root of a pseudo-periodic mapping class $F$ is $3q(F)(g+1)(g+2)$, where $q(F)$ is a unique positive integer associated with the conjugacy class of $F$. Moreover, this bound is realized by a root of a power of a Dehn twist about a separating curve of genus $[g/2]$ in $S_g$, where $g\equiv 0,9 \pmod{12}$. Finally, for $g\geq 3$, we show that any pseudo-periodic mapping class having a nontrivial periodic component that is not the hyperelliptic involution, normally generates $\mathrm{Mod}(S_g)$. Consequently, we establish that $\mathrm{Mod}(S_g)$ is normally generated by a root of a bounding pair map or a root of a nontrivial power of a Dehn twist.

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arXiv2026

Generating the liftable mapping class groups of cyclic covers of spheres

For $g\geq 2$, let $\text{Mod}(S_g)$ be the mapping class group of closed orientable surface $S_g$ of genus $g$. In this paper, we derive a finite generating set for the liftable mapping class groups corresponding to finite-sheeted regular branched cyclic covers of spheres. As an application, we provide an algorithm to derive presentations of these liftable mapping class groups, and the normalizers and centralizers of periodic mapping classes corresponding to these covers. Furthermore, we determine the isomorphism classes of the normalizers of irreducible periodic mapping classes in $\text{Mod}(S_g)$. Moreover, we derive presentations for the liftable mapping class groups corresponding to covers induced by certain reducible periodic mapping classes. Consequently, we derive a presentation for the centralizer and normalizer of a reducible periodic mapping class in $\text{Mod}(S_g)$ of the highest order $2g+2$. As final applications of our results, we recover the generating sets of the liftable mapping class groups of the hyperelliptic cover obtained by Birman-Hilden and the balanced superelliptic cover obtained by Ghaswala-Winarski.

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arXiv2026

Infinite metacyclic subgroups of the mapping class group

For $g\geq 2$, let $\text{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g$. In this paper, we provide necessary and sufficient conditions for a pair of elements in $\text{Mod}(S_g)$ to generate an infinite metacyclic subgroup. In particular, we provide necessary and sufficient conditions under which a pseudo-Anosov mapping class generates an infinite metacyclic subgroup of $\text{Mod}(S_g)$ with a nontrivial periodic mapping class. As applications of our main results, we establish the existence of infinite metacyclic subgroups of $\text{Mod}(S_g)$ isomorphic to $\mathbb{Z}\rtimes \mathbb{Z}_m, \mathbb{Z}_n \rtimes \mathbb{Z}$, and $\mathbb{Z} \rtimes \mathbb{Z}$. Furthermore, we derive bounds on the order of a nontrivial periodic generator of an infinite metacyclic subgroup of $\text{Mod}(S_g)$ that are realized. Finally, we show that the centralizer of an irreducible periodic mapping class $F$ is either $\langle F\rangle$ or $\langle F\rangle \times \langle i\rangle$, where $i$ is a hyperelliptic involution.

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arXiv2026

On a family of pseudo-Anosov-like maps on the infinite ladder surface

Let $S_g$ be the closed surface of genus $g$, $\mathcal{L}$ be the infinite Jacob's ladder surface, and $\mathrm{Map}(S)$ denote the mapping class group of a surface $S$. Let $q_g:\mathcal{L}\to S_g$ be the regular infinite-sheeted cover with deck transformation group $\mathbb{Z}$. In this paper, we show the existence of ``pseudo-Anosov-like'' maps on $\mathcal{L}$ that arise as the lifts of Penner-type pseudo-Anosov maps on $S_g$ under the cover $q_g$. Furthermore, we establish that these lifts are topologically transitive, mixing, and support null recurrent dynamics. Moreover, we present concrete examples of infinite families of such maps on $\mathcal{L}$.

