arXiv research

Study of Lagrangian submanifolds in pseudo-nearly Kähler SL(2,R)×SL(2,R).

The paper classifies topological properties of specific geometric forms on manifolds.

We study the Chabauty compactification of two families of closed subgroups of $SL(n,\mathbb{Q}_p)$. The first family is the set of all parahoric subgroups of $SL(n,\mathbb{Q}_p)$. Although the Chabauty compactification of parahoric subgroups is well studied, we give a different and more geometric proof using various Le…

The study proves non-existence of hypercomplex structures on SL(3,R) and finds one on SL(2n+1,C).

Study on cohomology of SL_n(Z) for n>=3, proving vanishing of certain cohomology groups.

Classifies real-analytic SL(n,R) actions on closed manifolds.

Odd-dimensional SL(n,Q) contains dense surface subgroups.

We characterize groups admitting Anosov representations into $\mathsf{SL}(3,\mathbb R)$, projective Anosov representations into $\mathsf{SL}(4,\mathbb R)$, and Borel Anosov representations into $\mathsf{SL}(4,\mathbb R)$. More generally, we obtain bounds on the cohomological dimension of groups admitting $P_k$-Anosov r…

Classifies limits of groups of involutions in SL(2,F) over local fields.

Let $Γ$ be a finitely generated group and $G$ a real form of $\mathrm{SL}_n(\mathbb{C})$. We propose a definition for the $G$-character variety of $Γ$ as a subset of the $\mathrm{SL}_n(\mathbb{C})$-character variety of $Γ$. We consider two anti-holomorphic involutions of the $\mathrm{SL}_n(\mathbb{C})$ character variet…

Two specific Einstein metrics found on a product of SL(2,R) groups.

Regular subgroups of SL3(R) are identified and ruled out.

The paper classifies totally geodesic Lagrangian submanifolds in a specific pseudo-nearly Kähler space.

Polynomial density theorem for specific subgroup orbits in quotient spaces.

Defines a new $SL_2(\mathbb{R})$-Casson invariant using Reidemeister torsions.

Polynomial error equidistribution for SL2 groups.

Study $SL(2,\mathbb{C})$ connections on Seifert-fibered spaces using gauge theory.

Proves the relative h-principle for SL(3,R)^2 3-forms on 6-manifolds.

Paper studies complex Lagrangian surfaces and their relation to $\mathrm{SL}(3,\mathbb{C})$-representations.

The paper classifies nilmanifolds with specific SL(3,C) structures.

Proves polynomial error rate for equidistribution of unipotent flows.

We describe a family of representations in SL(3,$\mathbb C$) of the fundamental group $π$ of the Whitehead link complement. These representations are obtained by considering pairs of regular order three elements in SL(3,$\mathbb C$) and can be seen as factorising through a quotient of $π$ defined by a certain exception…

Let $Γ$ be the fundamental group of a complete hyperbolic $3$-manifold $M$ with toric cusps. We define the $ω$-Borel invariant $β_n^ω(ρ_ω)$ associated to a representation $ρ_ω: Γ\rightarrow SL(n,\mathbb{C}_ω)$, where $\mathbb{C}_ω$ is a field which can be constructed as a quotient of a suitable subset of $\mathbb{C}^\m…

Study character variety of a special 3-manifold, finding a polynomial formula.

The purpose of this paper is to provide an octonionic description of the Lie group $SL(2,{\mathbb O})$. The main result states that it can be obtained as a free group generated by invertible and determinant preserving transformations from $\mathfrak{h}_2({\mathbb O})$ onto itself. An interesting characterization is giv…

Unified framework counts knot representations into SU(2) and SL(2,R).

The study counts units and eigenvalue patterns in SL_n(Z) and Sp_{2n}(Z) in thin tubes.

We compute higher moments of the Siegel--Veech transform over quotients of $SL(2,\mathbb{R})$ by the Hecke triangle groups. After fixing a normalization of the Haar measure on $SL(2,\mathbb{R})$ we use geometric results and linear algebra to create explicit integration formulas which give information about densities of…

New examples show embeddings not approximated by Anosov representations.

Study compares two pseudo-Kähler structures on a specific mathematical component.

Study exotic tori and their SL_d(Z) actions, proving many do not admit nontrivial actions.

In this continuation of \cite{BM}, we prove the following: Let $Γ\subset \text{SL}(2,{\mathbb C})$ be a cocompact lattice, and let $ρ: Γ\rightarrow \text{GL}(r,{\mathbb C})$ be an irreducible representation. Then the holomorphic vector bundle $E_ρ\longrightarrow \text{SL}(2,{\mathbb C})/Γ$ associated to $ρ$ is polystab…

Study geodesics on $SL(n, \mathbb{R})$ adjoint orbits.

In this paper we prove that every open Riemann surface properly embeds in the Special Linear group $SL_2(\mathbb{C})$ as a holomorphic Legendrian curve, where $SL_2(\mathbb{C})$ is endowed with its standard contact structure. As a consequence, we derive the existence of proper, weakly complete, flat fronts in the real …

Goldman symplectic form and complex structure compatible on $\mathrm{SL}(3,\mathbb R)$ Hitchin component.

Explicitly bounds the spectral gap for Schottky subgroups of SL(2,Z).

Holomorphic curves found in compact quotients of SL(2,C).

The paper classifies and decomposes quaternionic projective transformations.

We present an efficient method for computing the ${\rm SL}(3,\mathbb{C})$-character varieties of two-generator groups.

New proof for 3-manifolds with specific homology groups.

Study real slices of SL(r,C)-opers via Riemann surface involution.

Let SL(2, $\mathbb H$) be the group of $2 \times 2$ quaternionic matrices $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with quaternionic determinant $\det A=|ad-aca^{-1} b|=1$. This group acts by the orientation-preserving isometries of the five dimensional (real) hyperbolic space. We obtain discreteness criteria f…

We characterize the universal covering of connected analytic pseudo-Riemannian manifolds which admit a non-trivial and isometric action of the simple Lie group $SL(3,\mathbb{R})$ with a dense orbit preserving a finite volume. If such manifold is also weakly irreducible we prove that $M$ is isometric to, or a quotient s…

Authors determine the SL(2,C) character variety of a specific knot without computer aid.

Let ${\rm SL(2, \mathbb H)}$ be the group of $2 \times 2$ quaternionic matrices with Dieudonné determinant $1$. The group ${\rm SL(2, \mathbb H)}$ acts on the five dimensional hyperbolic space by isometries. We investigate extremality of Jørgensen type inequalities in ${\rm SL(2, \mathbb H)}$. Along the way, we derive …

The spaces of linear differential operators on ${\mathbb{R}}^n$ acting on tensor densities of degree $λ$ and the space of functions on $T^*{\mathbb{R}}^n$ which are polynomial on the fibers are not isomorphic as modules over the Lie algebra $\Vect({\mathbb{R}}^n)$ of vector fields on ${\mathbb{R}}^n$. However, these mo…

Study cohomology spaces of sl(2) acting on n-ary differential operators.

Character variety of Whitehead link described in detail.