## Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions (1997)

Venue: | PROC. STEKLOV INST. MATH |

Citations: | 44 - 24 self |

### BibTeX

@INPROCEEDINGS{Katok97differentialrigidity,

author = {A. Katok and R. J. Spatzier},

title = {Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions},

booktitle = {PROC. STEKLOV INST. MATH},

year = {1997},

pages = {21--6},

publisher = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

We show that most homogeneous Anosov actions of higher rank Abelian groups are locally C∞-rigid (up to an automorphism). This result is the main part in the proof of local C∞-rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the Anosov actions by automorphisms of tori and nil–manifolds, and (ii) the actions of cocompact lattices on Furstenberg boundaries, in particular, projective spaces. The main new technical ingredient in the proofs is the use of a proper “non–stationary” generalization of the classical theory of normal forms for local contractions.

### Citations

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Citation Context ...not help to allow the conjugacy in the definitions to be only a C 1 diffeomorphism. Structural stability on the other hand is a rather wide-spread (although not completely understood) phenomenon. See =-=[17]-=- for a general background and [41, 28] for the definitive results on structural stability in C 1 (but not C r ; rs2) topology. Since local C 1 -rigidity implies structural stability one can immediatel... |

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Citation Context ...ient is a point, we get ergodicity of one-parameter subgroups of A. Remark. A.Starkov pointed out to us an alternative argument, avoiding the discussion of admissibility, which uses Theorem 8.24 from =-=[40]-=- and certain results proved independently by himself and by D.Witte. 2.3.3 Proof of Corollary 4 Suppose Z k acts on a compact nilmanifold M . Then the suspension (M \Theta R k )=Z k factors over T k :... |

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Citation Context ...factors. Let \Gamma ae G be an irreducible cocompact lattice. Let P be a parabolic subgroup of G, and let H be its Levi subgroup. Thus P = H U + where U + is the unipotent radical of P . (We refer to =-=[48]-=- for details on parabolic subgroups and boundaries of G.) Then H acts on G=\Gamma by left translations. These actions are Anosov (cf. eg. [26]). Theorem 6 If the real rank of G is at least 2, then the... |

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Citation Context ...ssion of the ergodicity of homogeneous flows [4]. This condition is automatically satisfied if H is semisimple without compact factors andsis an irreducible lattice, as follows from Mautner's theorem =-=[50]-=-. If H is semisimple, then the linear part of the action is automatically semisimple. The action of a split Cartan subgroup A of H on KnH= wheresae H is a cocompact lattice in H , and K is the compact... |

72 | Corrections to ‘Invariant measures for higher-rank hyperbolic abelian actions’, Ergodic Theory Dynam. Systems 18
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Citation Context ...ined in the the closure of the orbit of a generic point x under one-parameter groups which translate these intersections by isometries. It is similar to the condition of the main technical theorem of =-=[23]-=-, Theorem 5.1, on invariant measures for algebraic Anosov R k -actions. Corollary 2 Let ae be an algebraic Anosov action of R k , for ks2, such that the linear part of ae is semisimple. Assume that ev... |

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Citation Context ... for ks2. In this paper we bring the study of the local differentiable rigidity of algebraic Anosov actions of Z k and R k on compact manifolds as well as orbit foliations of such actions, started in =-=[18, 21, 22, 24]-=-, to a near conclusion. Our results for the abelian group actions formulated in Section 2.1 allow us to obtain comprehensive local C 1 rigidity for two very different types of algebraic actions of irr... |

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Citation Context ...oach to the actions of higher--rank semisimple Lie groups and lattices in such groups initiated by Zimmer in the mid-eighties did these differences come to the attention of researchers beginning with =-=[18]-=-. In particular, it turned out that both local differentiable stability (or rigidity) and cocycle rigidity are present and in fact typical already for Anosov actions of higher--rank abelian groups, i.... |

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Citation Context ... in particular projective spaces (Theorem 17, Section 4). While the latter area was virtually unapproachable save for a special result by Kanai [13], the former was extensively studied beginning from =-=[11, 18]-=- and continuing in [20, 34, 37, 38, 19, 35, 36]. Our result (Theorem 15 in Section 3) substantially extends all these earlier works and brings the question to a final solution. It also gives further c... |

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Citation Context ...velopment which concerns us here deals with the notion of an Anosov action of a group (either discrete or continuous), more general than Zor R. This concept was originally introduced by Pugh and Shub =-=[33] in the ea-=-rly seventies (their notion coincides with ours for abelian groups and is different in general) and for many years fundamental differences between the "classical" (Zand R) and other cases we... |

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Citation Context ...ionary generalization of the classical normal form theory for local contractions whose origins go back to Poincar`e and which, in its modern form for the C 1 case, was developed by Sternberg and Chen =-=[45, 6]-=-. It is quite possible that one can find results very similar to the ones below in the vast literature on normal forms. The first author wrote the precise versions needed for our application in [15]. ... |

