arXiv research

In this work, we study generalized entropies and information geometry in a group-theoretical framework. We explore the conditions that ensure the existence of some natural properties and at the same time of a group-theoretical structure for a large class of entropies. In addition, a method for defining new entropies, u…

It is shown that a locally geometrical structure of arbitrarily curved Riemannian space is defined by a deformed group of its diffeomorphisms

In this paper we develop some group theoretical methods which are shown to be very useful for a better understanding of the properties of the Riccati equation and we discuss some of its integrability conditions from a group theoretical perspective. The nonlinear superposition principle also arises in a simple way.

Study explores associated groups of symmetric quandles and their properties.

The postulate of independence of cause and mechanism (ICM) has recently led to several new causal discovery algorithms. The interpretation of independence and the way it is utilized, however, varies across these methods. Our aim in this paper is to propose a group theoretic framework for ICM to unify and generalize the…

New group-theoretic Johnson classes applied to curves with torsion Ceresa classes.

Based on a general formula due to R.Bryant, we work out the topological structure of the space of torsion-free $G_2$-structures generating the same associated Riemannian metric on a compact $7$-manifold. We also identify a corresponding Lie group-theoretic structure of the space. These observations are then used to des…

This paper generalizes neural transport learning for free energy estimation in arbitrary state spaces.

Following the approach of Dahmani, Guirardel and Osin, we extend the group theoretical Dehn filling theorem to show that the pre-images of infinite order elements have a certain structure of a free product. We then apply this result to show that groups hyperbolic relative to residually finite groups satisfying the Farr…

New group theory insights on knot surgery results.

Given a manifold M, it is natural to ask in how many ways it fibers (we mean fibering in a general way, where the base might be an orbifold -- this could be described as Seifert fibering)There are group-theoretic obstructions to the existence of even one fibering, and in some cases (such as Kahler manifolds or three-di…

Study symmetries in smoothed polygonal links.

In this paper we study the group theoretic structures of colored HOMFLY polynomials in a specific limit. The group structures arise in the perturbative expansion of $SU(N)$ Chern-Simons Wilson loops, while the limit is $N \rightarrow 0$. The result of the paper is twofold. First, we explain the emergence of Kadomsev-Pe…

We compare the random group model of Gromov and the model of generic groups of Arzhantseva and Ol'shanskii.

In the article arXiv:1108.5443 we established a general group-theoretical approach to the construction of Bäcklund transformations. We then showed how this construction can be applied to construct Bäcklund transformation between equations which are Darboux integrable. Here we give a number of detailed examples and new …

Study of origamis' singularities for groups of prime-power order.

This paper shows that, away from 6, the kernel of the Witten genus is precisely the ideal consisting of (bordism classes of) Cayley plane bundles with connected structure group, but only after restricting the Witten genus to string bordism. It does so by showing that the divisibility properties of Cayley plane bundle c…

The paper studies the graph geometry of finite groups, creating a dataset and analyzing its properties.

We show that a certain linear representation of the singular braid monoid on three strands is faithful. Furthermore we will give a second - group theoretically motivated - solution to the word problem in this monoid.

In this article we give the realization of the Klein's Program for geometrical structures (Riemannian spaces and fiber bundles with connection) with arbitrary variable curvature within the framework of infinite deformed groups. These groups generalize gauge groups to the case of nontrivial action on the base space of b…

Study coning totally geodesic boundaries of hyperbolic manifolds.

We show that the stable commutator length vanishes for certain groups defined as infinite unions of smaller groups. The argument uses a group-theoretic analogue of the Mazur swindle, and goes back to the works of Anderson, Fisher, and Mather on homeomorphism groups.

This is an expository article of our work on analogies between knot theory and algebraic number theory. We shall discuss foundational analogies between knots and primes, 3-manifolds and number rings mainly from the group-theoretic point of view.

