Integrable orbit equivalence rigidity for free groups

It is shown that every accessible group which is integrable orbit equivalent to a free group is virtually free. Moreover, we also show that any integrable orbit-equivalence between finitely generated groups extends to their end compactifications.     

Integrable lattices

We propose a method for constructing integrable lattices starting from dynamic systems with two different parameterizations of the canonical variables and hence two independent Bäcklund flows. We construct integrable lattices corresponding to generalizations of the nonlinear Schrödinger equation. We discuss the Toda, Volterra, and Heisenberg models in detail. For these systems, as well as for the Landau-Lifshitz model, we obtain totally discrete Lagrangians. We also discuss the relation of these systems to the Hirota equations.     

Large hyperbolic lattices

For a fundamental group of a compact orientable manifold, a condition is specified that is sufficient to guarantee the presence of a “virtual” epimorphism onto a free non-Abelian group. A consequence is deriving a strong Tits alternative. An arbitrary noncompact finitely generated discrete subgroup in PO(3, 1) either is large or is virtually Abelian. An application is provided to the problem of uniform exponential growth for lattices in a 3-dimensional hyperbolic space and of growth of Betti numbers for lattices in a hyperbolic n-dimensional space, where n is an odd number. Supported by RFBR (project No. 08-01-00067), by DFG grant Gr 627-11, and by Forschergruppe “Spektrale Analysis, asymptotical Verteilungen und stochastische Dynamiken,” Billfold University. (G. A. Noskov) Translated from Algebra i Logika, Vol. 48, No. 2, pp. 174–189, March–April, 2009.     

Measure equivalence rigidity and bi-exactness of groups

We get three types of results on measurable group theory; direct product groups of Ozawa's class S groups, wreath product groups and amalgamated free products. We prove measure equivalence factorization results on direct product groups of Ozawa's class S groups. As consequences, Monod-Shalom type orbit equivalence rigidity theorems follow. We prove that if two wreath product groups A?G, B?Γ of non-amenable exact direct product groups G, Γ with amenable bases A, B are measure equivalent, then G and Γ are measure equivalent. We get Bass-Serre rigidity results on amalgamated free products of non-amenable exact direct product groups.     

On the rigidity of hyperbolic cone-manifolds

Starting with a compact hyperbolic cone-manifold of dimension $n?3$, we study the deformations of the metric with the aim of getting Einstein cone-manifolds. If the singular locus is a closed codimension 2 submanifold and all cone angles are smaller than 2π, we show that there is no non-trivial infinitesimal Einstein deformations preserving the cone angles. To cite this article: G. Montcouquiol, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Structure and rigidity in hyperbolic groups I

We introduce certain classes of hyperbolic groups according to their possible actions on real trees. Using these classes and results from the theory of (small) group actions on real trees, we study the structure of hyperbolic groups and their automorphism group.The second author was partially supported by an NSF grant.     

Differentiable rigidity for hyperbolic toral actions

Our aim is to extend existing results about differentiable rigidity of higher rank abelian actions by automorphisms of a torus. Previous proofs have required an assumption of semisimplicity, that is, that the action is by commuting diagonalizable matrices. Here we introduce a technique that utilizes the unipotent part of a non-semisimple action, which allows us to discard the semisimplicity assumption. In its place we will make a technical assumption that the spectrum of the action restricted to leaves of the coarse Lyapunov decomposition is sufficiently narrow.     

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.     

Elementary equivalence and relatively free products of lattices

It is shown that if is a non-trivial variety of lattices, then there existA, B, C ∈ such thatB≡C but notA*B≡A*C. Except in the case when is the variety of all distributive lattices,A can be taken to consist of just one element. For the varietyD of all distributive lattices, it is shown that for anyB, C and any finiteA, B≡C if and only ifA*B≡A*C. The work of the first author was supported in part by NSF Grant GP-29129, and the work of the second author by a grant from the N.R.C. of Canada. Presented by C. C. Chang.     

Scalar curvature rigidity of hyperbolic product manifolds

This paper presents a scalar curvature rigidity result of real hyperbolic product manifolds in analogy to M. Min–Oos result in [14]. In order to prove this, we consider Dirac bundles obtained from the spinor bundle, and we derive Killing equations trivializing these Dirac bundles. Moreover, an integrated Bochner–Weitzenböck formula is shown which allows the usage of the non–compact Bochner technique.Mathematics Subject Classification (2000): Primary 53C24, Sec. 53C21in final form: 11 August 2003     

Lipschitz equivalence of self-similar sets and hyperbolic boundaries

Kaimanovich (2003) [9] introduced the concept of augmented tree on the symbolic space of a self-similar set. It is hyperbolic in the sense of Gromov, and it was shown by Lau and Wang (2009)  [12] that under the open set condition, a self-similar set can be identified with the hyperbolic boundary of the tree. In the paper, we investigate in detail a class of simple augmented trees and the Lipschitz equivalence of such trees. The main purpose is to use this to study the Lipschitz equivalence problem of the totally disconnected self-similar sets which has been undergoing some extensive development recently.     

On projective equivalence of invariant subspace lattices

For i = 1,2, let A_i be a linear transformation on a complex vector space and let σ be a lattice isomorphism from the invariant subspace lattice of A₁ onto the invariant subspace lattice of A₂. We determine the conditions under which σ is implemented by a linear or conjugate linear transformation (or a sum of these two kinds).     

Integral Gassman equivalence of algebraic and hyperbolic manifolds

Mathematische Zeitschrift - In this paper we construct arbitrarily large families of smooth projective varieties and closed Riemannian manifolds that share many algebraic and analytic invariants....