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arXiv2026

Baumslag-Solitar subgroups of the mapping class group

For $g\geq 2$ and nonzero integers $p,q$, let $\mathrm{Mod}(S_g)$ be the mapping class group of a closed oriented surface $S_g$ of genus $g$, and let $\mathrm{BS}(p,q)$ be the Baumslag-Solitar group. We provide necessary and sufficient conditions under which two mapping classes $G,F\in \mathrm{Mod}(S_g)$ generate a subgroup isomorphic to $\mathrm{BS}(p,q)$. In particular, if $\mathrm{BS}(p,q)$ embeds in $\mathrm{Mod}(S_g)$, then $|p|=|q|$ and $G,F$ are reducible mapping classes of infinite order. We also construct subgroups isomorphic to $\mathrm{BS}(p,p)$ and $\mathrm{BS}(p,-p)$ for $p>1$. Finally, we show that every infinite metacyclic subgroup and every Baumslag-Solitar subgroup of $\mathrm{Mod}(S_g)$ lifts to an isomorphic subgroup of $\mathrm{Diff}^+(S_g)$, that is, these subgroups satisfy the generalized Nielsen realization.

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arXiv2026

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.

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arXiv2026

Teichmüller isometries induced by certain irreducible periodic mapping classes

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface of genus $g \geq 2$, and let $\mathrm{Teich}(S_g)$ be the Teichmüller space of $S_g$. In this paper, we provide an algorithm for determining the isometries induced in $\mathrm{Teich}(S_g)$ by certain irreducible periodic mapping classes in Fenchel-Nielsen coordinates. As a demonstration of this method, we provide a description of the isometry induced by periodic mapping class of order $4g+2$ and also derive some of its geometric properties.

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arXiv2025

Fenchel-Nielsen coordinates of the branch loci of cyclic actions

Let $S_g$ be a closed, connected, and oriented smooth surface of genus $g\geq 2$. Let the mapping class group of $S_g$ be denoted by $\mathrm{Mod}(S_g)$ and the Teichmüller space of $S_g$ by $\mathrm{Teich}(S_g)$. It is known that $\mathrm{Mod}(S_g)$ acts by isometries on $\mathrm{Teich}(S_g)$ with respect to the Weil-Petersson metric. In this paper, we develop algorithms to describe the Fenchel-Nielsen coordinates of fixed points of the actions of certain finite cyclic subgroups of $\mathrm{Mod}(S_g)$ on $\mathrm{Teich}(S_g)$. As applications of these algorithms, we compute the Fenchel-Nielsen coordinates of the fixed points of three cyclic subgroups of orders $10$, $8$, and $4$, in $\mathrm{Mod}(S_2)$.

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arXiv2025

Liftability of periodic mapping classes under alternating covers

Let $S_g$ be the closed orientable surface of genus $g \geq 2$, and let $\mathrm{Mod}(S_g)$ be the mapping class group of $S_g$. Let $A_n$ denote the alternating group on $n$ letters. We derive necessary and sufficient conditions under which a periodic mapping class has a conjugate that lifts under the branched cover $S_g \to S_g/A_n$ induced by an action of $A_n$ on $S_g$. This provides a classification of the subgroups of $\mathrm{Mod}(S_g)$ that are isomorphic to $A_n \rtimes \mathbb{Z}_m$, up to a certain equivalence that we call weak conjugacy. As an application, we show that for $n \geq 7$, such a subgroup of $\mathrm{Mod}(S_g)$ cannot have an irreducible periodic mapping class. Furthermore, we show that for $n \geq 5$ and $n \neq 6$, if the order of such a subgroup is greater than $5g-5$, then $m \leq 26$. Moreover, for $g \geq 2$ and $n \geq 5$, we establish that there exists no subgroup of $\mathrm{Mod}(S_g)$ that is isomorphic to $A_n \rtimes \mathbb{Z}$, where the $\mathbb{Z}$-component is generated by a power of a Dehn twist. Finally, we provide a complete classification of the weak conjugacy classes of such subgroups in $\mathrm{Mod}(S_{10})$ and $\mathrm{Mod}(S_{11})$.

