scholarly journals Diffeomorphism group valued cocycles over higher-rank abelian Anosov actions

2018 ◽  
Vol 40 (1) ◽  
pp. 117-141 ◽  
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.

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

2001 ◽  
Vol 21 (1) ◽  
pp. 121-164 ◽  
Author(s):  
GREGORY A. MARGULIS ◽  
NANTIAN QIAN
Keyword(s):  
Lie Groups ◽  
Left Translation ◽  
Real Rank ◽  
Semisimple Lie Groups ◽  
Rigidity Result ◽  
Global Rigidity ◽  
Local Rigidity ◽  
Algebraic Actions ◽  
Anosov Actions ◽  
Higher Rank

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.

scholarly journals Global rigidity of higher rank Anosov actions on tori and nilmanifolds

2012 ◽  
Vol 26 (1) ◽  
pp. 167-198 ◽  
Author(s):  
David Fisher ◽  
Boris Kalinin ◽  
Ralf Spatzier
Keyword(s):  
Global Rigidity ◽  
Anosov Actions ◽  
Higher Rank

Some new rigidity results for stable orbit equivalence

1995 ◽  
Vol 15 (2) ◽  
pp. 209-219 ◽  
Author(s):  
Scot Adams
Keyword(s):  
Hyperbolic Group ◽  
Finite Measure ◽  
Universal Cover ◽  
The Other ◽  
Stable Orbit ◽  
Rigidity Result ◽  
Rigidity Results ◽  
Word Hyperbolic Group ◽  
Orbit Equivalent ◽  
Rank One

AbstractBroadly speaking, we prove that an action of a group with very little commutativity cannot be stably orbit equivalent to an action of a group with enough commutativity, assuming both actions are free and finite measure preserving. For example, one group may be SL2(ℝ) and the other a group with infinite discrete center (e.g., the universal cover of SL2(ℝ)); I believe this is the first rigidity result of this type for a pair of simpleLie groups both of split rank one. Another example: one group may be any nonelementary word hyperbolic group, the other any group with infinite discrete center.

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.

scholarly journals A rank rigidity result for CAT(0) spaces with one-dimensional Tits boundaries

2019 ◽  
Vol 31 (5) ◽  
pp. 1317-1330
Author(s):  
Russell Ricks
Keyword(s):  
Symmetric Space ◽  
Group Action ◽  
Spherical Building ◽  
Rigidity Result ◽  
One Dimensional ◽  
Higher Rank ◽  
Geodesically Complete ◽  
Euclidean Building ◽  
Alternate Condition

AbstractWe prove the following rank rigidity result for proper {\operatorname{CAT}(0)} spaces with one-dimensional Tits boundaries: Let Γ be a group acting properly discontinuously, cocompactly, and by isometries on such a space X. If the Tits diameter of {\partial X} equals π and Γ does not act minimally on {\partial X}, then {\partial X} is a spherical building or a spherical join. If X is also geodesically complete, then X is a Euclidean building, higher rank symmetric space, or a nontrivial product. Much of the proof, which involves finding a Tits-closed convex building-like subset of {\partial X}, does not require the Tits diameter to be π, and we give an alternate condition that guarantees rigidity when this hypothesis is removed, which is that a certain invariant of the group action be even.

scholarly journals Revisiting Leighton’s theorem with the Haar measure

2020 ◽  
pp. 1-9
Author(s):  
DANIEL J. WOODHOUSE
Keyword(s):  
Haar Measure ◽  
Free Groups ◽  
Finite Cover ◽  
Covering Theorem ◽  
Rigidity Result ◽  
Graph Covering ◽  
Finite Graphs ◽  
Universal Covers ◽  
Line Patterns

Abstract Leighton’s graph covering theorem states that a pair of finite graphs with isomorphic universal covers have a common finite cover. We provide a new proof of Leighton’s theorem that allows generalisations; we prove the corresponding result for graphs with fins. As a corollary we obtain pattern rigidity for free groups with line patterns, building on the work of Cashen–Macura and Hagen–Touikan. To illustrate the potential for future applications, we give a quasi-isometric rigidity result for a family of cyclic doubles of free groups.

scholarly journals Homomorphisms between diffeomorphism groups

2013 ◽  
Vol 35 (1) ◽  
pp. 192-214 ◽  
Author(s):  
KATHRYN MANN
Keyword(s):  
Closed Manifold ◽  
Independent Interest ◽  
Diffeomorphism Group ◽  
Diffeomorphism Groups ◽  
One Dimensional ◽  
Compactly Supported ◽  
Trivial Group ◽  
Elementary Form ◽  
General Conditions

AbstractFor $r\geq 3$, $p\geq 2$, we classify all actions of the groups ${ \mathrm{Diff} }_{c}^{r} ( \mathbb{R} )$ and ${ \mathrm{Diff} }_{+ }^{r} ({S}^{1} )$ by ${C}^{p} $-diffeomorphisms on the line and on the circle. This is the same as describing all non-trivial group homomorphisms between groups of compactly supported diffeomorphisms on 1-manifolds. We show that all such actions have an elementary form, which we call topologically diagonal. As an application, we answer a question of Ghys in the 1-manifold case: if $M$ is any closed manifold, and ${\mathrm{Diff} }^{\infty } \hspace{-2.0pt} \mathop{(M)}\nolimits_{0} $ injects into the diffeomorphism group of a 1-manifold, must $M$ be one-dimensional? We show that the answer is yes, even under more general conditions. Several lemmas on subgroups of diffeomorphism groups are of independent interest, including results on commuting subgroups and flows.

Non-abelian cohomology of abelian Anosov actions

2000 ◽  
Vol 20 (1) ◽  
pp. 259-288 ◽  
Author(s):  
ANATOLE KATOK ◽  
VIOREL NIŢICĂ ◽  
ANDREI TÖRÖK
Keyword(s):  
Discrete Time ◽  
Lie Group ◽  
Global Information ◽  
Infinite Dimensional ◽  
Anosov Actions ◽  
Geometric Character ◽  
Higher Rank ◽  
Real Analytic ◽  
A New Technique ◽  
Discrete Time Case

We develop a new technique for calculating the first cohomology of certain classes of actions of higher-rank abelian groups (${\mathbb Z}^k$ and ${\mathbb R}^k$, $k\ge 2$) with values in a linear Lie group. In this paper we consider the discrete-time case. Our results apply to cocycles of different regularity, from Hölder to smooth and real-analytic. The main conclusion is that the corresponding cohomology trivializes, i.e. that any cocycle from a given class is cohomologous to a constant cocycle. The principal novel feature of our method is its geometric character; no global information about the action based on harmonic analysis is used. The method can be developed to apply to cocycles with values in certain infinite dimensional groups and to rigidity problems.

scholarly journals Global rigidity of higher rank abelian Anosov algebraic actions

2014 ◽  
Vol 198 (1) ◽  
pp. 165-209 ◽  
Author(s):  
Federico Rodriguez Hertz ◽  
Zhiren Wang
Keyword(s):  
Global Rigidity ◽  
Algebraic Actions ◽  
Higher Rank
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