Local rigidity of higher rank non-abelian action on torus

2017 ◽  
Vol 39 (06) ◽  
pp. 1668-1709
Author(s):  
ZHENQI JENNY WANG
Keyword(s):  
Iterative Scheme ◽  
Local Rigidity ◽  
Rank One ◽  
Higher Rank ◽  
Toral Automorphisms

In this paper, we show local smooth rigidity for higher rank ergodic nilpotent action by toral automorphisms. In former papers all examples for actions enjoying the local smooth rigidity phenomenon are higher rank and have no rank-one factors. In this paper we give examples of smooth rigidity of actions having rank-one factors. The method is a generalization of the KAM (Kolmogorov–Arnold–Moser) iterative scheme.

Local rigidity of Anosov higher-rank lattice actions

1998 ◽  
Vol 18 (3) ◽  
pp. 687-702 ◽  
Author(s):  
NANTIAN QIAN ◽  
CHENGBO YUE
Keyword(s):  
Abelian Group ◽  
Lyapunov Functional ◽  
Rigidity Result ◽  
Absolute Value ◽  
Local Rigidity ◽  
Higher Rank ◽  
Standard Action ◽  
Lattice Actions

Let $\rho_0$ be the standard action of a higher-rank lattice $\Gamma$ on a torus by automorphisms induced by a homomorphism $\pi_0:\Gamma\to SL(n,{\Bbb Z})$. Assume that there exists an abelian group ${\cal A}\subset \Gamma$ such that $\pi_0({\cal A})$ satisfies the following conditions: (1) ${\cal A}$ is ${\Bbb R}$-diagonalizable; (2) there exists an element $a\in {\cal A}$, such that none of its eigenvalues $\lambda_1,\dots,\lambda_n$ has unit absolute value, and for all $i,j,k=1,\dots,n$, $|\lambda_i\lambda_j|\neq|\lambda_k|$; (3) for each Lyapunov functional $\chi_i$, there exist finitely many elements $a_j\in {\cal A}$ such that $E_{\chi_i}=\cap_{j} E^u(a_j)$ (see \S1 for definitions). Then $\rho_0$ is locally rigid. This local rigidity result differs from earlier ones in that it does not require a certain one-dimensionality condition.

The spectrum and heat dynamics of locally symmetric spaces of higher rank

2014 ◽  
Vol 35 (5) ◽  
pp. 1524-1545 ◽  
Author(s):  
LIZHEN JI ◽  
ANDREAS WEBER
Keyword(s):  
Symmetric Spaces ◽  
Chaotic Behavior ◽  
Heat Semigroup ◽  
Locally Symmetric Spaces ◽  
Rank One ◽  
Higher Rank

The aim of this paper is to study the spectrum of the$L^{p}$Laplacian and the dynamics of the$L^{p}$heat semigroup on non-compact locally symmetric spaces of higher rank. Our work here generalizes previously obtained results in the setting of locally symmetric spaces of rank one to higher rank spaces. Similarly as in the rank-one case, it turns out that the$L^{p}$heat semigroup on$M$has a certain chaotic behavior if$p\in (1,2)$, whereas for$p\geq 2$such chaotic behavior never occurs.

scholarly journals Local rigidity of affine actions of higher rank groups and lattices

2009 ◽  
Vol 170 (1) ◽  
pp. 67-122 ◽  
Author(s):  
David Fisher ◽  
Gregory Margulis
Keyword(s):  
Local Rigidity ◽  
Higher Rank ◽  
Affine Actions

HYPERSURFACES IN NONCOMPACT COMPLEX GRASSMANNIANS OF RANK TWO

2012 ◽  
Vol 23 (10) ◽  
pp. 1250103 ◽  
Author(s):  
JÜRGEN BERNDT ◽  
YOUNG JIN SUH
Keyword(s):  
Geometric Structure ◽  
Symmetric Spaces ◽  
Second Fundamental Form ◽  
Kähler Structure ◽  
Noncompact Type ◽  
The Second Fundamental Form ◽  
Special Cases ◽  
Rank One ◽  
Higher Rank ◽  
Riemannian Symmetric Spaces

Consider a Riemannian manifold N equipped with an additional geometric structure, such as a Kähler structure or a quaternionic Kähler structure, and a hypersurface M in N. The geometric structure induces a decomposition of the tangent bundle TM of M into subbundles. A natural problem is to classify all hypersurfaces in N for which the second fundamental form of M preserves these subbundles. This problem is reasonably well understood for Riemannian symmetric spaces of rank one, but not for higher rank symmetric spaces. A general treatment of this problem for higher rank symmetric spaces is out of reach at present, and therefore it is desirable to understand this problem better in a few special cases. Due to some conceptual differences between symmetric spaces of compact type and of noncompact type it appears that one needs to consider these two cases separately. In this paper we investigate this problem for the rank two symmetric space SU 2, m/S(U2Um) of noncompact type.

scholarly journals Novikov Conjectures and Relative Hyperbolicity

1999 ◽  
Vol 85 (2) ◽  
pp. 169 ◽  
Author(s):  
Boris Goldfarb
Keyword(s):  
Symmetric Spaces ◽  
Large Family ◽  
Hyperbolic Groups ◽  
Fundamental Groups ◽  
Arithmetic Groups ◽  
Relatively Hyperbolic Groups ◽  
Rank One ◽  
Higher Rank ◽  
Relatively Hyperbolic ◽  
Relative Hyperbolicity

