A new method for constructing Anosov Lie algebras

Deré, Jonas

doi:10.1090/tran6655

A new method for constructing Anosov Lie algebras
HTML articles powered by AMS MathViewer

by Jonas Deré PDF

Trans. Amer. Math. Soc. 368 (2016), 1497-1516 Request permission

Abstract:

It is conjectured that every closed manifold admitting an Anosov diffeomorphism is, up to homeomorphism, finitely covered by a nilmanifold. Motivated by this conjecture, an important problem is to determine which nilmanifolds admit an Anosov diffeomorphism. The main theorem of this article gives a general method for constructing Anosov diffeomorphisms on nilmanifolds. As a consequence, we give new examples which were overlooked in a corollary of the classification of low-dimensional nilmanifolds with Anosov diffeomorphisms and a correction to this statement is proven. This method also answers some open questions about the existence of Anosov diffeomorphisms which are minimal in some sense.

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
Johannes Buchmann and Ulrich Vollmer, Binary quadratic forms, Algorithms and Computation in Mathematics, vol. 20, Springer, Berlin, 2007. An algorithmic approach. MR 2300780
S. G. Dani, Some two-step and three-step nilpotent Lie groups with small automorphism groups, Trans. Amer. Math. Soc. 355 (2003), no. 4, 1491–1503. MR 1946401, DOI 10.1090/S0002-9947-02-03178-1
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, Almost-Bieberbach groups: affine and polynomial structures, Lecture Notes in Mathematics, vol. 1639, Springer-Verlag, Berlin, 1996. MR 1482520, DOI 10.1007/BFb0094472
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, What an infra-nilmanifold endomorphism really should be$\ldots$, Topol. Methods Nonlinear Anal. 40 (2012), no. 1, 111–136. MR 3026104
Karel Dekimpe and Jonas Deré, Existence of anosov diffeomorphisms on infra-nilmanifolds modeled on free nilpotent lie groups, Topological Methods in Nonlinear Analysis (2014).
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
Karel Dekimpe and Kelly Verheyen, Anosov diffeomorphisms on nilmanifolds modelled on a free nilpotent Lie group, Dyn. Syst. 24 (2009), no. 1, 117–121. MR 2548819, DOI 10.1080/14689360802506177
Karel Dekimpe and Kelly Verheyen, Constructing infra-nilmanifolds admitting an Anosov diffeomorphism, Adv. Math. 228 (2011), no. 6, 3300–3319. MR 2844944, DOI 10.1016/j.aim.2011.08.008
David Fried, Nontoral pinched Anosov maps, Proc. Amer. Math. Soc. 82 (1981), no. 3, 462–464. MR 612740, DOI 10.1090/S0002-9939-1981-0612740-8
The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.7.4, 2014.
Fritz J. Grunewald, Daniel Segal, and Leon S. Sterling, Nilpotent groups of Hirsch length six, Math. Z. 179 (1982), no. 2, 219–235. MR 645498, DOI 10.1007/BF01214314
I. Martin Isaacs, Character theory of finite groups, Pure and Applied Mathematics, No. 69, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0460423
N. Jacobson, A note on automorphisms and derivations of Lie algebras, Proc. Amer. Math. Soc. 6 (1955), 281–283. MR 68532, DOI 10.1090/S0002-9939-1955-0068532-9
Jorge Lauret, Examples of Anosov diffeomorphisms, J. Algebra 262 (2003), no. 1, 201–209. MR 1970807, DOI 10.1016/S0021-8693(03)00030-9
Jorge Lauret, Rational forms of nilpotent Lie algebras and Anosov diffeomorphisms, Monatsh. Math. 155 (2008), no. 1, 15–30. MR 2434923, DOI 10.1007/s00605-008-0562-0
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
Jorge Lauret and Cynthia E. Will, Nilmanifolds of dimension $\leq 8$ admitting Anosov diffeomorphisms, Trans. Amer. Math. Soc. 361 (2009), no. 5, 2377–2395. MR 2471923, DOI 10.1090/S0002-9947-08-04757-0
Meera G. Mainkar and Cynthia E. Will, Examples of Anosov Lie algebras, Discrete Contin. Dyn. Syst. 18 (2007), no. 1, 39–52. MR 2276485, DOI 10.3934/dcds.2007.18.39
Anthony Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math. 96 (1974), 422–429. MR 358865, DOI 10.2307/2373551
Tracy L. Payne, Anosov automorphisms of nilpotent Lie algebras, J. Mod. Dyn. 3 (2009), no. 1, 121–158. MR 2481335, DOI 10.3934/jmd.2009.3.121
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
Daniel Segal, Polycyclic groups, Cambridge Tracts in Mathematics, vol. 82, Cambridge University Press, Cambridge, 1983. MR 713786, DOI 10.1017/CBO9780511565953
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1