arXiv subjects

Arunava Mandal

Publications and source records attributed to Arunava Mandal.

14 recordsLinked to original sources

arXiv2026

Polynomial Maps with Constants over Division Algebras and the Generalized Kaplansky--L'vov Conjecture

The Kaplansky--L'vov conjecture asserts that the image of a multilinear polynomial map on a full matrix algebra over a field is always a vector space. Although the conjecture remains open in general, substantial progress has been made for $2\times 2$ and $3\times 3$ matrix algebras over various fields. Recently, Panja, Saini, and Singh formulated a generalized Kaplansky--L'vov conjecture for polynomial maps with matrix coefficients over algebraically closed fields and verified it for $2\times 2$ matrices. In this work, we investigate an analogous problem for polynomial maps with constant matrix coefficients over an infinite division algebra. Specifically, we consider polynomials in the free algebra $M_2(\mathbb D)\langle x_1,\ldots,x_m\rangle$ of the form $\omega = A_1x_1^{k_1}+\cdots+A_mx_m^{k_m},$ where the $A_1,\ldots, A_m\in M_2(\mathbb D)$ are fixed matrices, $\mathbb D$ is an infinite division algebra, and $k_1,\ldots, k_m$ are positive integers. We prove that the corresponding generalized Kaplansky--L'vov conjecture holds for $2\times 2$ matrices over $\mathbb R$ and the quaternion division algebra $\mathbb H$. We also investigate the surjectivity of these polynomial maps. This can be viewed as a generalized Waring problem for matrix algebras.

math.RA↗

arXiv2026

Cartan subgroups in connected locally compact groups

We define Cartan subgroups in connected locally compact groups, which extends the classical notion of Cartan subgroups in Lie groups. We prove their existence and justify our choice of the definition which differs from the one given by Chevalley on general groups. Apart from proving some properties of Cartan subgroups, we show that the Cartan subgroups of the quotient groups are precisely the images of Cartan subgroups of the ambient group. We establish the so-called `Levi' decomposition of Cartan subgroups which extends Wüstner's decomposition theorem and our earlier results for Lie groups. We also show that the centraliser of any maximal torus of the radical is connected and its Cartan subgroups are also Cartan subgroups of the ambient group; moreover, every Cartan subgroup arises this way. We prove that Cartan subalgebras defined by Hofmann and Morris in pro-Lie algebras are the same as those corresponding to Cartan subgroups in case of pro-Lie algebras of connected locally compact groups, and that they are nilpotent. We characterise density of the image of a power map in a connected locally compact group in terms of its surjectivity on all Cartan subgroups, and show that weak exponentiality of the group is equivalent to the condition that all its Cartan subgroups are connected.

math.GR↗

arXiv2026

Determining $\mathbb R$-Rank in Semisimple Lie Groups via uniform approximate Lattice arising as Regular Model Sets

Let $G$ be a linear semisimple Lie group without compact factors. We show that uniform approximate lattices $Λ$ arising as regular model sets in $G$ determine the ambient group $G$ in a strong sense. Specifically, for every non-compact Cartan subgroup $C$ of $G$, there exists $g \in G$ such that the intersection $gCg^{-1} \cap Λ^2$ is non-empty and itself forms a uniform approximate lattice, extending a classical result of Mostow for lattices. The proof relies on a Moore-type ergodicity theorem for the hull of a strong approximate lattice, proved here as a key tool. Moreover, we prove that such approximate lattices determine the $\mathbb{R}$-rank of the ambient group $G$, drawing on ideas from the work of Prasad and Raghunathan on lattices.

math.GR↗

arXiv2026

Combination of locally quasiconvex hyperbolic TDLC groups and Cannon-Thurston maps

In this article, we study acylindrical graphs of groups, local quasiconvexity, and Cannon-Thurston maps in the setting of totally disconnected locally compact (TDLC) hyperbolic groups, extending several fundamental notions and results from discrete hyperbolic groups to this broader context. Leveraging Dahmani's technique and a topological characterization of hyperbolic TDLC groups in terms of uniform convergence groups given by Carette-Dreesen, we prove a combination theorem for an acylindrical graph of hyperbolic TDLC groups and give an explicit construction of the Gromov boundary of the fundamental group of the given graph of groups. Using the description of the Gromov boundary, we prove our main result: a combination theorem for an acylindrical graph of locally quasiconvex hyperbolic TDLC groups. Further, we generalise the work of Mosher, proving the existence of quasiisometric sections for a given short exact sequence of hyperbolic TDLC groups. This leads us to prove the existence of a Cannon-Thurston map for a normal hyperbolic subgroup of a hyperbolic TDLC group, generalising a theorem of Mj.

