First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity

This is the first in a series of papers exploring rigidity properties of hyperbolic actions ofZ ^k orR ^k fork ≥ 2. We show that for all known irreducible examples, the cohomology of smooth cocycles over these actions is trivial. We also obtain similar Hölder and C¹ results via a generalization of the Livshitz theorem for Anosov flows. As a consequence, there are only trivial smooth or Hölder time changes for these actions (up to an automorphism). Furthermore, small perturbations of these actions are Hölder conjugate and preserve a smooth volume.     

Tangential flatness and global rigidity of higher rank lattice actions

We establish the continuous tangential flatness for orientable
weakly Cartan actions of higher rank lattices. As a corollary, we obtain the global rigidity of Anosov Cartan actions.

     

Cocycle rigidity of abelian partially hyperbolic actions

Suppose G is a higher-rank connected semisimple Lie group with finite center and without compact factors. Let G = G or G = G ? V, where V is a finite-dimensional vector space V. For any unitary representation (π,H) of G, we study the twisted cohomological equation π(a)f ? λf = g for partially hyperbolic element a ∈ G and λ ∈ U(1), as well as the twisted cocycle equation π(a₁)f ? λ1f = π(a₂)g ? λ₂g for commuting partially hyperbolic elements a₁, a₂ ∈ G. We characterize the obstructions to solving these equations, construct smooth solutions and obtain tame Sobolev estimates for the solutions. These results can be extended to partially hyperbolic flows in parallel.As an application, we prove cocycle rigidity for any abelian higher-rank partially hyperbolic algebraic actions. This is the first paper exploring rigidity properties of partially hyperbolic that the hyperbolic directions don’t generate the whole tangent space. The result can be viewed as a first step toward the application of KAM method in obtaining differential rigidity for these actions in future works.     

Affine actions of higher rank lattices

We consider actions of lattices in certain higher rank simple Lie groups by affine (i.e. connection-preserving) transformations of a compact Riemannian manifold. When the dimension of the manifold is not too large, such actions are partially described here in terms of affine actions on the flat torus and isometric actions. The main tools are Marguils' and Zimmer's rigidity theorems.     

Quasi-isometric rigidity of nonuniform lattices in higher rank symmetric spaces

We compute the quasi-isometry group of an irreducible nonuniform lattice in a semisimple Lie group with finite center and no rank one factors, and show that any two such lattices are quasi-isometric if and only if they are commensurable up to conjugation.

     

Rigidity of higher rank abelian cocycles with values in diffeomorphism groups

We consider cocycles over certain hyperbolic actions, , and show rigidity properties for cocycles with values in a Lie group or a diffeomorphism group, which are close to identity on a set of generators, and are sufficiently smooth. The actions we consider are Cartan actions of or , for , and Γ torsion free cocompact lattice. The results in this paper rely on a technique developed recently by D. Damjanović and A. Katok.      

Joinings of higher rank torus actions on homogeneous spaces

Publications mathématiques de l'IHÉS - We show that joinings of higher rank torus actions on $S$ -arithmetic quotients of semi-simple or perfect algebraic groups must be algebraic.     

Higher order mixing and rigidity of algebraic actions on compact Abelian groups

Let Γ be a discrete group and fori=1,2; letα _i be an action of Γ on a compact abelian groupX _i by continuous automorphisms ofX _i. We study measurable equivariant mapsf: (X ₁,α ₁)→(X ₂,α ₂), and prove a rigidity result under certain assumption on the order of mixing of the underlying actions.     

New geometric examples of Anosov actions

We study a new class of Anosov actions (in the sense of Hirsch, Pugh and Shub) of reductive Lie groups, which are related to Riemannian symmetric spaces of non-compact type. The orbits of these actions can be identified with unions of parallel geodesics and the resulting orbit spaces are symplectic manifolds. For symmetric spaces of rank 1 all actions coincide with the geodesic flow.     