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arXiv2025

Generating the liftable mapping class groups of regular cyclic covers

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface of genus $g \geq 1$, and let $\mathrm{LMod}_{p}(X)$ be the liftable mapping class group associated with a finite-sheeted branched cover $p:S \to X$, where $X$ is a hyperbolic surface. For $k \geq 2$, let $p_k: S_{k(g-1)+1} \to S_g$ be the standard $k$-sheeted regular cyclic cover. In this paper, we show that $\{\mathrm{LMod}_{p_k}(S_g)\}_{k \geq 2}$ forms an infinite family of self-normalizing subgroups in $\mathrm{Mod}(S_g)$, which are also maximal when $k$ is prime. Furthermore, we derive explicit finite generating sets for $\mathrm{LMod}_{p_k}(S_g)$ for $g \geq 3$ and $k \geq 2$, and $\mathrm{LMod}_{p_2}(S_2)$. For $g \geq 2$, as an application of our main result, we also derive a generating set for $\mathrm{LMod}_{p_2}(S_g) \cap C_{\mathrm{Mod}(S_g)}(ι)$, where $C_{\mathrm{Mod}(S_g)}(ι)$ is the centralizer of the hyperelliptic involution $ι\in \mathrm{Mod}(S_g)$. Let $\mathcal{L}$ be the infinite ladder surface, and let $q_g : \mathcal{L} \to S_g$ be the standard infinite-sheeted cover induced by $\langle h^{g-1} \rangle$ where $h$ is the standard handle shift on $\mathcal{L}$. As a final application, we derive a finite generating set for $\mathrm{LMod}_{q_g}(S_g)$ for $g \geq 3$.

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arXiv2025

Estimating the distances between hyperbolic structures in the moduli space

Let $\mathrm{Mod}(S_g)$ be 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 $\mathrm{Fix}(H)$ be the set of all fixed points induced by the action of $H$ on the Teichmüller space $\mathrm{Teich}(S_g)$ of $S_g$. This paper provides a method to estimate the distance between the unique fixed points of certain irreducible cyclic actions on $S_g$. We begin by deriving an explicit description of a pants decomposition of $S_g$, the length of whose curves are bounded above by the Bers' constant. To obtain the estimate, our method then uses the quasi-isometry between $\mathrm{Teich}(S_g)$ and the pants graph $\mathcal{P}(S_g)$.

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arXiv2024

Liftable mapping class groups of regular abelian covers

Let $S_g$ be the closed oriented surface of genus $g \geq 0$, and let $\mathrm{Mod}(S_g)$ be the mapping class group of $S_g$. For $g\geq 2$, we develop an algorithm to obtain a finite generating set for the liftable mapping class group $\mathrm{LMod}_p(S_g)$ of a regular abelian cover $p$ of $S_g$. A key ingredient of our method is a result that provides a generating set of a group $G$ acting on a connected graph $X$ such that the quotient graph $X/G$ is finite. As an application of our algorithm, when $k$ is prime, we provide a finite generating set for $\mathrm{LMod}_{p_k}(S_2)$ for cyclic cover $p_k:S_{k+1}\to S_2$. Using the Birman-Hilden theory, when $k=2,3$ and $g=2$, we also obtain a finite generating set for the normalizer of the Deck transformation group of $p_k$ in $\mathrm{Mod}(S_{k+1})$. We conclude the paper with an application of our algorithm that gives a finite generating set for $\mathrm{LMod}_p(S_2)$, where $p:S_5\to S_2$ is a cover with deck transformation group isomorphic to $\mathbb{Z}_2\oplus \mathbb{Z}_2$.

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arXiv2023

Alternating and symmetric actions on surfaces

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface of genus $g \geq 2$. In this article, we derive necessary and sufficient conditions under which two torsion elements in $\mathrm{Mod}(S_g)$ will have conjugates that generate a finite symmetric or an alternating subgroup of $\mathrm{Mod}(S_g)$. Furthermore, we characterize when an involution would lift under the branched cover induced by an alternating action on $S_g$. Moreover, up to conjugacy, we derive conditions under which a given periodic mapping class is contained in a symmetric or an alternating subgroup of $\mathrm{Mod}(S_g)$. In particular, we show that symmetric or alternating subgroups can not contain irreducible mapping classes and hyperelliptic involutions. Finally, we classify the symmetric and alternating actions on $S_{10}$ and $S_{11}$ up to a certain equivalence we call weak conjugacy.

