Jarek Kędra

Crossed flux homomorphisms and vanishing theorems for Flux groups

Abstract.We study the flux homomorphism for closed forms of arbitrary degree, with special emphasis on volume forms and on symplectic forms. The volume flux group is an invariant of the underlying manifold, whose non-vanishing implies that the manifold resembles one with a circle action with homologically essential orbits.

On the autonomous metric on the group of area-preserving diffeomorphisms of the 2–disc

EDRA Let D 2 be the open unit disc in the Euclidean plane and let GWD Diff.D 2 ; area/ be the group of smooth compactly supported area-preserving diffeomorphisms of D 2 . For every natural number k we construct an injective homomorphism Z k ! G , which is bi-Lipschitz with respect to the word metric on Z k and the autonomous metric on G . We also show that the space of homogeneous quasimorphisms vanishing on all autonomous diffeomorphisms in the above group is infinite-dimensional. 57S05

ON BI-INVARIANT WORD METRICS

We prove that bi-invariant word metrics are bounded on certain Chevalley groups. As an application we provide restrictions on Hamiltonian actions of such groups.

Concordance group and stable commutator length in braid groups

We define a quasihomomorphism from braid groups to the concordance group of knots and examine its properties and consequences of its existence. In particular, we provide a relation between the stable four ball genus in the concordance group and the stable commutator length in braid groups, and produce examples of infinite families of concordance classes of knots with uniformly bounded four ball genus. We also provide applications to the geometry of the infinite braid group. In particular, we show that its commutator subgroup admits a stably unbounded conjugation invariant norm. This answers an open problem posed by Burago, Ivanov and Polterovich.

THE CANCELLATION NORM AND THE GEOMETRY OF BI-INVARIANT WORD METRICS

We study bi-invariant word metrics on groups. We provide an efficient algorithm for computing the bi-invariant word norm on a finitely generated free group and we construct an isometric embedding of a locally compact tree into the bi-invariant Cayley graph of a nonabelian free group. We investigate the geometry of cyclic subgroups. We observe that in many classes of groups, cyclic subgroups are either bounded or detected by homogeneous quasimorphisms. We call this property the bq-dichotomy and we prove it for many classes of groups of geometric origin.

On the algebraic independence of Hamiltonian characteristic classes

We prove that Hamiltonian characteristic classes defined as fibre integrals of powers of the coupling class are algebraically independent for generic coadjoint orbits.

On the autonomous norm on the group of Hamiltonian diffeomorphisms of the torus

We prove that the autonomous norm on the group of Hamiltonian diffeomorphisms of the two-dimensional torus is unbounded. We provide explicit examples of Hamiltonian diffeomorphisms with arbitrarily large autonomous norm. For the proofs we construct quasimorphisms on

A two-cocycle on the group of symplectic diffeomorphisms

Ham(T^2)

On the Cohomology of Classifying Spaces of Groups of Homeomorphisms

and some of them are Calabi.

Remarks on monotone Lagrangians in

We investigate the properties of a two-cocycle on the group of symplectic diffeomorphisms of an exact symplectic manifold defined by Ismagilov, Losik, and Michor. We provide both vanishing and nonvanishing results and applications to foliated symplectic bundles and to Hamiltonian actions of finitely generated groups.