The higher rank rigidity theorem for manifolds with no focal points

We say that a Riemannian manifold M has rank M ≥ k if every geodesic in M admits at least k parallel Jacobi fields. The Rank Rigidity Theorem of Ballmann and Burns–Spatzier, later generalized by Eberlein–Heber, states that a complete, irreducible, simply connected Riemannian manifold M of rank k ≥ 2 (the “higher rank” assumption) whose isometry group Γ satisfies the condition that the Γ-recurrent vectors are dense in SM is a symmetric space of noncompact type. This includes, for example, higher rank M which admit a finite volume quotient. We adapt the method of Ballmann and Eberlein–Heber to prove a generalization of this theorem where the manifold M is assumed only to have no focal points. We then use this theorem to generalize to no focal points a result of Ballmann–Eberlein stating that for compact manifolds of nonpositive curvature, rank is an invariant of the fundamental group.     

A foliated metric rigidity theorem for higher rank irreducible symmetric spaces

LetF be a foliation of a compact manifold with a transverse invariant measure of finite total mass. We prove that ifF admits a leafwise metric such that every leaf is an irreducible symmetric space of noncompact type and higher rank, then any other leafwise metric of nonpositive curvature is also symmetric along any leaf in the support of the transverse measure. A rank one version of this result is also exposed.The second author is partially supported, by a Seed Grant from The Ohio State University.     

Anosov actions of nilpotent Lie groups

We study Anosov actions of nilpotent Lie groups on closed manifolds. Our main result is a generalization to the nilpotent case of a classical theorem by J.F. Plante in the 70's. More precisely, we prove that, for what we call a good Anosov action of a nilpotent Lie group on a closed manifold, if the non-wandering set is the entire manifold, then the closure of stable strong leaves coincide with the closure of the strong unstable leaves. This implies the existence of an equivariant fibration of the manifold onto a homogeneous space of the Lie group, having as fibers the closures of the leaves of the strong foliation.     

Global rigidity results for lattice actions on tori and new examples of volume-preserving actions

Any action of a finite index subgroup in SL(n,ℤ),n≥4 on then-dimensional torus which has a finite orbit and contains an Anosov element which splits as a direct product is smoothly conjugate to an affine action. We also construct first examples of real-analytic volume-preserving actions of SL(n,ℤ) and other higher-rank lattices on compact manifolds which are not conjugate (even topologically) to algebraic models. This work was partially supported by NSF grant DMS9017995.     

Virtual boundaries of Hadamard spaces with admissible actions of higher rank

In this paper, we prove that every conformal minimal immersion of an open Riemann surface into \({\mathbb {R}}^n\) for \(n\ge 5\) can be approximated uniformly on compacts by conformal minimal embeddings (see Theorem 1.1). Furthermore, we show that every open Riemann surface carries a proper conformal minimal embedding into \({\mathbb {R}}^5\) (see Theorem 1.2). One of our main tools is a Mergelyan approximation theorem for conformal minimal immersions to \({\mathbb {R}}^n\) for any \(n\ge 3\) which is also proved in the paper (see Theorem 5.3).     

Singular abelian covers of algebraic surfaces

In this paper we generalise F. Cataneses work on singular (/2)²-covers to arbitrary finite abelian covers of algebraic surfaces. We determine the contribution of singularities to the invariants , K ², q, p _g , P _n and the canonical sheaf. We use these computations to construct a surface of general type with birational canonical map ₁, p _g =4 and K ² =31. Mathematics Subject Classification (2000):14J29, 14J17, 14J25, 14E20     

Minimal rank of abelian group matrices

The minimal rank of abelian group matrices with positive integral entries is determined.The corresponding problem for circulant matrices have been investigated by Ingleton and more recently by Shiu-Ma-Fang. Our work can be viewed as a generalization of their results, since a group matrix becomes circulant when the group is cyclic.     

Differentiability,rigidity and Godbillon-Vey classes for Anosov flows

Supported in part by grants from the NSF and the Sloan Foundation.