arXiv research

Study proves surface group quotients have non-virtually solvable profinite completions.

A new presentation of a quotient of braid groups leads to a new type of Burnside group.

We define and study the Burnside quotient Green ring of a Mackey functor. Some refinements of Dress induction theory are presented, together with applications to computation results for $K$-theory and $L$-theory of finite and infinite groups.

Refines quantum annular homology using stable homotopy methods.

The paper introduces $n$-moves for links and studies their Burnside groups.

Novel groups exhibit contradictory behaviors with respect to Burnside laws.

Extends quantum annular homology to infinite sets.

New Euler characteristic and Burnside group defined for definable groupoids.

Yasutaka Nakanishi asked in 1981 whether a 3-move is an unknotting operation. In Kirby's problem list, this question is called `The Montesinos-Nakanishi 3-move conjecture'. We define the n-th Burnside group of a link and use the 3rd Burnside group to answer Nakanishi's question; ie, we show that some links cannot be re…

Refines Khovanov homology using signed Burnside categories.

In a pair of recent papers (one to appear and one forthcoming), the author develops a general version of small cancellation theory applicable in higher dimensions, and then applies this theory to the Burnside groups of sufficiently large exponent. The present article gives a brief introduction to the methods and techni…

This paper is motivated by a general question: for which values of k and n is the universal Burnside kei of k generators and Kei "exponent" n, $\bar Q(k,n)$, finite? It is known (starting from the work of M. Takasaki (1942)) that $\bar Q(2,n)$ is isomorphic to the dihedral quandle Z_n and $\bar Q(3,3)$ is isomorphic to…

The study of periodic subgroups in homeomorphism groups of manifolds.

Constructs Khovanov spectra for periodic links, proving rank inequalities.

We compute Steenrod squares on Khovanov homology.

We define for arbitrary modules over a finite von Neumann algebra $\cala$ a dimension taking values in $[0,\infty]$ which extends the classical notion of von Neumann dimension for finitely generated projective $\cala$-modules and inherits all its useful properties such as additivity, cofinality and continuity. This all…

Functor decomposes Khovanov spectra for non-alternating diagrams.

In Classical Knot Theory and in the new Theory of Quantum Invariants substantial effort was directed toward the search for unknotting moves on links. We solve, in this note, several classical problems concerning unknotting moves. Our approach uses a new concept, Burnside groups of links, which establishes unexpected re…

There are (at least) two different approaches to define equivariant analogue of the Euler charateristic for a space with a finite group action. The first one defines it as an element of the Burnside ring of the group. The second approach emerged from physics and includes the orbifold Euler characteristic and its higher…

We classify 3-braids up to (2,2)-move equivalence and in particular we show how to adjust the Harikae-Nakanishi-Uchida conjecture so it holds for closed 3-braids. As important steps to classify 3-braids up to (2,2)-move equivalence we prove the conjecture for 2-algebraic links and classify (2,2)-equivalence classes for…

We discuss several open problems in classical Knot Theory and we develop techniques that allow us to study them: Lagrangian tangles, skein modules and Burnside groups.

Researchers found all embeddings of Kuratowski graphs on a double torus.

The purpose of this expository paper is to present new directions in the classical Nielsen-Reidemeister fixed point theory. We describe twisted Burnside-Frobenius theorem, groups with $R_\infty$ \emph{property} and a connection between Nielsen fixed point theory and symplectic Floer homology.

The purpose of the present paper is to prove for finitely generated groups of type I the following conjecture of A.Fel'shtyn and R.Hill, which is a generalization of the classical Burnside theorem. Let G be a countable discrete group, f one of its automorphisms, R(f) the number of f-conjugacy classes, and S(f)=# Fix (f…

In this note we present a combinatorial link invariant that underlies some recent stable homotopy refinements of Khovanov homology of links. The invariant takes the form of a functor between two combinatorial 2-categories, modulo a notion of stable equivalence. We also develop some general properties of such functors.

In a previous paper, the authors defined an equivariant version of the so-called Saito duality between the monodromy zeta functions as a sort of Fourier transform between the Burnside rings of an abelian group and of its group of characters. Here a so-called enhanced Burnside ring $\widehat{B}(G)$ of a finite group $G$…

Let G be a compact Lie group and A(G) its Burnside Ring. For a compact smooth n-dimensional G-manifold X equipped with a generic G-invariant vector field v, we prove an equivariant analog of the Morse formula Ind^G(v) = \sum_{k = 0}^{n} (-1)^k χ^G(\d_k^+X) which takes its values in A(G). Here Ind^G(v) denotes the equiv…

New proof confirms subgroup commensurators for $N\geq 3$.

Study shows equivariant Khovanov homotopy types are equivalent.

In this paper, we give a new construction of a Khovanov homotopy type. We show that this construction gives a space stably homotopy equivalent to the Khovanov homotopy types constructed in [LS14a] and [HKK] and, as a corollary, that those two constructions give equivalent spaces. We show that the construction behaves w…

The paper classifies Kleinian groups with Hausdorff dimension less than 1.

We study a coarse homology theory with prescribed growth conditions. For a finitely generated group G with the word length metric this homology theory turns out to be related to amenability of G. We characterize vanishing of a certain fundamental class in our homology in terms of an isoperimetric inequality on G and sh…

In this article we extend the classical definitions of equivariant cohomotopy theory to the setting of proper actions of Lie groups. We combine methods originally developed in the analysis of nonlinear differential equations, mainly in connection with Leray-Schauder theory, and on the other hand from developments of eq…

Study on realizing subgroup twists in 3-manifolds.

New operations match Steenrod squares on Khovanov homology.

Study examines quotients of affine connection control systems.

The paper studies hyperbolic quotients of projection complexes and their actions.

The paper introduces pseudo-quotients for algebraic actions and applies them to character varieties.

Study non-arithmetic orbifold ball quotients using Jacobian of Bolza curve.

Sharp bounds found on nonabelian quotients of surface braid groups.

Finite quotients of braid groups have large sizes.

Study quotients of curve complex actions by mapping class group.

Study finds smallest non-trivial quotients of braid groups and commutator subgroups.

New infinite discrete group without finite quotients found.

Maximal cubic quotient of braid algebra studied for n ≤ 5.

Paper compares total quotient curvature and proves bounds for Einstein metric.

Researchers show how complex structures vary in symplectic quotients.

Establishes a concavity property for positive Hessian quotient operators.