Hyperbolic automorphisms and Anosov diffeomorphisms on nilmanifolds

Author:
Karel Dekimpe

Journal:
Trans. Amer. Math. Soc. **353** (2001), 2859-2877

MSC (2000):
Primary 37D20; Secondary 20F18, 20F34

DOI:
https://doi.org/10.1090/S0002-9947-01-02683-6

Published electronically:
March 14, 2001

MathSciNet review:
1828476

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Abstract | References | Similar Articles | Additional Information

We translate the problem of finding Anosov diffeomorphisms on a nilmanifold which is covered by a free nilpotent Lie group into a problem of constructing matrices in whose eigenvalues satisfy certain conditions. Afterwards, we show how this translation can then be solved in some specific situations. The paper starts with a section on polynomial permutations of , a subject which is of interest on its own.

**1.**Dekimpe, K. and Igodt, P.*Polynomial Alternatives for the Group of Affine Motions.*Math. Zeit. 234 (2000), 457-485. CMP**2000:16****2.**Dekimpe, K. and Malfait, W.*A special class of nilmanifolds admitting an Anosov diffeomorphism*. Proc. Amer. Math. Soc. 128 (2000), 2171-2179. MR**2000m:37029****3.**Malfait, W.*Anosov diffeomorphisms on nilmanifolds of dimension at most six.*Geometriae Dedicata, (3) 79 (2000), 291-298. CMP**2000:12****4.**Manning, A.*There are no new Anosov diffeomorphisms on tori*. Amer. J. Math., 1974, 96 (3), pp. 422-429. MR**50:11324****5.**Passi, I. B. S.*Group rings and their augmentation ideals*, volume 715, of Lecture Notes in Math. Springer-Verlag, 1979. MR**80k:20009****6.**Passman, D. S.*The Algebraic Structure of Group Rings*. Pure and Applied Math. John Wiley & Sons, Inc. New York, 1977. MR**81d:16001****7.**Porteous, H. L.*Anosov diffeomorphisms of flat manifolds.*Topology, 1972, 11, pp. 307-315. MR**45:6035****8.**Szczepanski, A.*Outer automorphism groups of Bieberbach groups*. Bull. of Belg. Math. Soc. (Simon Stevin), 1996, 3, pp. 585-593. MR**97k:57050**

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Additional Information

**Karel Dekimpe**

Affiliation:
Katholieke Universiteit Leuven, Campus Kortrijk, B-8500 Kortrijk, Belgium

Email:
Karel.Dekimpe@kulak.ac.be

DOI:
https://doi.org/10.1090/S0002-9947-01-02683-6

Received by editor(s):
January 17, 1999

Received by editor(s) in revised form:
January 31, 2000

Published electronically:
March 14, 2001

Additional Notes:
Postdoctoral Fellow of the Fund for Scientific Research - Flanders (F.W.O.)

Article copyright:
© Copyright 2001
American Mathematical Society