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Tushar Kanta Naik

Publications and source records attributed to Tushar Kanta Naik.

14 recordsLinked to original sources
arXiv2026

Symmetry and Rigidity of Star-Shaped Coxeter Systems

We provide a complete description of the automorphism group $\Aut (W)$ of a Coxeter group $W$ admitting a star-shaped finite Coxeter diagram. We prove that each automorphism decomposes as a product of inner and diagram automorphisms, along with three additional types: transvections and two families of partial conjugations. Furthermore, we investigate the natural short exact sequence $1 \to \Inn (W) \to \Aut (W) \to \Out (W) \to 1$. Using Moussong's criteria for hyperbolicity, we show that these groups possess the $R_\infty$-property. Finally, we establish rigidity properties for these groups using known techniques and provide a solution to the isomorphism problem within the class of star-shaped Coxeter systems.

math.GR↗
arXiv2025

Linear representations, crystallographic quotients, and twisted conjugacy of virtual Artin groups

Virtual Artin groups were recently introduced by Bellingeri, Paris, and Thiel as broad generalizations of the well-known virtual braid groups. For each Coxeter graph $Γ$, they defined the virtual Artin group $VA[Γ]$, which is generated by the corresponding Artin group $A[Γ]$ and the Coxeter group $W[Γ]$, subject to certain mixed relations inspired by the action of $W[Γ]$ on its root system $Φ[Γ]$. There is a natural surjection $ \mathrm{VA}[Γ] \rightarrow W[Γ]$, with the kernel $PVA[Γ]$ representing the pure virtual Artin group. In this paper, we explore linear representations, crystallographic quotients, and twisted conjugacy of virtual Artin groups. Inspired from the work of Cohen, Wales, and Krammer, we construct a linear representation of the virtual Artin group $VA[Γ]$. As a consequence of this representation, we deduce that if $W[Γ]$ is a spherical Coxeter group, then $VA[Γ]/PVA[Γ]'$ is a crystallographic group of dimension $ |Φ[Γ]|$ with the holonomy group $W[Γ]$. We also classify the torsion elements in $VA[Γ]/PVA[Γ]'$ and determine precisely when two elements are conjugate in this group. Further, we investigate twisted conjugacy, and prove that each right-angled virtual Artin group admit the $R_\infty$-property.

math.GR↗
arXiv2023

Virtual planar braid groups and permutations

Twin groups and virtual twin groups are planar analogues of braid groups and virtual braid groups, respectively. These groups play the role of braid groups in the Alexander-Markov correspondence for the theory of stable isotopy classes of immersed circles on orientable surfaces. Motivated by the general idea of Artin and a recent work of Bellingeri and Paris \cite{BellingeriParis2020}, we obtain a complete description of homomorphisms between virtual twin groups and symmetric groups, which as an application gives us the precise structure of the automorphism group of the virtual twin group $VT_n$ on $n \ge 2$ strands. This is achieved by showing the existence of an irreducible right-angled Coxeter group $KT_n$ inside $VT_n$. As a by-product, it also follows that the twin group $T_n$ embeds inside the virtual twin group $VT_n$, which is an analogue of a similar result for braid groups.

math.GR↗
arXiv2023

Congruence subgroups and crystallographic quotients of small Coxeter groups

Small Coxeter groups are precisely the ones for which the Tits representation is integral, which makes the study of their congruence subgroups relevant. The symmetric group $S_n$ has three natural extensions, namely, the braid group $B_n$, the twin group $T_n$ and the triplet group $L_n$. The latter two groups are small Coxeter groups, and play the role of braid groups under the Alexander-Markov correspondence for appropriate knot theories, with their pure subgroups admitting suitable hyperplane arrangements as Eilenberg-MacLane spaces. In this paper, we prove that the congruence subgroup property fails for infinite small Coxeter groups which are not virtually abelian. As an application, we deduce that the congruence subgroup property fails for both $T_n$ and $L_n$ when $n \ge 4$. We also determine subquotients of principal congruence subgroups of $T_n$, and identify the pure twin group $PT_n$ and the pure triplet group $PL_n$ with suitable principal congruence subgroups. Further, we investigate crystallographic quotients of these two families of small Coxeter groups, and prove that $T_n /PT_n^{'}$, $T_n/T_n^{''}$ and $L_n /PL_n^{'}$ are crystallographic groups. We also determine crystallographic dimensions of these groups and identify the holonomy representation of $T_n/T_n^{''}$.

math.GR↗
arXiv2023

Commutator subgroups and crystallographic quotients of virtual extensions of symmetric groups

The virtual braid group $VB_n$, the virtual twin group $VT_n$ and the virtual triplet group $VL_n$ are extensions of the symmetric group $S_n$, which are motivated by the Alexander-Markov correspondence for virtual knot theories. The kernels of natural epimorphisms of these groups onto the symmetric group $S_n$ are the pure virtual braid group $VP_n$, the pure virtual twin group $PVT_n$ and the pure virtual triplet group $PVL_n$, respectively. In this paper, we investigate commutator subgroups, pure subgroups and crystallographic quotients of these groups. We derive explicit finite presentations of the pure virtual triplet group $PVL_n$, the commutator subgroup $VT_n^{'}$ of $VT_n$ and the commutator subgroup $VL_n^{'}$ of $VL_n$. Our results complete the understanding of these groups, except that of $VB_n^{'}$, for which the existence of a finite presentations is not known for $n \ge 4$. We also prove that $VL_n/PVL_n^{'}$ is a crystallographic group and give an explicit construction of infinitely many torsion elements in it.

