arXiv research

Study on braid groups' congruence subgroups and their crystallographic quotients.

Solved a specific case of Salter's question on Burau representation.

By using computer assistance, we prove that the fundamental group of the complement of a real complexified line arrangement is not determined by its intersection lattice, providing a counter-example for a problem of Falk and Randell. We also deduce that the torsion of the lower central series quotients is not combinato…

Power quandles improve group invariants and allow group presentations.

Counterexamples show Salter's question on Burau image is negative for n=4.

We prove that the braid group $B_4$ on 4 strings, as well as its central quotient $B_4/< z>$, have the property RD of Haagerup-Jolissaint. It follows that the automorphism group $\Aut(F_2)$ of the free group $F_2$ on 2 generators has property RD. We also prove that the braid group $B_4$ is a group of intermediate rank …

Constructs infinite-dimensional Siegel disc as symplectic and Kaehler quotient.

We show that if the lower central series of the fundamental group of a closed oriented $3$-manifold stabilizes then the maximal nilpotent quotient is a cyclic group, a quaternion $2$-group cross an odd order cyclic group, or a Heisenberg group. These groups are well known to be precisely the nilpotent fundamental group…

We give a new interpretation of the Faddeev-Mickelsson anomaly in certain Yang-Mills theories in terms of S^1-central extensions of Lie groupoids.

We give a refined value group for the collection of triple linking numbers of links in the 3-sphere. Given two links with the same pairwise linking numbers we show that they have the same refined triple linking number collection if and only if the links admit homeomorphic surface systems. Moreover these two conditions …

Our main result is that the image of the quantum representation of a central extension of the mapping class group of the genus $g\geq 3$ closed orientable surface at a prime $p\geq 5$ is a Zariski dense discrete subgroup of some higher rank algebraic semi-simple Lie group $\mathbb G_p$ defined over $\Q$. As an applicat…

The paper studies surface quotients of Fuchsian buildings.

The paper analyzes the excess risk of PCA and provides a precise characterization.

We study groups of some virtual knots with small number of crossings and prove that there is a virtual knot with long lower central series which, in particular, implies that there is a virtual knot with residually nilpotent group. This gives a possibility to construct invariants of virtual knots using quotients by term…

Geometrically constructs Virasoro-Bott group from circle diffeomorphisms.

Extends Milnor's invariants to knots and links in 3-manifolds.

The study proves residual finiteness for certain lattice extensions and negatively curved projective varieties.

We construct a series of homomorphisms from the $Y$-filtration on the monoid of homology cylinders to torsion modules via the mod $\mathbb{Z}$ reduction of the LMO functor. The restriction of our homomorphism to the lower central series of the Torelli group does not factor through Morita's refinement of the Johnson hom…

In his 1957 paper, John Milnor introduced link invariants which measure the homotopy class of the longitudes of a link relative to the lower central series of the link group. Consequently, these invariants determine the lower central series quotients of the link group. This work has driven decades of research with prof…

By using a Borel density theorem for algebraic quotients, we prove a theorem concerning isometric actions of a Lie group $G$ on a smooth or analytic manifold $M$ with a rigid $\mathrm{A}$-structure $σ$. It generalizes Gromov's centralizer and representation theorems to the case where $R(G)$ is split solvable and $G/R(G…

New computations show symplectic groups and mapping class groups have different properties regarding torsion.

Study on Milnor fibrations of arrangements with trivial algebraic monodromy.

We determine the lower central and derived series of the n-string braid groups B_n(RP^2) of the real projective plane. We are motivated in part by the study of Fadell-Neuwirth short exact sequences, but the problem is interesting in its own right. For n=1,2, B_n(RP^2) is finite and its lower central and derived series …

The approaches to quantum field theories based in the so called loop representation deserved much attention recently. In it, closed curves and holonomies around them play a central role. In this framework the group of loops and the group of hoops have been defined, the first one consisting in closed curves quotient wit…

Polyhedral surfaces can be broken down into parallelograms.

The paper establishes analogs of Stallings' theorem for group homomorphisms and their nilpotent quotients.

