### Abstract

Let Γ = SL(n, ℤ) or any subgroup of finite index, n ≥ 4. We show that the standard action of Γ on {Mathematical expression}^{ n} is locally rigid, i.e., every action of Γ on {Mathematical expression}^{ n} by C^{∞} diffeomorphisms which is sufficiently close to the standard action is conjugate to the standard action by a C^{∞} diffeomorphism. In the course of the proof, we obtain a global rigidity result (Theorem 4.12) for actions of free abelian subgroups of maximal rank in SL(n, ℤ).

Original language | English (US) |
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Pages (from-to) | 203-241 |

Number of pages | 39 |

Journal | Israel Journal of Mathematics |

Volume | 75 |

Issue number | 2-3 |

DOIs | |

State | Published - Oct 1 1991 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Israel Journal of Mathematics*,

*75*(2-3), 203-241. https://doi.org/10.1007/BF02776025

}

*Israel Journal of Mathematics*, vol. 75, no. 2-3, pp. 203-241. https://doi.org/10.1007/BF02776025

**Local rigidity for certain groups of toral automorphisms.** / Katok, Anatoly; Lewis, J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Local rigidity for certain groups of toral automorphisms

AU - Katok, Anatoly

AU - Lewis, J.

PY - 1991/10/1

Y1 - 1991/10/1

N2 - Let Γ = SL(n, ℤ) or any subgroup of finite index, n ≥ 4. We show that the standard action of Γ on {Mathematical expression} n is locally rigid, i.e., every action of Γ on {Mathematical expression} n by C∞ diffeomorphisms which is sufficiently close to the standard action is conjugate to the standard action by a C∞ diffeomorphism. In the course of the proof, we obtain a global rigidity result (Theorem 4.12) for actions of free abelian subgroups of maximal rank in SL(n, ℤ).

AB - Let Γ = SL(n, ℤ) or any subgroup of finite index, n ≥ 4. We show that the standard action of Γ on {Mathematical expression} n is locally rigid, i.e., every action of Γ on {Mathematical expression} n by C∞ diffeomorphisms which is sufficiently close to the standard action is conjugate to the standard action by a C∞ diffeomorphism. In the course of the proof, we obtain a global rigidity result (Theorem 4.12) for actions of free abelian subgroups of maximal rank in SL(n, ℤ).

UR - http://www.scopus.com/inward/record.url?scp=51249171985&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=51249171985&partnerID=8YFLogxK

U2 - 10.1007/BF02776025

DO - 10.1007/BF02776025

M3 - Article

AN - SCOPUS:51249171985

VL - 75

SP - 203

EP - 241

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 2-3

ER -