arXiv research

Homogeneous magnetic trajectories in a special linear group proven.

Let $G$ be a virtually special group. Then the residual finiteness growth of $G$ is at most linear. This result cannot be found by embedding $G$ into a special linear group. Indeed, the special linear group $\text{SL}_k(\mathbb{Z})$, for $k > 2$, has residual finiteness growth $n^{k-1}$.

Proves hyperbolized groups are virtually compact special and linear.

Study on Hermitian curvature flow on special linear groups, disproving a conjecture and finding non-algebraic solitons.

Special linear group actions on certain sphere products are trivial.

Invariant minimal surfaces in the real special linear group of degree 2 with canonical Riemannian and Lorentzian metrics are studied. Constant mean curvature surfaces with vertically harmonic Gauß map are classified.

Non-Euclidean number rings have non-integral Steinberg modules.

We characterize helix surfaces (constant angle surfaces) in the special linear group $\mathrm{SL}(2,\r)$. In particular, we give an explicit local description of these surfaces in terms of a suitable curve and a 1-parameter family of isometries of $\mathrm{SL}(2,\r)$.

A special linear Lie group over the real number field and the quarternion field admits a projectivley flat affine connection. We show that parabolic subgroups are autoparallel submanifolds and give a criterion the induced connection is projectively equivalent to a flat affine connection.

We characterize the biharmonic curves in the special linear group $SL(2,R)$. In particular, we show that all proper biharmonic curves in $SL(2,R)$are helices and we give their explicit parametrizations as curves in the pseudo-Euclidean space $R^4_2$.

Formanek and Procesi have demonstrated that Aut(F_n) is not linear for n >2. Their technique is to construct nonlinear groups of a special form, which we call FP-groups, and then to embed a special type of automorphism group, which we call a poison group, in Aut(F_n), from which they build an FP-group. We first prove t…

The paper classifies and describes translators in $SL(2,\mathbb{R})$ under specific symmetry conditions.

New thin subgroups found in special linear groups via bending techniques.

Classifies special homogeneous curves with polynomial equations.

The paper uses singularity theory to find normal forms for sub-Riemannian exponential maps.

Classifies reversible and strongly reversible elements in quaternionic groups.

Authors compute Weingarten map and curvatures for SL(n, R).

Let M be a graph manifold. We prove that fundamental groups of embedded incompressible surfaces in M are separable in the fundamental group of M, and that the double cosets for crossing surfaces are also separable. We deduce that if there is a "sufficient" collection of surfaces in M, then the fundamental group of M is…

The paper finds many thin subgroups isomorphic to Gromov-Piatetski-Shapiro lattices.

Constructs special Kähler structures on Lie groups.

Study of modular representations in homology of congruence subgroups.

Characterizes decomposable torus fiber bundles in low dimensions.

The paper develops theory for holomorphic null curves in SL2(C).

Let $G$ be a connected Lie group and $\mathfrak{g}$ its Lie algebra. We denote by $\nabla^0$ the torsion free bi-invariant linear connection on $G$ given by $\nabla^0_XY=\frac12[X,Y],$ for any left invariant vector fields $X,Y$. A Poisson structure on $\mathfrak{g}$ is a commutative and associative product on $\mathfra…

The study examines group actions of SL_n(Z) on aspherical manifolds and their implications.

Study of magnetic curves in SL(2,R) with quantization and horocycle projections.

Holomorphic curves in $SL_2(\mathbb{C})$ embed open surfaces.

The study examines Lorentzian metrics on specific Lie groups and their curvatures.

For a Coxeter group $W$ we have an associating bi-linear form $B$ on a real vector space. We assume that $B$ has the signature $(n-1,1)$. In this case we have the Cannon-Thurston map for $W$, that is, a $W$-equivariant continuous surjection from the Gromov boundary of $W$ to the limit set of $W$. We focus on the case w…

Study shows $L^{2}$-Betti numbers vanish for certain matrix groups over rings, proving non-acylindrical hyperbolicity.

We study the group of symplectic birational transformations of the plane. It is proved that this group is generated by $\mathrm{SL}(2,\mathbb{Z})$, the torus and a special map of order $5$, as it was conjectured by A. Usnich. Then we consider a special subgroup $H$, of finite type, defined over any field which admits a…

It is proved that each of compact linear groups of one special type admits a polynomial factorization map onto a real vector space. More exactly, the group is supposed to be non-commutative one-dimensional and to have two connected components, and its representation should be the direct sum of three irreducible two-dim…

A free action of a finite group on an odd-dimensional sphere is said to be almost linear if the action restricted to each cyclic or 2-hyperelementary subgroup is conjugate to a free linear action. We begin this survey paper by reviewing the status of almost linear actions on the 3-sphere. We then discuss almost linear …

I show that the adjoint variety of the complex special linear group is rigid to order three.

Classifies finite orbits of mapping class group actions on surface group representations.

Proves one-relator groups with negative immersions are hyperbolic and virtually special.

A complete classification is obtained of continuous, translation invariant, Minkowski valuations on an m-dimensional complex vector space which are covariant under the complex special linear group.

Study local system points on surfaces using group descent.

Motivated by the physical concept of special geometry two mathematical constructions are studied, which relate real hypersurfaces to tube domains and complex Lagrangean cones respectively. Me\-thods are developed for the classification of homogeneous Riemannian hypersurfaces and for the classification of linear transit…

Study on cohomology of special linear groups over Euclidean number rings.

We prove that the discriminant of a nonsingular space curve of genus $g\geq 2$ is stable with respect to the standard action of the special linear group.

The study establishes conditions for groups acting on polygonal complexes to contain virtually free subgroups.

An algorithm for efficient computation of equivariant neural network layers.

The special linear groups, the mapping class groups of surfaces, the outer autormorphism groups of free groups appear in numerous domains. Their analogies, developped in particular in K. Vogtmann's work, have been written about a lot. In this report, we concentrate on the contractible spaces on which these groups act i…

We give a complete characterization of countable primitive groups in several settings including linear groups, subgroups of mapping class groups, groups acting minimally on trees and convergence groups. The latter category includes as a special case Kleinian groups as well as subgroups of word hyperbolic groups. As an …

We prove that the conjugacy problem in right-angled Artin groups (RAAGs), as well as in a large and natural class of subgroups of RAAGs, can be solved in linear-time. This class of subgroups contains, for instance, all graph braid groups (i.e. fundamental groups of configuration spaces of points in graphs), many hyperb…

In this paper we construct a multivariable link invariant arising from the quantum group associated to the special linear Lie superalgebra sl(2|1). The usual quantum group invariant of links associated to (generic) representations of sl(2|1) is trivial. However, we modify this construction and define a nontrivial link …

Characterizes almost Abelian Lie algebras with special $H$-structures.