math.GR↗

arXiv2025

On rational and real elements in a class of Lie groups

For a class of groups $G$ over a field $\mathbb{F}$, including certain Lie groups, Algebraic groups and finite groups, we develop a general method to determine rational and real elements, thereby unifying earlier group-specific results into a wider framework. As an application, we classify all real and rational elements in the semidirect product ${\rm SL}(2,\mathbb{R}) \ltimes \mathrm{Sym}^n(\mathbb{R}^2)$. Furthermore, for affine groups of the form ${\rm GL}(n,\mathbb{R}) \ltimes \mathbb{R}^n$, we show that if $x \in {\rm GL}(n,\mathbb{R})$ is rational, then $(x,v)$ is rational for every $v \in \mathbb{R}^n$.

math.GR↗

arXiv2025

On combination theorems and Bowditch boundaries of relatively hyperbolic TDLC groups

Based on the work of Farb, Bowditch, and Groves-Manning on discrete relatively hyperbolic groups, we introduce an approach to relative hyperbolicity for totally disconnected locally compact (TDLC) groups. For compactly generated TDLC groups, we prove that this notion is equivalent to the one introduced by Arora-Pedroza. Let $G=A\ast_C B$ or $G=A\ast_C$ where $A$ and $B$ are relatively hyperbolic TDLC groups and $C$ is compact. We prove that $G$ is a relatively hyperbolic TDLC group and give a construction of the Bowditch boundary of $G$. As a consequence, we prove that if the rough ends of $G$ are infinite, then the topology of the Bowditch boundary of $G$ is uniquely determined by the topology of the Bowditch boundary of $A$ and $B$. Further, we show that if a relatively hyperbolic TDLC group has one rough end, then its Bowditch boundary is connected. Finally, we show that if the Gromov boundary of a hyperbolic TDLC group $G$ is totally disconnected, then $G$ splits as a finite graph of compact groups.

math.GR↗

arXiv2025

On stable Cartan subgroups of Lie groups

Let $G$ be a connected real Lie group with associated Lie algebra $\mathfrak g$, and let ${\rm Aut}(G)$ be the group of (Lie) automorphisms of $G$. It is noted here that, given a super-solvable subgroup $Γ\subset {\rm Aut}(G)$ of semisimple automorphisms, there exists a $Γ$-stable Cartan subgroup, by using a result of Borel and Mostow. We characterize the $Γ$-stable Cartan subgroups (with induced action) in the quotient group modulo a $Γ$-stable closed normal subgroup as the images of the $Γ$-stable Cartan subgroups in the ambient group. It is well known that a semisimple automorphism of $\mathfrak g$ always fixes a Cartan subalgebra of $\mathfrak g$. Conversely, if we take a representative from each non-conjugate class of Cartan subalgebras in a real Lie algebra, we show that there exists a non-identity automorphism that fixes these representatives. We explicitly identify such automorphisms in the case of classical simple Lie algebras. As a consequence, we deduce an analogous result for semisimple Lie groups. Moreover, given a $Γ$-stable Cartan subgroup $H$ of $G$, and a $Γ$-stable closed connected normal subgroup $M$ of $G$, we prove that there exists a $Γ$-stable Cartan subgroup $H_M$ of $M$ such that $H\cap M\subset H_M$.

math.GR↗

arXiv2025

Roots of elements for groups over local fields

Let $\mathbb F$ be a local field and $G$ be a linear algebraic group defined over $\mathbb F$. For $k\in\mathbb N$, let $g\to g^k$ be the $k$-th power map $P_k$ on $G(\mathbb F)$. The purpose of this article is two-fold. First, we study the power map on real algebraic group. We characterise the density of the images of the power map $P_k$ on $G(\mathbb R)$ in terms of Cartan subgroups. Next we consider the linear algebraic group $G$ over non-Archimedean local field $\mathbb F$ with any characteristic. If the residual characteristic of $\mathbb F$ is $p$, and an element admits $p^k$-th root in $G(\mathbb F)$ for each $k$, then we prove that some power of the element is unipotent. In particular, we prove that an element $g\in G(\mathbb F)$ admits roots of all orders if and only if $g$ is contained in a one-parameter subgroup in $G(\mathbb F)$. Also, we extend these results to all linear algebraic groups over global fields.