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Citation Context ...paces (Theorem 17, Section 4). While the latter area was virtually unapproachable save for a special result by Kanai [13], the former was extensively studied beginning from [11, 18] and continuing in =-=[20, 34, 37, 38, 19, 35, 36]-=-. Our result (Theorem 15 in Section 3) substantially extends all these earlier works and brings the question to a final solution. It also gives further credence to the global conjecture that all Anoso... |

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Citation Context ...eorem of W. Parry, an affine automorphism of a nilmanifold H= is weakly mixing precisely when the quotient of the linear part on the abelianization h [h;h] does not have roots of unity as eigenvalues =-=[31]-=-. As an orbit equivalence close to the identity between suspensions of Z k -actions automatically produces a conjugacy between the actions themselves, we get the following corollary. Corollary 4 Let a... |

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Citation Context ... equivalence along these intersections. The essential new technique used here is a suitable non-stationary generalization of the classical theory of normal forms for local differentiable contractions =-=[15]-=-. This technique is summarized in the following section. Once we have smoothness of the orbit equivalence along the foliations of the splitting, standard elliptic theory shows smoothness. Theorem 6 de... |

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Citation Context ...cients in any finite dimensional representation of \Gamma vanishes [29, Theorem 3']. Hence, by a theorem of D. Stowe, a sufficiently close perturbed action ~ ae still has a fixed point o 0 for \Gamma =-=[46]. Since ae-=- and ~ ae coincide on \Delta, it follows that \Delta " \Gamma fixes both o and o 0 . Since \Delta " \Gamma + contains Anosov elements and o and o 0 are close, this forces o = o 0 . Hence we ... |

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Citation Context ...bolic action on \Gamma nG. We would like to thank E. Ghys for suggesting this approach. In fact, Ghys used a similar duality to obtain the smooth classification of boundary actions of Fuchsian groups =-=[7]-=-. In the special case of a projective action of a cocompact lattice in SL(n+1;R) on the n-sphere S n , M. Kanai established a local rigidity result for small perturbations in the C 4 -topology for all... |

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Citation Context ... the global conjecture that all Anosov actions of such lattices are smoothly equivalent to one of the above. Such a classification was established for the much more special class of Cartan actions in =-=[8]-=-. The objects considered in this paper, i.e. group actions and foliations, are assumed to be of class C 1 . Accordingly the basic notions, including the structural stability are adapted to this case w... |

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Citation Context ...e(fl)(�� (x)) = ��(~ae(fl)(x)) ff(fl; x): 18 Then ff is a cocycle over ~ ae with values in H . Furthermore, H is always reductive with compact center as was shown for uniform lattices by R. Zi=-=mmer in [51]-=- and for non-uniform lattices by J. Lewis in [27]. In order to be able to apply Zimmer's superrigidity theorem for cocycles [50], the algebraic hull needs to be (Zariski) connected. This can be achiev... |

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Citation Context ...p of R k acts ergodically is equivalent to the R k -action being weakly mixing, as one easily sees. We refer to Brezin and Moore for a more extensive discussion of the ergodicity of homogeneous flows =-=[4]-=-. This condition is automatically satisfied if H is semisimple without compact factors andsis an irreducible lattice, as follows from Mautner's theorem [50]. If H is semisimple, then the linear part o... |

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Citation Context ...paces (Theorem 17, Section 4). While the latter area was virtually unapproachable save for a special result by Kanai [13], the former was extensively studied beginning from [11, 18] and continuing in =-=[20, 34, 37, 38, 19, 35, 36]-=-. Our result (Theorem 15 in Section 3) substantially extends all these earlier works and brings the question to a final solution. It also gives further credence to the global conjecture that all Anoso... |

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Citation Context ...paces (Theorem 17, Section 4). While the latter area was virtually unapproachable save for a special result by Kanai [13], the former was extensively studied beginning from [11, 18] and continuing in =-=[20, 34, 37, 38, 19, 35, 36]-=-. Our result (Theorem 15 in Section 3) substantially extends all these earlier works and brings the question to a final solution. It also gives further credence to the global conjecture that all Anoso... |

8 |
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Citation Context ... for ks2. In this paper we bring the study of the local differentiable rigidity of algebraic Anosov actions of Z k and R k on compact manifolds as well as orbit foliations of such actions, started in =-=[18, 21, 22, 24]-=-, to a near conclusion. Our results for the abelian group actions formulated in Section 2.1 allow us to obtain comprehensive local C 1 rigidity for two very different types of algebraic actions of irr... |

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Citation Context ...jecting x 2 ~ N simultaneously to oe a (x) and oe \Gammaa (x) for a regular element a. Since a topological automorphism of \Delta is C 1 (for the smooth structure on \Delta induced by the ~ G-action) =-=[5]-=- and the maps oe a and oe \Gammaa are C 1 we see that ~ OE is C 1 transversally to the R k -orbits. As ~ OE is C 1 along the orbits, ~ OE is C 1 . Consider the group ~ Aut(ff) of lifts of automorphism... |

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Citation Context ...pology for all ns21 [13]. His method is completely different and relies on the vanishing of a certain cohomology. We begin with a brief review of foliated bundles and the suspension construction (cf. =-=[10]-=- and also [49] for a more detailed description). Let M be a compact manifold with cover ~ M (let us emphasize that ~ M need not be the universal cover). Let \Gamma be the group of covering transformat... |

