Nilmanifolds of dimension ≤8 admitting Anosov diffeomorphisms

Lauret, Jorge; Will, Cynthia

doi:10.1090/S0002-9947-08-04757-0

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).