math.NT↗

arXiv2022

On universal subspaces for Lie groups

Let $U$ be a finite dimentional vector space over $\mathbb R$ or $\mathbb C$, and let $ρ:G\to GL(U)$ be a representation of a connected Lie group $G$. A linear subspace $V\subset U$ is called universal if every orbit of $G$ meets $V$. We study universal subspaces for Lie groups, especially compact Lie groups. Jinpeng and Doković approached universality for compact groups through a certain topological obstruction. They showed that the non-vanishing of the obstruction class is sufficient for the universality of $V$, and asked whether it is also necessary under certain conditions. In this article, we show that the answer to the question is negative in general, but we discuss some important situations where the answer is positive. We show that if $G$ is solvable and $ρ:G\to GL(U)$ is a complex representation, then the only universal complex subspace is $U$ itself. We also investigate the question of universality for a Levi subgroup.

math.RT↗

arXiv2021

Dense images of the power maps for a disconnected real algebraic group

Let $G$ be a complex algebraic group defined over $\mathbb R$, which is not necessarily Zariski connected. In this article, we study the density of the images of the power maps $g\to g^k$, $k\in\mathbb N$, on real points of $G$, i.e., $G(\mathbb R)$ equipped with the real topology. As a result, we extend a theorem of P. Chatterjee on surjectivity of the power map for the set of semisimple elements of $G(\mathbb R)$. We also characterize surjectivity of the power map for a disconnected group $G(\mathbb R)$. The results are applied in particular to describe the image of the exponential map of $G(\mathbb R).$

math.GR↗

arXiv2021

Group actions and power maps for groups over non-Archimedean local fields

We consider linear groups and Lie groups over a non-Archimedean local field $\mathbb F$ for which the power map $x\mapsto x^k$ has a dense image or it is surjective. We prove that the group of $\mathbb F$-points of such algebraic groups is a compact extension of unipotent groups with the order of the compact group being relatively prime to $k$. This in particular shows that the power map is surjective for all $k$ is possible only when the group is unipotent or trivial depending on whether the characteristic of $\mathbb F$ is zero or positive. Similar results are proved for Lie groups via the adjoint representation. To a large extent, these results are extended to linear groups over local fields and global fields.

math.GR↗

arXiv2020

The Structure of Cartan Subgroups in Lie Groups

We study properties and the structure of Cartan subgroups in a connected Lie group. We obtain a characterisation of Cartan subgroups which generalises Wüstner's structure theorem for the same. We show that Cartan subgroups are same as those of the centralizers of maximal compact subgroups of the radical. Moreover, we describe a recipe for constructing Cartan subgroups containing certain nilpotent subgroups in a connected solvable Lie group. We characterise the Cartan subgroups in the quotient group modulo a closed normal subgroup as the images of the Cartan subgroups in the ambient group. We also study the density of the images of power maps on a connected Lie group and show that the image of any $k$-th power map has dense image if its restriction to a closed normal subgroup and the corresponding map on the quotient group have dense images.

math.GR↗

arXiv2017

On the density of images of the power maps in Lie groups

Let $G$ be a connected Lie group. In this paper, we study the density of the images of individual power maps $P_k:G\to G:g\mapsto g^k$. We give criteria for the density of $P_k(G)$ in terms of regular elements, as well as Cartan subgroups. In fact, we prove that if ${\rm Reg}(G)$ is the set of regular elements of $G$, then $P_k(G)\cap {\rm Reg}(G)$ is closed in ${\rm Reg}(G)$. On the other hand, the weak exponentiality of $G$ turns out to be equivalent to the density of all the power maps $P_k$. In linear Lie groups, weak exponentiality reduces to the density of $P_2(G)$. We also prove that the density of the image of $P_k$ for $G$ implies the same for any connected full rank subgroup.

math.GR↗

arXiv2016

On the surjectivity of the power maps of a class of solvable groups

Let $G$ be a group containing a nilpotent normal subgroup $N$ with central series $\{N_j\}$, such that each $N_j/N_{j+1}$ is a $\mathbb{F}$-vector space over a field $\mathbb{F}$ and the action of $G$ on $N_j/N_{j+1}$ induced by the conjugation action is $\mathbb{F}$-linear. For $k\in \mathbb N$ we describe a necessary and sufficient condition for all elements from any coset $xN$, $x\in G$, to admit $k$-th roots in $G$, in terms of the action of $x$ on the quotients $N_j/N_{j+1}.$ This yields in particular a condition for surjectivity of the power maps, generalising various results known in special cases. For $\mathbb{F}$-algebraic groups we also characterise the property in terms of centralizers of elements. For a class of Lie groups, it is shown that surjectivity of the $k$-th power map, $k\in \mathbb N$, implies the same for the restriction of the map to the solvable radical of the group. The results are applied in particular to the study of exponentiality of Lie groups.

math.GR↗