Nilmanifolds of dimension $\leq 8$ admitting Anosov diffeomorphisms
HTML articles powered by AMS MathViewer
- by Jorge Lauret and Cynthia E. Will PDF
- Trans. Amer. Math. Soc. 361 (2009), 2377-2395 Request permission
Abstract:
After more than thirty years, the only known examples of Anosov diffeomorphisms are topologically conjugated to hyperbolic automorphisms of infranilmanifolds, and even the existence of an Anosov automorphism is a really strong condition on an infranilmanifold. Any Anosov automorphism determines an automorphism of the rational Lie algebra determined by the lattice, which is hyperbolic and unimodular (and conversely ...). These two conditions together are strong enough to make of such rational nilpotent Lie algebras (called Anosov Lie algebras) very distinguished objects. In this paper, we classify Anosov Lie algebras of dimension less than or equal to 8.
As a corollary, we obtain that if an infranilmanifold of dimension $n\leq 8$ admits an Anosov diffeomorphism $f$ and it is not a torus or a compact flat manifold (i.e. covered by a torus), then $n=6$ or 8 and the signature of $f$ necessarily equals $\{ 3,3\}$ or $\{ 4,4\}$, respectively.
References
- Louis Auslander and John Scheuneman, On certain automorphisms of nilpotent Lie groups, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 9–15. MR 0270395
- Yves Benoist and François Labourie, Sur les difféomorphismes d’Anosov affines à feuilletages stable et instable différentiables, Invent. Math. 111 (1993), no. 2, 285–308 (French, with French summary). MR 1198811, DOI 10.1007/BF01231289
- Charles Cassidy, Neil Kennedy, and Dirk Scevenels, Hyperbolic automorphisms for groups in $\scr T(4,2)$, Crystallographic groups and their generalizations (Kortrijk, 1999) Contemp. Math., vol. 262, Amer. Math. Soc., Providence, RI, 2000, pp. 171–175. MR 1796132, DOI 10.1090/conm/262/04174
- S. G. Dani, Nilmanifolds with Anosov automorphism, J. London Math. Soc. (2) 18 (1978), no. 3, 553–559. MR 518242, DOI 10.1112/jlms/s2-18.3.553
- S. G. Dani and Meera G. Mainkar, Anosov automorphisms on compact nilmanifolds associated with graphs, Trans. Amer. Math. Soc. 357 (2005), no. 6, 2235–2251. MR 2140439, DOI 10.1090/S0002-9947-04-03518-4
- Karel Dekimpe, Hyperbolic automorphisms and Anosov diffeomorphisms on nilmanifolds, Trans. Amer. Math. Soc. 353 (2001), no. 7, 2859–2877. MR 1828476, DOI 10.1090/S0002-9947-01-02683-6
- Karel Dekimpe and Sandra Deschamps, Anosov diffeomorphisms on a class of 2-step nilmanifolds, Glasg. Math. J. 45 (2003), no. 2, 269–280. MR 1997705, DOI 10.1017/S001708950300123X
- John Franks, Anosov diffeomorphisms, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 61–93. MR 0271990
- Étienne Ghys, Holomorphic Anosov systems, Invent. Math. 119 (1995), no. 3, 585–614. MR 1317651, DOI 10.1007/BF01245193
- Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. MR 623534, DOI 10.1007/BF02698687
- Kiyoshi Ito, Classification of nilmanifolds $M^n$ $(n\le 6)$ admitting Anosov diffeomorphisms, The study of dynamical systems (Kyoto, 1989) World Sci. Adv. Ser. Dynam. Systems, vol. 7, World Sci. Publ., Teaneck, NJ, 1989, pp. 31–49. MR 1117284
- Jorge Lauret, Examples of Anosov diffeomorphisms, J. Algebra 262 (2003), no. 1, 201–209. MR 1970807, DOI 10.1016/S0021-8693(03)00030-9
- J. Lauret, On rational forms of nilpotent Lie algebras, Monatsh. Math. 155 (2008), 15-30.
- Jorge Lauret and Cynthia E. Will, On Anosov automorphisms of nilmanifolds, J. Pure Appl. Algebra 212 (2008), no. 7, 1747–1755. MR 2400740, DOI 10.1016/j.jpaa.2007.11.011
- Wim Malfait, Anosov diffeomorphisms on nilmanifolds of dimension at most six, Geom. Dedicata 79 (2000), no. 3, 291–298. MR 1755730, DOI 10.1023/A:1005264730096
- Wim Malfait, An obstruction to the existence of Anosov diffeomorphisms on infra-nilmanifolds, Crystallographic groups and their generalizations (Kortrijk, 1999) Contemp. Math., vol. 262, Amer. Math. Soc., Providence, RI, 2000, pp. 233–251. MR 1796136, DOI 10.1090/conm/262/04178
- Anthony Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math. 96 (1974), 422–429. MR 358865, DOI 10.2307/2373551
- Gregory Margulis, Problems and conjectures in rigidity theory, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 161–174. MR 1754775
- Hugh L. Porteous, Anosov diffeomorphisms of flat manifolds, Topology 11 (1972), 307–315. MR 296976, DOI 10.1016/0040-9383(72)90016-X
- John Scheuneman, Two-step nilpotent Lie algebras, J. Algebra 7 (1967), 152–159. MR 217134, DOI 10.1016/0021-8693(67)90052-X
- Michael Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math. 91 (1969), 175–199. MR 240824, DOI 10.2307/2373276
- S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
- A. Verjovsky, Sistemas de Anosov, Notas de Curso, IMPA (Rio de Janeiro).
Additional Information
- Jorge Lauret
- Affiliation: FaMAF and CIEM, Universidad Nacional de Córdoba, Córdoba, Argentina
- MR Author ID: 626241
- Email: lauret@mate.uncor.edu
- Cynthia E. Will
- Affiliation: FaMAF and CIEM, Universidad Nacional de Córdoba, Córdoba, Argentina
- MR Author ID: 649211
- Email: cwill@mate.uncor.edu
- Received by editor(s): March 22, 2007
- Published electronically: November 25, 2008
- Additional Notes: This research was supported by CONICET fellowships and grants from FONCyT and Fundación Antorchas.
- © Copyright 2008 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 361 (2009), 2377-2395
- MSC (2000): Primary 37D20; Secondary 22E25, 20F34
- DOI: https://doi.org/10.1090/S0002-9947-08-04757-0
- MathSciNet review: 2471923