arXiv subjects

Pankaj Kapari

Publications and source records attributed to Pankaj Kapari.

7 recordsLinked to original sources

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

Dehn quandles of surfaces and their bounded cohomology

We introduce new families of quandles that serve as invariants for classifying closed orientable surfaces. These families generalize the classical Dehn quandle and are defined, respectively, on isotopy classes of unoriented closed curves and on integral weighted multicurves. We establish their fundamental algebraic properties and construct a natural quandle covering that relates them. We then analyze their metric properties, showing that these quandles are unbounded with respect to the quandle metric. Next, we compute their second bounded quandle cohomology, proving it to be infinite-dimensional. We also establish a version of the Gromov Mapping Theorem, showing that the natural map from an abelian quandle extension onto the original quandle induces an injection on bounded quandle cohomology in every dimension. Finally, inspired by recent developments in quandle rings, we analyze idempotents in the integral quandle rings arising from the classical Dehn quandle of a surface.

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arXiv2025

Liftable mapping class groups of certain branched covers of torus

Let $S_{g,n}$ be a closed oriented hyperbolic surface of genus $g$ with $n$ marked points, with the understanding that $S_{g,0}=S_g$. Let $\mathrm{Mod}(S_{h,n})$ be the mapping class group of $S_{h,n}$ and $\mathrm{LMod}_p(S_{h,n})$ be the liftable mapping class group associated to a cover $p:S_g\to S_{h,n}$. For the cover $p_k:S_k\to S_{1,2}$, Ghaswala, in his PhD thesis, derived a finite presentation for $\mathrm{LMod}_{p_k}(S_{1,2})$ when $k=2,3,4$ and a finite generating set when $k=5,6$ using the Reidemeister-Schreier rewriting process. In this paper, we derive a finite generating set for $\mathrm{LMod}_{p_k}(S_{1,2})$ for all $k\geq 2$. In the process, we also prove that the kernel of the homology representation $\Psi:\mathrm{Mod}(S_{1,2})\to \mathrm{GL}_3(\Z)$ is normally generated by a Dehn twist about a separating simple closed curve, and it is free with a countable basis. We also provide an explicit countable basis for $\ker\Psi$ consisting of separating Dehn twists. As an application of Birman-Hilden theory, we provide a finite generating set for the normalizer of the Deck group of $p_k$ in $\mathrm{Mod}(S_k)$ when $k=2,3$. We conclude the paper by proving that $\mathrm{LMod}_{p_k}(S_{1,2})$ is maximal in $\mathrm{Mod}(S_{1,2})$ if and only if $k$ is prime.

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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

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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arXiv2022

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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arXiv2021

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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