We present a new conjectural symmetry of the colored Alexander polynomial, that is the specialization of the quantum $\mathfrak{sl}_N$ invariant widely known as the colored HOMFLY-PT polynomial. We provide arguments in support of the existence of the symmetry by studying the loop expansion and the character expansion o…

In 2004, Sormani and Wei introduced the covering spectrum: a geometric invariant that isolates part of the length spectrum of a Riemannian manifold. In their paper they observed that certain Sunada isospectral manifolds share the same covering spectrum, thus raising the question of whether the covering spectrum is a sp…

We continue our study of ends of non-compact manifolds, with a focus on the inward tameness condition. For manifolds with compact boundary, inward tameness, has significant implications. For example, such manifolds have stable homology at infinity in all dimensions. We show that these manifolds have 'almost perfectly s…

The A-B slice problem, a reformulation of the 4-dimensional topological surgery conjecture for free groups, is shown to admit a link-homotopy+ solution. The proof relies on geometric applications of the group-theoretic 2-Engel relation. Implications for the surgery conjecture are discussed.

Let O be a compact orientable 3-orbifold with non-empty singular locus and a finite volume hyperbolic structure. (Equivalently, O is the quotient of hyperbolic 3-space by a lattice in PSL(2,C) with torsion.) Then we prove that O has a tower of finite-sheeted covers {O_i} with linear growth of p-homology, for some prime…

Classifies good involutions in conjugation subquandles and racks.

A trisection of a smooth, closed, oriented 4-manifold is a decomposition into three 4-dimensional 1-handlebodies meeting pairwise in 3-dimensional 1-handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the 3-dimensional handlebodies, the 4-dimensional handlebodies, and the clo…

Every closed orientable surface S has the following property: any two connected covers of S of the same degree are homeomorphic (as spaces). In this, paper we give a complete classification of compact 3-manifolds with empty or toroidal boundary which have the above property. We also discuss related group-theoretic ques…

It is known that Garside groups are strongly translation discrete. In this paper, we show that the translation numbers in a Garside group are rational with uniformly bounded denominators and can be computed in finite time. As an application, we give solutions to some group-theoretic problems.

The paper classifies holomorphic maps between Riemann surface configuration spaces.

New Garside structures derived from groups, leading to new group properties.

A group-theoretical method, via Wada's representations, is presented to distinguish Kishino's virtual knot from the unknot. Biquandles are constructed for any group using Wada's braid group representations. Cocycle invariants for these biquandles are studied. These invariants are applied to show the non-existence of Al…

Abstraction plays a key role in concept learning and knowledge discovery; this paper is concerned with computational abstraction. In particular, we study the nature of abstraction through a group-theoretic approach, formalizing it as symmetry-driven---as opposed to data-driven---hierarchical clustering. Thus, the resul…

Novel symmetry found in colored HOMFLY polynomials from superalgebras.

We consider the deformation of a discontinuous group acting on the Euclidean space by affine transformations. A distinguished feature here is that even a `small' deformation of a discrete subgroup may destroy proper discontinuity of its action. In order to understand the local structure of the deformation space of disc…

Finite type invariants (also known as Vassiliev invariants) of pure braids are considered from a group-theoretic point of view. New results include a construction of a universal invariant with integer coefficients based on the Magnus expansion of a free group and a calculation of numbers of independent invariants of ea…

The paper extends quasimorphisms on subgroups to larger groups.

New methods reveal symmetries in Chern-Simons theory.

The results of Culler and Shalen for 2,3 or 4-free hyperbolic 3-manifolds are contingent on properties specific to and special about rank two subgroups of a free group. Here we determine what construction and algebraic information is required in order to make a geometric statement about $M$, a closed, orientable hyperb…

For right-angled Coxeter groups $W_Γ$, we obtain a condition on $Γ$ that is necessary and sufficient to ensure that $W_Γ$ is thick and thus not relatively hyperbolic. We show that Coxeter groups which are not thick all admit canonical minimal relatively hyperbolic structures; further, we show that in such a structure, …

We study complexities of 3-manifolds defined from triangulations, Heegaard splittings, and surgery presentations. We show that these complexities are related by linear inequalities, by presenting explicit geometric constructions. We also show that our linear inequalities are asymptotically optimal. Our results are used…

Surveying methods to create spaces with non-trivial self covers.

The paper proposes conjectures about moduli space rigidity.

We prove that an irreducible 3-manifold whose fundamental group satisfies a certain group-theoretic property called RFRS is virtually fibered. As a corollary, we show that 3-dimensional reflection orbifolds and arithmetic hyperbolic orbifolds defined by a quadratic form virtually fiber. These include the Seifert Weber …

Researchers identify graph components for unicellular collections.