Related papers: Global rigidity of smooth ${\mathbb Z}\ltimes_\lam…

Let $\Lambda_0$ be an ordered abelian group. We show how an $\mathrm{ATF}(\mathbb{Z}\times\Lambda_0)$ group -- that is, a group admitting a free affine action without inversions on a $\mathbb{Z}\times\Lambda_0$-tree -- admits a natural…

We show the following dichotomy for a linear parabolic $\mathbb Z^2$-action $\rho_L$ on the torus with at least one step-2 generator: (i) Any affine $\mathbb Z^2$-action with linear part $\rho_L$ has a $\mathbb Z$-factor that is either…

Let $\Lambda$ be an ordered abelian group, $\mathrm{Aut}^+(\Lambda)$ the group of order-preserving automorphisms of $\Lambda$, $G$ a group and $\alpha:G\to\mathrm{Aut}^+(\Lambda)$ a homomorphism. An $\alpha$-affine action of $G$ on a…

We establish a strong form of local rigidity for hyperbolic automorphisms of the 3-torus with real spectrum. Namely, let $L\colon\mathbb T^3\to\mathbb T^3$ be a hyperbolic automorphism of the 3-torus with real spectrum and let $f$ be a…

We consider actions of cocompact lattices in semisimple Lie groups of the noncompact type on their boundaries $G/Q$, $Q$ a parabolic group, the so-called standard actions. We show that perturbations of the standard action in the…

Let X' be a smooth contractible three-dimensional affine algebraic variety with a free algebraic C_+ -action on it such that S =X'// C_+ is smooth. We prove that X' is isomorphic to $S \times \C$ and the action is induced by a translation…

Consider a smooth, locally free, codimension-one action of a higher-rank, simple, split Lie group $G$ on a closed manifold $M$. Let $P$ be a minimal parabolic subgroup of $G$. If the action admits a $P$-invariant probability measure that is…

Let $\Lambda$ be a countably infinite property (T) group, and let $D$ be UHF-algebra of infinite type. We prove that there exists a continuum of pairwise non (weakly) cocycle conjugate, strongly outer actions of $\Lambda$ on $D$. The proof…

Consider a free ergodic measure preserving profinite action $\Gamma\curvearrowright X$ (i.e. an inverse limit of actions $\Gamma\curvearrowright X_n$, with $X_n$ finite) of a countable property (T) group $\Gamma$ (more generally of a group…

The heterotic and type II superstring actions are identified in different anomaly-free decompositions of a single topological sigma-model action depending on bosonic and fermionic coordinates, $X^\mu$ and $\r^A$ respectively, and of their…

We deal with the general problem of connectedness for the space of $\mathbb{Z}^d$ actions by (orientation-preserving) diffeomorphisms of a compact 1-manifold. We prove two results. First, the space of $\mathbb{Z}^d$ actions by $C^2$…

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…

Let $G$ be the group of orientation-preserving isometries of a rank-one symmetric space $X$ of non-compact type. We study local rigidity of certain actions of a solvable subgroup $\Gamma \subset G$ on the boundary of $X$, which is…

Motivated by Popa's seminal work \cite{Po04}, in this paper, we provide a fairly large class of examples of group actions $\Gamma \curvearrowright X$ satisfying the extended Neshveyev-St{\o}rmer rigidity phenomenon \cite{NS03}: whenever…

We construct affine uniformly Lipschitz actions on $\ell^1$ and $L^1$ for certain groups with hyperbolic features. For acylindrically hyperbolic groups, our actions have unbounded orbits, while for residually finite hyperbolic groups and…

We show that ergodic affine abelian discrete actions on the torus, that have a rank-one factor in their linear part, are locally rigid in a KAM sense if and only if the rank one factor is trivial and the action is higher-rank transversally…

The unique irreducible representation of $\SL_2(\R)$ on $\R^n$ induces an action, called the \textit{linear action}, of $\SL_2(\Z)$ on the torus $\T^n$ for every $n\geq 2$. For $n$ odd, it factors through $\PSL_2(\Z)$, so we denote by $G_n$…

In this paper we study rigidity properties of abelian \hyphenation{break-able}actions with weak or no hyperbolicty. We introduce a general strategy for proving $C^\infty$ local rigidity of algebraic actions. As a consequence, we show…

Let $G$ be a locally compact abelian group. By modifying a theorem of Pedersen, it follows that actions of $G$ on $C^*$-algebras $A$ and $B$ are outer conjugate if and only if there is an isomorphism of the crossed products that is…

An analog of the Baumslag-Solitar group BS(1,k) naturally acts on the sphere by conformal transformations. The action is not locally rigid in higher dimension, but exhibits a weak form of local rigidity. More precisely, any perturbation…