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arXiv2022

Factoring periodic maps into Dehn twists

Let $\text{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g \geq 1$. In this paper, we develop various methods for factoring periodic mapping classes into Dehn twists, up to conjugacy. As applications, we develop methods for factoring certain roots of Dehn twists as words in Dehn twists. We will also show the existence of conjugates of periodic maps of order $4g$ and $4g+2$, for $g\geq 2$, whose product is pseudo-Anosov.

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arXiv2022

Metacyclic actions on surfaces

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$. In this paper, we derive necessary and sufficient conditions under which two torsion elements in $\mathrm{Mod}(S_g)$ will have conjugates that generate a finite metacyclic subgroup of $\mathrm{Mod}(S_g)$. This yields a complete solution to the problem of liftability of periodic mapping classes under finite cyclic covers. As applications of the main result, we show that $4g$ is a realizable upper bound on the order of a non-split metacyclic action on $S_g$ and this bound is realized by the action of a dicyclic group. Moreover, we give a complete characterization of the dicyclic subgroups of $\mathrm{Mod}(S_g)$ up to a certain equivalence that we will call weak conjugacy. Furthermore, we show that every periodic mapping class in a non-split metacyclic subgroup of $\mathrm{Mod}(S_g)$ is reducible. We provide necessary and sufficient conditions under which a non-split metacyclic action on $S_g$ factors via a split metacyclic action. Finally, we provide a complete classification of the weak conjugacy classes of the finite non-split metacyclic subgroups of $\mathrm{Mod}(S_{10})$ and $\mathrm{Mod}(S_{11})$.

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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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arXiv2021

Split metacyclic actions on surfaces

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$. In this paper, we derive necessary and sufficient conditions under which two torsion elements in $\mathrm{Mod}(S_g)$ will have conjugates that generate a finite split non-abelian metacyclic subgroup of $\mathrm{Mod}(S_g)$. As applications of the main result, we give a complete characterization of the finite dihedral and the generalized quaternionic subgroups of $\mathrm{Mod}(S_g)$ up to a certain equivalence that we will call weak conjugacy. Furthermore, we show that any finite-order mapping class whose corresponding orbifold is a sphere, has a conjugate that lifts under certain finite-sheeted regular cyclic covers of $S_g$. Moreover, for $g \geq 5$, we show the existence of an infinite dihedral subgroup of $\mathrm{Mod}(S_g)$ that is generated by an involution and a root of a bounding pair map of degree $3$. Finally, we provide a complete classification of the weak conjugacy classes of the non-abelian finite split metacyclic subgroups of $\mathrm{Mod}(S_3)$ and $\mathrm{Mod}(S_5)$. We also describe nontrivial geometric realizations of some of these actions.

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arXiv2021

Liftable mapping class groups of regular cyclic covers

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface of genus $g \geq 1$. For $k \geq 2$, we consider the standard $k$-sheeted regular cover $p_k: S_{k(g-1)+1} \to S_g$, and analyze the liftable mapping class group $\mathrm{LMod}_{p_k}(S_g)$ associated with the cover $p_k$. In particular, we show that $\mathrm{LMod}_{p_k}(S_g)$ is the stabilizer subgroup of $\mathrm{Mod}(S_g)$ with respect to a collection of vectors in $H_1(S_g,\mathbb{Z}_k)$, and also derive a symplectic criterion for the liftability of a given mapping class under $p_k$. As an application of this criterion, we obtain a normal series of $\mathrm{LMod}_{p_k}(S_g)$, which generalizes a well known normal series of congruence subgroups in $\mathrm{SL}(2,\mathbb{Z})$. Among other applications, we describe a procedure for obtaining a finite generating set for $\mathrm{LMod}_{p_k}(S_g)$ and examine the liftability of certain finite-order and pseudo-Anosov mapping classes.

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