Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A new method for constructing Anosov Lie algebras


Author: Jonas Deré
Journal: Trans. Amer. Math. Soc. 368 (2016), 1497-1516
MSC (2010): Primary 37D20; Secondary 22E25, 20F34
DOI: https://doi.org/10.1090/tran6655
Published electronically: June 15, 2015
MathSciNet review: 3430371
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] 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 (42 #5284)
  • [2] Johannes Buchmann and Ulrich Vollmer, Binary quadratic forms, An algorithmic approach, Algorithms and Computation in Mathematics, vol. 20, Springer, Berlin, 2007. MR 2300780 (2008b:11046)
  • [3] 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 (2004b:22010), https://doi.org/10.1090/S0002-9947-02-03178-1
  • [4] 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 (2006j:22005), https://doi.org/10.1090/S0002-9947-04-03518-4
  • [5] Karel Dekimpe, Almost-Bieberbach groups: affine and polynomial structures, Lecture Notes in Mathematics, vol. 1639, Springer-Verlag, Berlin, 1996. MR 1482520 (2000b:20066)
  • [6] Karel Dekimpe, Hyperbolic automorphisms and Anosov diffeomorphisms on nilmanifolds, Trans. Amer. Math. Soc. 353 (2001), no. 7, 2859-2877 (electronic). MR 1828476 (2002c:37043), https://doi.org/10.1090/S0002-9947-01-02683-6
  • [7] Karel Dekimpe, What an infra-nilmanifold endomorphism really should be$ \ldots $, Topol. Methods Nonlinear Anal. 40 (2012), no. 1, 111-136. MR 3026104
  • [8] Karel Dekimpe and Jonas Deré, Existence of anosov diffeomorphisms on infra-nilmanifolds modeled on free nilpotent lie groups, Topological Methods in Nonlinear Analysis (2014).
  • [9] 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 (2004i:37054), https://doi.org/10.1017/S001708950300123X
  • [10] 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 (2010i:37062), https://doi.org/10.1080/14689360802506177
  • [11] Karel Dekimpe and Kelly Verheyen, Constructing infra-nilmanifolds admitting an Anosov diffeomorphism, Adv. Math. 228 (2011), no. 6, 3300-3319. MR 2844944, https://doi.org/10.1016/j.aim.2011.08.008
  • [12] David Fried, Nontoral pinched Anosov maps, Proc. Amer. Math. Soc. 82 (1981), no. 3, 462-464. MR 612740 (82e:58079), https://doi.org/10.2307/2043961
  • [13] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.7.4, 2014.
  • [14] 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 (83d:20025), https://doi.org/10.1007/BF01214314
  • [15] 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 (57 #417)
  • [16] N. Jacobson, A note on automorphisms and derivations of Lie algebras, Proc. Amer. Math. Soc. 6 (1955), 281-283. MR 0068532 (16,897e)
  • [17] Jorge Lauret, Examples of Anosov diffeomorphisms, J. Algebra 262 (2003), no. 1, 201-209. MR 1970807 (2004g:37033a), https://doi.org/10.1016/S0021-8693(03)00030-9
  • [18] Jorge Lauret, Rational forms of nilpotent Lie algebras and Anosov diffeomorphisms, Monatsh. Math. 155 (2008), no. 1, 15-30. MR 2434923 (2009f:17019), https://doi.org/10.1007/s00605-008-0562-0
  • [19] Jorge Lauret and Cynthia E. Will, On Anosov automorphisms of nilmanifolds, J. Pure Appl. Algebra 212 (2008), no. 7, 1747-1755. MR 2400740 (2009a:37058), https://doi.org/10.1016/j.jpaa.2007.11.011
  • [20] 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 (2010b:37082), https://doi.org/10.1090/S0002-9947-08-04757-0
  • [21] Meera G. Mainkar and Cynthia E. Will, Examples of Anosov Lie algebras, Discrete Contin. Dyn. Syst. 18 (2007), no. 1, 39-52. MR 2276485 (2008a:37033), https://doi.org/10.3934/dcds.2007.18.39
  • [22] Anthony Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math. 96 (1974), 422-429. MR 0358865 (50 #11324)
  • [23] Tracy L. Payne, Anosov automorphisms of nilpotent Lie algebras, J. Mod. Dyn. 3 (2009), no. 1, 121-158. MR 2481335 (2010k:37047), https://doi.org/10.3934/jmd.2009.3.121
  • [24] Hugh L. Porteous, Anosov diffeomorphisms of flat manifolds, Topology 11 (1972), 307-315. MR 0296976 (45 #6035)
  • [25] John Scheuneman, Two-step nilpotent Lie algebras, J. Algebra 7 (1967), 152-159. MR 0217134 (36 #225)
  • [26] Daniel Segal, Polycyclic groups, Cambridge Tracts in Mathematics, vol. 82, Cambridge University Press, Cambridge, 1983. MR 713786 (85h:20003)
  • [27] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. MR 0228014 (37 #3598)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 37D20, 22E25, 20F34

Retrieve articles in all journals with MSC (2010): 37D20, 22E25, 20F34


Additional Information

Jonas Deré
Affiliation: KU Leuven Kulak, E. Sabbelaan 53, 8500 Kortrijk, Belgium
Email: jonas.dere@kuleuven-kulak.be

DOI: https://doi.org/10.1090/tran6655
Received by editor(s): December 10, 2013
Received by editor(s) in revised form: June 26, 2014, and November 18, 2014
Published electronically: June 15, 2015
Additional Notes: The author was supported by a Ph.D. fellowship of the Research Foundation – Flanders (FWO). Research supported by the research Fund of the KU Leuven
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society