Rigidity of weakly hyperbolic actions of higher real rank semisimple Lie groups and their lattices

Under some weak hyperbolicity conditions, we establish C^0- and C^\infty-local rigidity theorems for two classes of standard algebraic actions: (1) left translation actions of higher real rank semisimple Lie groups and their lattices on quotients of Lie groups by uniform lattices; (2) higher rank lattice actions on nilmanifolds by affine diffeomorphisms. The proof relies on an observation that local rigidity of the standard actions is a consequence of the local rigidity of some constant cocycles. The C^0-local rigidity for weakly hyperbolic standard actions follows from a cocycle C^0-local rigidity result proved in the paper. The main ingredients in the proof of the latter are Zimmer's cocycle superrigidity theorem and stability properties of partially hyperbolic vector bundle maps. The C^\infty-local rigidity is deduced from the C^0-local rigidity following a procedure outlined by Katok and Spatzier.Using similar considerations, we also establish C^0-global rigidity of volume preserving, higher rank lattice Anosov actions on nilmanifolds with a finite orbit.

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Diffeomorphism group valued cocycles over higher-rank abelian Anosov actions

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2018.22 ◽

2018 ◽

Vol 40 (1) ◽

pp. 117-141 ◽

Cited By ~ 2

Author(s):

DANIJELA DAMJANOVIĆ ◽

DISHENG XU

Keyword(s):

Fixed Point ◽

Universal Cover ◽

Diffeomorphism Group ◽

Finite Cover ◽

Diffeomorphism Groups ◽

Rigidity Result ◽

Global Rigidity ◽

Rigidity Results ◽

Anosov Actions ◽

Higher Rank

We prove that every smooth diffeomorphism group valued cocycle over certain$\mathbb{Z}^{k}$Anosov actions on tori (and more generally on infranilmanifolds) is a smooth coboundary on a finite cover, if the cocycle is center bunched and trivial at a fixed point. For smooth cocycles which are not trivial at a fixed point, we have smooth reduction of cocycles to constant ones, when lifted to the universal cover. These results on cocycle trivialization apply, via the existing global rigidity results, to maximal Cartan$\mathbb{Z}^{k}$($k\geq 3$) actions by Anosov diffeomorphisms (with at least one transitive), on any compact smooth manifold. This is the first rigidity result for cocycles over$\mathbb{Z}^{k}$actions with values in diffeomorphism groups which does not require any restrictions on the smallness of the cocycle or on the diffeomorphism group.

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Local rigidity of Anosov higher-rank lattice actions

Ergodic Theory and Dynamical Systems ◽

10.1017/s014338579810826x ◽

1998 ◽

Vol 18 (3) ◽

pp. 687-702 ◽

Cited By ~ 2

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.

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