We introduce a 2-cocycle for symplectic and skew-hermitian hyperbolic groups over arbitrary fields and skew-fields, with values in the Witt group of hermitian forms. This cocycle has good functorial properties: it is natural under extension of scalars and stable, and so it can be viewed as a universal 2-dimensional characteristic class for these groups. Over R and C, it coincides with the first...

For the space of long knots in R, Vassiliev’s theory defines the so called finite order cocycles. Zero degree cocycles are finite type knot invariants. The first non-trivial cocycle of positive dimension in the space of long knots has dimension one and order three. We apply Vassiliev’s combinatorial formula, given in [1], and find the value mod2 of this cocycle on the 1-cycles that are obtained...

We establish braided tensor equivalences among module categories over the twisted quantum double of a finite group defined by an extension of a group G by an abelian group, with 3-cocycle inflated from a 3-cocycle on G. We also prove that the canonical ribbon structure of the module category of any twisted quantum double of a finite group is preserved by braided tensor equivalences. We give two...

Let T be the unit circle with an irrational rotation T : x 7→ x+α mod 1, F a real or circle valued cocycle (i.e. we consider cylinder flows and Anzai skew products). A0 and L0 are the Banach spaces of zero mean absolutely continuous, respectively Lipschitz (real or T-valued) cocycles. Two cocycles on T are said to be cohomologous if they differ in a coboundary, i.e. a function G − G ◦ T where G...

A double complex consisting of superspace forms under the action of the de Rham differential and a ghost field differential is introduced. A naturally associated single complex with a total differential D is defined. The Wess-Zumino term becomes a potential for a D cocycle associated with the p-brane. The algebras of generators of the left and right group actions are modified in the presence of...

Cocycles of Zm-actions on compact metric spaces provide a means for constructing Rm-actions or flows, called suspension flows. It is known that all Rm flows with a free dense orbit have an almost one-to-one extension which is a suspension flow. In this paper we investigate when the space for a suspension flow depends only on the given Zm-action and not on the actual cocycle. The identity map I ...

This text is an expanded version of the lecture notes of a minicourse (with the same title of this text) delivered by the authors in the Bedlewo school “Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory” (from 4 to 16 July, 2011). In the first part of this text, i.e., from Sections 1 to 5, we discuss the Teichmüller and moduli space of translation surfaces, the Teich...

In this paper we use the description of free group factors as the von Neumann algebras of Berezin’s deformation of the upperhalf plane, modulo PSL2(Z). The derivative, in the deformation parameter, of the product in the corresponding algebras, is a positive — 2 Hochschild cocycle, defined on a dense subalgebra. By analyzing the structure of the cocycle we prove that there is a generator, L, for...

We discuss different types of almost periodicity of trajectories of cocycles on compact solenoidal groups in relation to various notions of triviality of these cocycles. Introduction Let G be a compact solenoidal group, that is, a compact Abelian group containing, as a dense subgroup, the image of IR by a continuous homomorphism, say a. We shall assume that G is not a circle by requiring a to b...

We prove the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for a measure supported on abelian differentials which come from nonorientable quadratic differentials through a standard orienting, double cover construction. The proof uses Forni’s criterion [For11] for non-uniform hyperbolicity of the cocycle for SL(2,R)-invariant measures. We apply these results to the study of deviatio...