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 PSL(2,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, fo...

We prove that a generic linear cocycle over a minimal base dynamics of finite dimension has the property that the Oseledets splitting with respect to any invariant probability coincides almost everywhere with the finest dominated splitting. Therefore the restriction of the generic cocycle to a subbundle of the finest dominated splitting is uniformly subexponentially quasiconformal. This extends...

let $mathcal{r}$ be a  commutative ring with identity, let $a$ and $b$ be two $mathcal{r}$-algebras and $varphi:blongrightarrow a$ be an $mathcal{r}$-additive algebra homomorphism. we introduce a new algebra $atimes_varphi b$, and give some basic properties of this algebra. generalized $2$-cocycle derivations on $atimes_varphi b$ are studied. accordingly, $atimes_varphi b$ is considered from th...

We extend the Yang-Baxter cocycle invariants for virtual knots by augmenting Yang-Baxter 2-cocycles with cocycles from a cohomology theory associated to a virtual biquandle structure. These invariants coincide with the classical Yang-Baxter cocycle invariants for classical knots but provide extra information about virtual knots and links. In particular, they provide a method for detecting non-c...

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, so it can be viewed as a universal 2-dimensional characteristic class for these groups. Over R and C, it coincides with the first Che...

We describe infinite clusters which arise in nearest-neighbour percolation for so-called cocycle measures on the square lattice. These measures arise naturally in the study of random transformations. We show that infinite clusters have a very specific form and direction. In concrete situations, this leads to a quick decision whether or not a certain cocycle measure percolates. We illustrate thi...

Résumé. — Un cocycle lisse est un produit gauche qui agit par des difféomorphismes dans les fibres. Si les exposants de Lyapounov extremaux du cocycle coincident alors les fibres possèdent certaines structures qui sont invariantes, à la fois, par la dynamique et par un pseudo-groupe canonique de transformations d’holonomie. Nous démontrons ce principe d’ invariance pour les cocycles lisses au d...

We consider 1-cocycles with values in locally compact, second countable abelian groups on discrete, nonsingular, ergodic equivalence relations. If such a cocycle is invariant under certain automorphisms of these relations we show that the skew product extension deened by the cocycle is ergodic. As an application we obtain an extension of many of the results in 9] to higher-dimensional shifts of...