If G and H are finitely generated, residually nilpotent metabelian groups, H is termed para-G if there is a homomorphism of G into H which induces an isomorphism between the corresponding terms of their lower central quotient groups. We prove that this is an equivalence relation. It is a much coarser relation than isom…

For a oriented genus g surface with one boundary component, S, the Torelli group is the group of orientation preserving homeomorphisms of S that induce the identity on homology. The Magnus representation of the Torelli group represents the action on F/F" where F=pi_1(S) and F" is the second term of the derived series. …

We establish a criterion for when an abelian extension of infinite-dimensional Lie algebras integrates to a corresponding Lie group extension $\hat{G}$ of $G$ by $A$, where $G$ is a connected, simply connected Lie group and $A$ is a quotient of its Lie algebra by some discrete subgroup. When $G$ is non-simply connected…

Study on 4-manifolds for special Kähler metrics with constant Ricci determinant.

We prove that groups that are mod-p-homology equivalent are isomorphic modulo any term of their derived p-series, in precise analogy to Stallings' 1963 result for the lower-central p-series. Similarly spaces that are mod-p-homology equivalent have fundamental groups that are isomorphic modulo any term of their p-derive…

In this paper, we classify the three-dimensional contact partially hyperbolic diffeomorphisms whose stable, unstable and central distributions are smooth, and whose non-wandering set equals the whole manifold. We prove that up to a finite quotient or a finite power, they are smoothly conjugated either to the time-one m…

In the paper of Yu. A. Mikhalchishina for an arbitrary virtual link $L$ three groups $G_{1,r}(L)$, $r>0$, $G_{2}(L)$ and $G_{3}(L)$ were defined. In the present paper these groups for the virtual trefoil are investigated. The structure of these groups are found out and the fact that some of them are not isomorphic to e…

New sub-Riemannian structures fail synthetic curvature bounds.

We describe an algebraic proof of the well-known topological fact that $π_1(SO(n)) \cong Z/2Z$. The fundamental group of $SO(n)$ appears in our approach as the center of a certain finite group defined by generators and relations. The latter is a factor group of the braid group $B_n$, obtained by imposing one additional…

Study shows conjugacy of torsion in genus 2 surfaces.

We show, finitely generated rational $\mathsf{VIC}_{\mathbb Q}$-modules and $\mathsf{SI}_{\mathbb Q}$-modules are uniformly representation stable and all their submodules are finitely generated. We use this to prove two conjectures of Church and Farb, which state that the quotients of the lower central series of the To…

We prove that for a relatively hyperbolic group G there is a sequence of relatively hyperbolic proper quotients such that their growth rates converge to the growth rate of G. Under natural assumptions, the same conclusion holds for the critical exponent of a cusp-uniform action of G on a hyperbolic metric space. As a c…

Constructing exponential families from statistical manifolds.

The paper proves residual finiteness of certain lattices and constructs surfaces with specific fundamental groups.

Affine deformations of cotangent groupoids

The study proves conjecture for specific Artin groups.

Let F_n be the free group on n generators. Define IA_n to be group of automorphisms of F_n that act trivially on first homology. The Johnson homomorphism in this setting is a map from IA_n to its abelianization. The first goal of this paper is to determine how much this map contributes to the second rational cohomology…

The study explores splitting conditions for mixed braid group sequences.

Let $G$ be a Garside group with Garside element $Δ$. An element $g$ in $G$ is said to be \emph{periodic} if some power of $g$ lies in the cyclic group generated by $Δ$. This paper shows the following. (i) The periodicity of an element does not depend on the choice of a particular Garside structure if and only if the ce…

This paper extends previous work on genus two fibrations by studying and resolving singular fibers.

A homogeneous nilpotent Lie group has a scaling automorphism determined by a grading of its Lie algebra. Many proofs of upper bounds for the Dehn function of such a group depend on being able to fill curves with discs compatible with this grading; the area of such discs changes predictably under the scaling automorphis…

In a previous work it is shown that every finite group $G$ of diffeomorphisms of a connected smooth manifold $M$ of dimension $\geq 2$ equals, up to quotient by the flow, the centralizer of the group of smooth automorphisms of a $G$-invariant complete vector field $X$ (shortly $X$ describes $G$). Here the foregoing res…