math.GR↗
arXiv2022

Nilpotent Lie algebras with two centralizer dimensions over a finite field

A result of Barnea and Isaacs states that if $L$ is a finite dimensional nilpotent Lie algebra with exactly two distinct centralizer dimensions, then nilpotency class of $L$ is either $2$ or $3$. In this article, we classify all such finite dimensional $3$-step nilpotent Lie algebras over a finite field.

math.RA↗
arXiv2022

Structure and automorphisms of pure virtual twin groups

Study of stable isotopy classes of a finite collection of immersed circles without triple or higher intersections on closed oriented surfaces is considered as a planar analogue of virtual knot theory, a far reaching generalisation of classical knot theory. Recent works have established Alexander and Markov theorems in the planar setting. In the classical case, the role of groups is played by twin groups, a class of right-angled Coxeter groups. A new class of groups called virtual twin groups, that extends twin groups in a natural way, plays the role of groups in the virtual case. The virtual twin group $VT_n$ contains the pure virtual twin group $PVT_n$, a planar analogue of the pure Artin braid group. In this paper, we prove that the pure virtual twin group $PVT_n$ is an irreducible right-angled Artin group with trivial center and give it's precise presentation. We show that $PVT_n$ has a decomposition as an iterated semi-direct product of infinite rank free groups. We give a complete description of the automorphism group of $PVT_n$ and establish splitting of natural exact sequences of automorphism groups. As applications, we show that $VT_n$ is residually finite and $PVT_n$ has the $R_\infty$-property.

math.GR↗
arXiv2021

Nilpotent Lie Algebras of breadth type $(0,3)$

For a natural number $m$, a Lie algebra $L$ over a field $k$ is said to be of breadth type $(0, m)$ if the co-dimension of the centralizer of every non-central element is of dimension $m$. In this article, we classify finite dimensional nilpotent Lie algebras of breadth type $(0, 3)$ over $\mathbb F_q$ of odd characteristics up to isomorphism. We also give a partial classification of the same over finite fields of even characteristic, $\mathbb C$ and $\mathbb R$. We also discuss $2$-step nilpotent Camina Lie algebras.

math.RA↗
arXiv2020

Automorphisms of odd Coxeter groups

An odd Coxeter group $W$ is one which admits a Coxeter system $(W,S)$ for which all the exponents $m_{ij}$ are either odd or infinity. The paper investigates the family of odd Coxeter groups whose associated labeled graphs $\mathcal{V}_{(W,S)}$ are trees. It is known that two Coxeter groups in this family are isomorphic if and only if they admit Coxeter systems having the same rank and the same multiset of finite exponents. In particular, each group in this family is isomorphic to a group that admits a Coxeter system whose associated labeled graph is a star shaped tree. We give the complete description of the automorphism group of this group, and derive a sufficient condition for the splitting of the automorphism group as a semi-direct product of the inner and the outer automorphism groups. As applications, we prove that Coxeter groups in this family satisfy the $R_\infty$-property and are (co)-Hopfian. We compare structural properties, automorphism groups, $\R_\infty$-property and (co)-Hopfianity of a special odd Coxeter group whose only finite exponent is three with the braid group and the twin group.

math.GR↗
arXiv2020

Some remarks on twin groups

The twin group $T_n$ is a right angled Coxeter group generated by $n- 1$ involutions and having only far commutativity relations. These groups can be thought of as planar analogues of Artin braid groups. In this note, we study some properties of twin groups whose analogues are well-known for Artin braid groups. We give an algorithm for two twins to be equivalent under individual Markov moves. Further, we show that twin groups $T_n$ have $R_\infty$-property and are not co-Hopfian for $n \ge 3$.

math.GR↗
arXiv2020

Conjugacy classes and automorphisms of twin groups

The twin group $T_n$ is a right angled Coxeter group generated by $n-1$ involutions and the pure twin group $PT_n$ is the kernel of the natural surjection from $T_n$ onto the symmetric group on $n$ symbols. In this paper, we investigate some structural aspects of these groups. We derive a formula for the number of conjugacy classes of involutions in $T_n$, which quite interestingly, is related to the well-known Fibonacci sequence. We also derive a recursive formula for the number of $z$-classes of involutions in $T_n$. We give a new proof of the structure of $\Aut(T_n)$ for $n \ge 3$, and show that $T_n$ is isomorphic to a subgroup of $\Aut(PT_n)$ for $n \geq 4$. Finally, we construct a representation of $T_n$ to $\Aut(F_n)$ for $n \ge 2$.

math.GR↗
arXiv2018

On the probability distribution associated to commutator word map in finite groups \rom{2}

Let $P(G)$ denotes the set of sizes of fibers of non-trivial commutators of the commutator word map. Here, we prove that $|P(G)|=1$, for any finite group $G$ of nilpotency class $3$ with exactlly two conjugacy class sizes. We also show that for given $n\geq 1$, there exists a finite group $G$ of nilpotency class $2$ with exactlly two conjugacy class sizes such that $|P(G)|=n$.

math.GR↗
arXiv2017

Finite $p$-Groups of Nilpotency Class $3$ with Two Conjugacy Class Sizes

It is proved that, for a prime $p>2$ and integer $n\geq 1$, finite $p$-groups of nilpotency class $3$ and having only two conjugacy class sizes $1$ and $p^n$ exist if and only if $n$ is even; moreover, for a given even positive integer, such a group is unique up to isoclinism (in the sense of Philip Hall).

math.GR↗