Related papers: Cocycle rigidity of abelian partially hyperbolic a…

We show local and cocycle rigidity for $\R^k \times \Z^l$ partially hyperbolic translation actions on homogeneous spaces $\mc G/ \Lambda$. We consider a large class of actions whose geometric properties are more complicated than previously…

We extend the recent progress on the cocycle rigidity of partially hyperbolic homogeneous abelian actions to the setting with rank 1 factors in the universal cover. The method of proof relies on the periodic cycle functional and analysis of…

Let $\mathbb{G}$ be a higher-rank connected semisimple Lie group with finite center and without compact factors. In any unitary representation $(\pi, \mathcal{H})$ of $\mathbb{G}$ without non-trivial $\mathbb{G}$-fixed vectors, we study the…

We give a complete solution to the local classification program of higher rank partially hyperbolic algebraic actions. We show $C^\infty$ local rigidity of abelian ergodic algebraic actions for symmetric space examples, twisted symmetric…

We consider partially hyperbolic abelian algebraic high-rank actions on compact homogeneous spaces obtained from simple indefinite orthogonal and unitary groups. In the first part of the paper, we show local differentiable rigidity for such…

We study the twisted cohomoligical equation over the geodesic flow on $SL(2,\mathbb{R})/\Gamma$. We characterize the obstructions to solving the twisted cohomological equation, construct smooth solution and obtain the tame Sobolev estimates…

Let $\mathbb{G}$ denote a higher rank $\mathbb R$-split simple Lie group of the following type: $SL(n,\mathbb R)$, $SO_o(m,m)$, $E_{6(6)}$, $E_{7(7)}$ and $E_{8(8)}$, where $m\geq 4$ and $n \geq 3$. We study the cohomological equation for…

We prove the locally differentiable rigidity of generic partially hyperbolic abelian algebraic high-rank actions on compact homogeneous spaces obtained from split symplectic Lie groups. We also gave a non-generic action rigidity example on…

For any unitary representation $(\pi,\mathcal{H})$ of $G=SL(n,\RR)$, $n\geq 3$ without non-trivial $G$-invariant vectors, we study smooth solutions of the cohomological equation $\mathfrak{u}f=g$ where $\mathfrak{u}$ is a vector in the root…

We prove absolute continuity of "high entropy" hyperbolic invariant measures for smooth actions of higher rank abelian groups assuming that there are no proportional Lyapunov exponents. For actions on tori and infranilmanifolds existence of…

We obtain a global rigidity result for abelian partially hyperbolic higher rank actions on certain $2-$step nilmanifolds $X_{\Gamma}$. We show that, under certain natural assumptions, all such actions are $C^{\infty}-$conjugated to an…

Suppose $L$ is a semisimple Levi subgroup of a connected Lie group~$G$, $X$ is a Borel $G$-space with finite invariant measure, and $\alpha \colon X \times G \to \GL_n(\real)$ is a Borel cocycle. Assume $L$ has finite center, and that the…

We study actions by higher-rank abelian groups on quotients of semisimple Lie groups with finite center. First, we consider actions arising from the flows of two commuting elements of the Lie algebra--one nilpotent, and the other…

Suppose $G$ is a finitely generated group and $H$ is a subgroup of $G$. Let $\partial_{c}^{\mathcal{F}\mathcal{Q}}G$ denote the contracting boundary of $G$ with the topology of fellow travelling quasi-geodesics defined by Cashen-Mackay…

Let G be a group admitting a non-elementary acylindrical action on a Gromov hyperbolic space (for example, a non-elementary relatively hyperbolic group, or the mapping class group of a closed hyperbolic surface, or Out(F_n) for n>1). We…

Let $G$ be a compactly generated locally compact group and $(\pi, \mathcal H)$ a unitary representation of $G.$ The $1$-cocycles with coefficients in $\pi$ which are harmonic (with respect to a suitable probability measure on $G$) represent…

In this paper we study perturbations of constant cocycles for actions of higher rank semi-simple algebraic groups and their lattices. Roughly speaking, for ergodic actions, Zimmer's cocycle superrigidity theorems implies that the perturbed…

This is a further investigation of our approach to group actions in homological algebra in the settings of homology of {\Gamma}-simplicial groups, particularly of {\Gamma}-equivariant homology and cohomology of {\Gamma}-groups. This…

Let G be a group, H a hyperbolically embedded subgroup of G, V a normed G-module, U an H-invariant submodule of V. We propose a general construction which allows to extend 1-quasi-cocycles on H with values in U to 1-quasi-cocycles on G with…

Let $A$ be a Hopf algebra over a field $K$ of characteristic 0 and suppose there is a coalgebra projection $\pi$ from $A$ to a sub-Hopf algebra $H$ that splits the inclusion. If the projection is $H$-bilinear, then $A$ is isomorphic to a…