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Citation Context ...tion 3), and (ii) the actions on Furstenberg boundaries, in particular projective spaces (Theorem 17, Section 4). While the latter area was virtually unapproachable save for a special result by Kanai =-=[13]-=-, the former was extensively studied beginning from [11, 18] and continuing in [20, 34, 37, 38, 19, 35, 36]. Our result (Theorem 15 in Section 3) substantially extends all these earlier works and brin... |

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Citation Context ... the definitions to be only a C 1 diffeomorphism. Structural stability on the other hand is a rather wide-spread (although not completely understood) phenomenon. See [17] for a general background and =-=[41, 28]-=- for the definitive results on structural stability in C 1 (but not C r , r ≥ 2) topology. Since local C ∞ -rigidity implies structural stability one can immediately see from the necessary conditions ... |

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Citation Context ...al hyperbolic behavior which renders itself to qualitative analysis relatively easily while the global classification remains to a large extent mysterious. The concept was introduced by D.V.Anosov in =-=[1]-=- and the fundamentals of the theory were developed in his classic monograph [2] which during almost thirty years that passed after its publication has been serving as a source of ideas and inspiration... |

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Citation Context ...ladly acknowledges the support from Erwin Schroedinger institute for mathematical physics where the final preparation of this paper was completed. The principal results of this paper are announced in =-=[25]. 4 2 Anos-=-ov actions of higher rank Abelian groups 2.1 The main results We will consider "essentially algebraic" Anosov actions of either R k or Z k . We recall that an action of a group G on a compac... |

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Citation Context ...abelian groups and is different in general) and for many years fundamental differences between the "classical" (Zand R) and other cases went mostly unnoticed, except for the one--dimensinal =-=situation [42]. Only wit-=-h the progress of the "geometric rigidity" approach to the actions of higher--rank semisimple Lie groups and lattices in such groups initiated by Zimmer in the mid-eighties did these differe... |

3 |
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Citation Context ...Kazhdan's property and ae preserves Haar measure, a perturbation ~ ae sufficiently close to ae in the C 1 -topology preserves an absolutely continuous probability measure (cf. e.g. [20, Lemma 2.6] or =-=[43]). N-=-ow let us apply Theorem 14 to ~ ae and its extension by derivatives to the frame bundle of M . In particular, we let M 0 denote the finite cover of M for the ~ ae action and �� be a superrigidity ... |

3 |
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Citation Context ... ns21 [13]. His method is completely different and relies on the vanishing of a certain cohomology. We begin with a brief review of foliated bundles and the suspension construction (cf. [10] and also =-=[49]-=- for a more detailed description). Let M be a compact manifold with cover ~ M (let us emphasize that ~ M need not be the universal cover). Let \Gamma be the group of covering transformations. Given a ... |

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Citation Context ...oots of unity as eigenvalues in the abelianization. This easily follows from the following unpublished algebraic result of G. Prasad and A. Rapinchuk which strengthens a result of V. E. Voskresenskii =-=[47]. Th-=-eorem (Prasad-Rapinchuk) Let G and \Gamma be as before. Let �� : G ! GL(N;R) be a linear representation of G. Then there exists a Cartan subgroup \Delta of \Gamma such that for all ffi 6= 1 2 \Del... |

1 |
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Citation Context ... + where U + is the unipotent radical of P . (We refer to [48] for details on parabolic subgroups and boundaries of G.) Then H acts on G=\Gamma by left translations. These actions are Anosov (cf. eg. =-=[26]-=-). Theorem 6 If the real rank of G is at least 2, then the orbit foliation O of H is C 1 -locally rigid. Moreover, the orbit equivalence can be chosen C 1 -close to the identity. We will actually use ... |

1 |
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Citation Context ... ff is a cocycle over ~ ae with values in H . Furthermore, H is always reductive with compact center as was shown for uniform lattices by R. Zimmer in [51] and for non-uniform lattices by J. Lewis in =-=[27]-=-. In order to be able to apply Zimmer's superrigidity theorem for cocycles [50], the algebraic hull needs to be (Zariski) connected. This can be achieved by passing to a suitable finite cover M 0 of M... |

1 |
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Citation Context ... the definitions to be only a C 1 diffeomorphism. Structural stability on the other hand is a rather wide-spread (although not completely understood) phenomenon. See [17] for a general background and =-=[41, 28]-=- for the definitive results on structural stability in C 1 (but not C r ; rs2) topology. Since local C 1 -rigidity implies structural stability one can immediately see from the necessary conditions fo... |

1 |
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Citation Context ... the definitions to be only a C 1 diffeomorphism. Structural stability on the other hand is a rather wide-spread (although not completely understood) phenomenon. See [17] for a general background and =-=[41, 28]-=- for the definitive results on structural stability in C 1 (but not C r ; rs2) topology. Since local C 1 -rigidity implies structural stability one can immediately see from the necessary conditions fo... |