arXiv research

Two commutators generate mapping class groups of surfaces with genus ≥5.

Study on infinite-type surfaces shows stable commutator length is continuous and defines open subgroups.

We give new upper bounds on the stable commutator lengths of Dehn twists in mapping class groups and new lower bounds on the stable commutator lengths of Dehn twists in hyperelliptic mapping class groups. In particular, we show that the stable commutator lengths of Dehn twists about a nonseparating and a separating cur…

New bounds on twist commutators for separating curves on surfaces.

We give a new upper bound on the stable commutator length of Dehn twists in hyperelliptic mapping class groups, and determine the stable commutator length of some elements. We also calculate values and the defects of homogeneous quasimorphisms derived from ω-signatures, and show that they are linearly independent in th…

We show that, in compact semisimple Lie groups and Lie algebras, any neighbourhood of the identity gets mapped, under the commutator map, to a neighbourhood of the identity.

Reformulates divergence map for Turaev cobracket in non-commutative geometry.

The paper examines conditions for commuting conjugates of finite-order mapping classes.

The paper finds a family of 2D surfaces where Laplacian and Dirichlet-to-Neumann commute.

Billiards in confocal quadrics show pluri-Lagrangian systems in action.

The quaternions are non-commutative. The deviation from commutativity is encapsulated in the commutator of unit quaternions. It is known that the k-th power of the commutator is null-homotopic if and only if k is divisible by 12. The main purpose of this paper is to construct a concrete null-homotopy of the 12-th power…

Constructs non-commutative modular vector fields for Poisson manifolds.

Maps commuting with sub-Laplacians on Carnot groups are conformal.

Study characteristic classes for TC structures on principal G-bundles.

Let $Γ$ be a finite index subgroup of the mapping class group $MCG(Σ)$ of a closed orientable surface $Σ$, possibly with punctures. We give a precise condition (in terms of the Nielsen-Thurston decomposition) when an element $g\inΓ$ has positive stable commutator length. In addition, we show that in these situations th…

We show that on a nonorientable surface of genus at least 7 any power of a Dehn twist is equal to a single commutator in the mapping class group and the same is true, under additional assumptions, for the twist subgroup, and also for the extended mapping class group of an orientable surface of genus at least 3.

It is proved that the stable commutator length of a Dehn twist in the mapping class group is positive and the tenth power of a Dehn twist about a nonseparating simple closed curve is a product of two commutators. As an application a new proof of the fact that the growth rate of a Dehn twist is linear is given.

New quasimorphisms show stable commutator lengths are not equivalent.

We introduce some chain maps between Khovanov complexes. Each of the chain maps commutes with a chain homotopy map and a retraction maps which obtain a Reidemeister invariance of Khovanov homology.

Commutes Pansu pullback with spectral complexes in Carnot groups.

We solved a conjecture about braid group quotients being alternating groups.

Assume that all spaces and maps are localised at a fixed prime $p$. We study the possibility of generating a universal space $U(X)$ from a space $X$ which is universal in the category of homotopy associative, homotopy commutative H-spaces in the sense that any map f:X->Y to a homotopy associative, homotopy commutative …

Smooth structures on infinite dimensional Grassmannians and non-commutative cross-ratios.

The paper studies surface bundles and Dehn twists, providing new bounds and factorizations.

Infinitesimal calculations link fundamental groups to Lie algebras.

We show that stable commutator length is rational on free products of free Abelian groups amalgamated over $\mathbb{Z}^k$, a class of groups containing the fundamental groups of all torus knot complements. We consider a geometric model for these groups and parameterize all surfaces with specified boundary mapping to th…

For any group, there is a natural (pseudo-)norm on the vector space B1 of real (group) 1-boundaries, called the stable commutator length norm. This norm is closely related to, and can be thought of as a relative version of, the Gromov (pseudo)-norm on (ordinary) homology. We show that for a free group, the unit ball of…

Introduces $\mathbb{K}$-framings for surfaces, generalizing quadratic forms.

Study stable commutator length in free products using surface maps.

The paper studies stability of commutativity properties of the Dirichlet-to-Neumann map.

We combine concepts from random matrix theory and free probability together with ideas from the theory of commutator length in groups and maps from surfaces, and establish new connections between the two. More particularly, we study measures induced by free words on the unitary groups $U(n)$. Every word $w$ in the free…

Let G be a connected Lie group with Lie algebra g. The Duflo map is a vector space isomorphism between the symmetric algebra S(g) and the universal enveloping algebra U(g) which, as proved by Duflo, restricts to a ring isomorphism from invariant polynomials onto the center of the universal enveloping algebra. The Duflo…

No almost complex structures on the six-sphere satisfy certain commutation conditions.

The pentagram map preserves Poncelet polygons in convex cases.

We outline the notions and concepts of the calculus of variational multivectors within the Poisson formalism over the spaces of infinite jets of mappings from commutative (non)graded smooth manifolds to the factors of noncommutative associative algebras over the equivalence under cyclic permutations of the letters in t…

The paper derives inequalities for Riemannian maps and submersions involving quaternionic space forms.

We give examples of foliations that answer two questions posed by Mitsumatsu and Vogt about the genus minimising properties of closed leaves of 2-dimensional foliations on 4-manifolds. By studying stable commutator lengths in certain stable mapping class groups, we also answer an asymptotic version of another question …

Study on homeomorphism groups of telescoping 2-manifolds showing strong distortion.

Paper analyzes covering monopole maps over compact four-manifolds, inducing homomorphisms and Sobolev estimates.

Constructs symplectic surface bundles with positive signatures.

We give manifolds in both the Riemannian and in the higher signature settings whose Riemann curvature operators commute, i.e. which satisfy R(a,b)R(c,d)=R(c,d)R(a,b) for all tangent vectors. These manifolds have global geometric phenomena which are quite different for higher signature manifolds than they are for Rieman…

Boundary Dehn twists become trivial after abelianization.

Quantum Teichmüller theory constants confirmed for cluster varieties.

We describe explicit presentations of all stable and the first nonstable homotopy groups of the unitary groups. In particular, for each n >= 2 we supply n homotopic maps that each represent the (n-1)!-th power of a suitable generator of pi_2n(U(n)) = Z_{n!}. The product of these n commuting maps is the constant map to …

Study the commutativity of reduction and symplectification in contact Hamiltonian systems.

Let S be a closed surface with nonzero Euler characteristic. We prove the existence of an open neighborhood V of the identity map of S in the C^1-topology with the following property: if G is an abelian subgroup of Diff^1(S) generated by any family of elements in V then the elements of G have common fixed points. This …

New projection operators for multipatch spaces with stable properties.

Study of homology commutativity in separable metrizable spaces.