We consider a class of relatively hyperbolic groups in the sense of Gromov and use an argument modeled after Carlsson-Pedersen to prove Novikov conjectures for these groups. This proof is related to [16,17] which dealt with arithmetic lattices in rank one symmetric spaces and some other arithmetic groups of higher rank. Here whe view the rank one lattices in this different larger context of relativve hyperbolicity which also inclues fundamental groups of pinched hyperbolic manifolds. Another large family of groups from this class is produced using combinatorial hyperbolization techniques.

scholarly journals An Area Theorem for Joint Harmonic Functions on the Product of Homogeneous Trees

2021 ◽  
Author(s):  
Laura Atanasi ◽  
Massimo A. Picardello
Keyword(s):  
Harmonic Functions ◽  
Symmetric Spaces ◽  
Cartesian Product ◽  
Measurable Subset ◽  
Hyperbolic Spaces ◽  
Area Theorem ◽  
Area Function ◽  
Homogeneous Trees ◽  
Rank One ◽  
Higher Rank

AbstractFor harmonic functions v on the disc, it has been known for a long time that non-tangential boundedness a.e.is equivalent to finiteness a.e. of the integral of the area function of v (Lusin area theorem). This result also hold for functions that are non-tangentially bounded only in a measurable subset of the boundary, and has been extended to rank-one hyperbolic spaces, and also to infinite trees (homogeneous or not). No equivalent of the Lusin area theorem is known on higher rank symmetric spaces, with the exception of the degenerate higher rank case given by the cartesian product of rank-one hyperbolic spaces. Indeed, for products of two discs, an area theorem for jointly harmonic functions was proved by M.P. and P. Malliavin, who introduced a new area function; non-tangential boundedness a.e. is a sufficient condition, but not necessary, for the finiteness of this area integral. Their result was later extended to general products of rank-one hyperbolic spaces by Korányi and Putz. Here we prove an area theorem for jointly harmonic functions on the product of a finite number of infinite homogeneous trees; for the sake of simplicity, we give the proofs for the product of two trees. This could be the first step to an area theorem for Bruhat–Tits affine buildings, thereby shedding light on the higher rank continuous set-up.

scholarly journals Prime II1 factors arising from irreducible lattices in products of rank one simple Lie groups

2019 ◽  
Vol 2019 (757) ◽  
pp. 197-246 ◽  
Author(s):  
Daniel Drimbe ◽  
Daniel Hoff ◽  
Adrian Ioana
Keyword(s):  
Lie Groups ◽  
Hyperbolic Groups ◽  
Arithmetic Groups ◽  
Semisimple Lie Groups ◽  
Infinite Groups ◽  
Prime Factorization ◽  
Finite Center ◽  
Simple Lie Groups ◽  
Rank One ◽  
Higher Rank

AbstractWe prove that if Γ is an icc irreducible lattice in a product of connected non-compact rank one simple Lie groups with finite center, then the {\mathrm{II}_{1}} factor {L(\Gamma)} is prime. In particular, we deduce that the {\mathrm{II}_{1}} factors associated to the arithmetic groups {\mathrm{PSL}_{2}(\mathbb{Z}[\sqrt{d}])} and {\mathrm{PSL}_{2}(\mathbb{Z}[S^{-1}])} are prime for any square-free integer {d\geq 2} with {d\not\equiv 1~{}(\operatorname{mod}\,4)} and any finite non-empty set of primes S. This provides the first examples of prime {\mathrm{II}_{1}} factors arising from lattices in higher rank semisimple Lie groups. More generally, we describe all tensor product decompositions of {L(\Gamma)} for icc countable groups Γ that are measure equivalent to a product of non-elementary hyperbolic groups. In particular, we show that {L(\Gamma)} is prime, unless Γ is a product of infinite groups, in which case we prove a unique prime factorization result for {L(\Gamma)}.

scholarly journals Local rigidity of Lyapunov spectrum for toral automorphisms

2020 ◽  
Vol 238 (1) ◽  
pp. 389-403
Author(s):  
Andrey Gogolev ◽  
Boris Kalinin ◽  
Victoria Sadovskaya
Keyword(s):  
Lyapunov Spectrum ◽  
Local Rigidity ◽  
Toral Automorphisms

scholarly journals Moment Functions on Groups

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Żywilla Fechner ◽  
Eszter Gselmann ◽  
László Székelyhidi
Keyword(s):  
Commutative Group ◽  
Bell Polynomials ◽  
Exponential Polynomials ◽  
Bell Polynomial ◽  
Rank One ◽  
Moment Functions ◽  
Binomial Type ◽  
Higher Rank ◽  
Generalized Moment

AbstractIn this paper generalized moment functions are considered. They are closely related to the well-known functions of binomial type which have been investigated on various abstract structures. The main purpose of this work is to prove characterization theorems for generalized moment functions on commutative groups. At the beginning a multivariate characterization of moment functions defined on a commutative group is given. Next the notion of generalized moment functions of higher rank is introduced and some basic properties on groups are listed. The characterization of exponential polynomials by means of complete (exponential) Bell polynomials is given. The main result is the description of generalized moment functions of higher rank defined on a commutative group as the product of an exponential and composition of multivariate Bell polynomial and an additive function. Furthermore, corollaries for generalized moment function of rank one are also stated. At the end of the paper some possible directions of further research are discussed.

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