arXiv subjects
Publications and source records attributed to Krishnendu Gongopadhyay.
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.
In this note, we initiate the concept of Dehn functions for a family of finite groups. We investigate the Dehn function for some specific families of finite polycyclic groups. We also consider related notions of spherical Dehn function and mean Dehn function for family of finite simple cyclic groups.
We consider the isometries of the complex hyperbolic bidisk, that is, the product space $\mathbb{H}^2_{\mathbb{C}} \times \mathbb{H}^2_{\mathbb{C}} $, where each factor $ \mathbb{H}^2_{\mathbb{C}} $ denotes the complex hyperbolic plane. We investigate the Dirichlet domain formed by the action of a cyclic subgroup $(g_1, g_2)$, where each $g_i$ is loxodromic. We prove that such a Dirichlet domain has two sides.
Let $Γ_p$ 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 $Γ_p$. We prove that for $l \to \infty$, $$|\mathcal{N}_l| = \mathcal{O}\left(\left\lfloor \tfrac{l+1}{2} \right\rfloor^{s-1} ρ^{\left\lfloor \tfrac{l+1}{2} \right\rfloor} \right), $$ where $\mathcal O$ is the `big O', $ρ\in [\sqrt{2}, 2]$ 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 $Γ_p$, 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.
Let $Γ$ be the fundamental group of a $k$-punctured, $k \geq 0$, closed connected orientable surface of genus $g \geq 2$. We show that the character variety of the $(Q^+, Q^-)$-Anosov irreducible representations, resp. the character variety of the $(P^+, P^-)$-Anosov Zariski dense representations of $Γ$ into $\SL(n , \C)$, $n \geq 2$, is a complex manifold of complex dimension \hbox{$(2g+k-2)(n^2-1)$}. For $Γ=π_1(Σ_g)$, we also show that these character varieties are holomorphic symplectic manifolds.
An element of a group is called $\textit{strongly reversible}$ or $\textit{strongly real}$ if it can be expressed as a product of two involutions. We provide necessary and sufficient conditions for an element of $\mathrm{SL}(n,\mathbb{C})$ to be a product of two involutions. In particular, we classify the strongly reversible conjugacy classes in $\mathrm{SL}(n,\mathbb{C})$.
An element of a group is called reversible if it is conjugate to its own inverse. Reversible elements are closely related to strongly reversible elements, which can be expressed as a product of two involutions. In this paper, we classify the reversible and strongly reversible elements in the quaternionic special linear group $\mathrm{SL}(n,\mathbb{H})$ and quaternionic projective linear group $ \mathrm{PSL}(n,\mathbb{H})$. We prove that an element of $ \mathrm{SL}(n,\mathbb{H})$ (resp. $ \mathrm{PSL}(n,\mathbb{H})$) is reversible if and only if it is a product of two skew-involutions (resp. involutions).
In this paper, we prove that the Dehn function of the palindromic automorphism group $ΠA(F_n)$ is exponential.
Let $G$ be a Lie group with Lie algebra $\mathfrak g$. In the paper "Reality of unipotent elements in simple Lie groups, Bull. Sci. Math., 185, 2023, 103261" by K. Gongopadhyay and C. Maity, an infinitesimal version of the notion of classical reality, namely adjoint reality, has been introduced. An element $X \in \mathfrak g$ is adjoint real if $-X$ belongs to the adjoint orbit of $X$ in $\mathfrak g$. In this paper, we investigate the adjoint real and the strongly adjoint real semisimple elements in complex simple classical Lie algebras. We also prove that every element in a complex symplectic Lie algebra is adjoint real.
The aim of this paper is to give a classification of real and strongly real unipotent elements in a classical simple Lie group. To do this, we will introduce an infinitesimal version of the notion of classical reality in a Lie group. This notion has been applied to classify real and strongly real unipotent elements in a classical simple Lie group.
Let $ G $ be a Lie group with Lie algebra $ \mathfrak{g} $. An element $ X \in \mathfrak{g} $ is called $\mathrm{Ad}_G$-real if $ -X=gXg^{-1} $ for some $ g \in G $. Moreover, if $ -X=gXg^{-1} $ holds for some involution $ g\in G $, then $ X $ is called strongly $\mathrm{Ad}_G$-real. We have classified the $\mathrm{Ad}_G$-real and the strongly $\mathrm{Ad}_G$-real orbits in the special linear Lie algebra $\mathfrak{sl}(n,\mathbb{F}) $ for $ \mathbb{F}=\mathbb{C}$ or $\mathbb{H} $.
In this article, we consider the ball model of an infinite dimensional complex hyperbolic space, i.e. the open unit ball of a complex Hilbert space centered at the origin equipped with the Caratheodory metric. We consider the group of holomorphic automorphisms of the ball and classify the conjugacy classes of automorphisms. We also compute the centralizers for elements in the group of automorphisms.
An element $g$ in a group $G$ is called reversible if $g$ is conjugate to $g^{-1}$ in $ G $. An element $g$ in $G$ is strongly reversible if $ g $ is conjugate to $g^{-1}$ by an involution in $G$. The group of affine transformations of $\mathbb{D}^n$ may be identified with the semi-direct product $\mathrm{GL}(n, \mathbb{D}) \ltimes \mathbb{D}^n $, where $\mathbb{D}:=\mathbb{R}, \mathbb{C}$ or $ \mathbb{H} $. This paper classifies reversible and strongly reversible elements in the affine group $\mathrm{GL}(n, \mathbb{D}) \ltimes \mathbb{D}^n $.
We consider the projective linear group $\mathrm{PSL}(3,\mathbb{H})$. We have investigated the reversibility problem in this group and use the reversibility to offer an algebraic characterization of the dynamical types of $\mathrm{PSL}(3,\mathbb{H})$. We further decompose elements of $\mathrm{SL}(3,\mathbb{H})$ as products of simple elements, where an element $g$ in $\mathrm{SL}(3,\mathbb{H})$ is called $\textit{simple}$ if it is conjugate to an element of $\mathrm{SL}(3,\mathbb{R})$. We have also revisited real projective transformations and following Goldman's ideas, have offered a complete classification for elements of $\mathrm{SL }(3,\mathbb{R})$.
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 $Γ_p$ for $p\geq 3$. This generalizes a result by Sarnak for reciprocal elements in the modular group.
In this paper, we consider the natural action of $\mathrm{SL}(3, \mathbb{H})$ on the quaternionic projective space $ \mathbb{P}_{\mathbb{H}}^2$. Under this action, we investigate limit sets for cyclic subgroups of $\mathrm{SL}(3, \mathbb{H})$. We compute two types of limit sets, which were introduced by Kulkarni and Conze-Guivarc'h, respectively.
We examine the group of isometries of the open unit ball of a complex Banach space of certain bounded linear operators equipped with the Carathéodory metric. Therein we obtain a charactrization of the normal isometries in terms of their special type of fixed points.
We extend classical results on the classification of reversible elements of the group $\mathrm{GL}(n, \mathbb{C})$ (and $\mathrm{GL}(n, \mathbb{R})$) to $\mathrm{GL}(n, \mathbb{H})$ using an infinitesimal version of the classical reversibility, namely adjoint reality in the Lie algebra set-up. We also provide a new proof of such a classification for the general linear groups over $\mathbb{R}$ and $\mathbb{C}$. Further, we classify the real adjoint orbits in the Lie algebra $\mathfrak{gl}(n, \mathbb{D})$ for $ \mathbb{D}=\mathbb{R}, \mathbb{C}$ or $\mathbb{H} $.