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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Minkowski's conjecture, well-rounded lattices and topological dimension


Author: Curtis T. McMullen
Journal: J. Amer. Math. Soc. 18 (2005), 711-734
MSC (2000): Primary 11H31; Secondary 11E57, 11J83, 55M10, 55N30
Published electronically: March 24, 2005
MathSciNet review: 2138142
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $A \subset {\operatorname{SL}}_n({\mathbb R})$ be the diagonal subgroup, and identify ${\operatorname{SL}}_n({\mathbb R})/ {\operatorname{SL}}_n({\mathbb Z})$ with the space of unimodular lattices in ${\mathbb R}^n$. In this paper we show that the closure of any bounded orbit

\begin{displaymath}A \cdot L \subset {\operatorname{SL}}_n({\mathbb R})/{\operatorname{SL}}_n({\mathbb Z}) \end{displaymath}

meets the set of well-rounded lattices. This assertion implies Minkowski's conjecture for $n=6$ and yields bounds for the density of algebraic integers in totally real sextic fields.

The proof is based on the theory of topological dimension, as reflected in the combinatorics of open covers of ${\mathbb R}^n$ and $T^n$.


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

  • [Ash] A. Ash.
    Small-dimensional classifying spaces for arithmetical subgroups of general linear groups.
    Duke Math. J. 51(1984), 459-468. MR 0747876 (85k:22027)
  • [AM] A. Ash and M. McConnell.
    Cohomology at infinity and the well-rounded retract for general linear groups.
    Duke Math. J. 90(1997), 549-576. MR 1480546 (98h:11063)
  • [BW] R. P. Bambah and A. C. Woods.
    Minkowski's conjecture for $n=5$; a theorem of Skubenko.
    J. Number Theory 12(1980), 27-48. MR 0566866 (81g:10043)
  • [Ba1] E. Bayer-Fluckiger.
    Lattices and number fields.
    In Algebraic Geometry: Hirzebruch 70 (Warsaw, 1998), pages 69-84. Amer. Math. Soc., 1999. MR 1718137 (2000j:11049)
  • [Ba2] E. Bayer-Fluckiger.
    Ideal lattices.
    In A Panorama of Number Theory, pages 168-184. Cambridge Univ. Press, 2002.MR 1975451 (2004k:11108)
  • [BN] E. Bayer-Fluckiger and G. Nebe.
    On the Euclidean minimum of some real number fields.
    Preprint, 2004.
  • [BiS] B. J. Birch and H. P. F. Swinnerton-Dyer.
    On the inhomogeneous minimum of the product of $n$linear forms.
    Mathematika 3(1956), 25-39. MR 0079049 (18:22a)
  • [BoS] Z. I. Borevich and I. R. Shafarevich.
    Number Theory.
    Academic Press, 1966. MR 0195803 (33:4001)
  • [BT] R. Bott and L. W. Tu.
    Differential Forms in Algebraic Topology.
    Springer-Verlag, 1982. MR 0658304 (83i:57016)
  • [CaS] J. W. S. Cassels and H. P. F. Swinnerton-Dyer.
    On the product of three homogeneous linear forms and the indefinite ternary quadratic forms.
    Philos. Trans. Roy. Soc. London. Ser. A. 248(1955), 73-96. MR 0070653 (17:14f)
  • [CoS] J. H. Conway and N. J. A. Sloane.
    Sphere Packings, Lattices and Groups.
    Springer-Verlag, 1999. MR 1662447 (2000b:11077)
  • [Da] H. Davenport.
    A simple proof of Remak's theorem on the product of $n$ linear forms.
    J. London Math. Soc. 14(1939), 47-51.
  • [Dy] F. J. Dyson.
    On the product of four non-homogeneous linear forms.
    Annals of Math. 49(1948), 82-109. MR 0025515 (10:19a)
  • [EM] S. Eilenberg and N. Steenrod.
    Foundations of Algebraic Topology.
    Princeton University Press, 1952.MR 0050886 (14:398b)
  • [Gr] M. Gromov.
    Volume and bounded cohomology.
    IHES Publ. Math. 56(1982), 5-100. MR 0686042 (84h:53053)
  • [GL] P. M. Gruber and C. G. Lekkerkerker.
    Geometry of Numbers.
    Elsevier, 1987. MR 0893813 (88j:11034)
  • [HW] W. Hurewicz and H. Wallman.
    Dimension Theory.
    Princeton University Press, 1941. MR 0006493 (3:312b)
  • [Ko] J. F. Koksma.
    Diophantische Approximationen.
    Springer-Verlag, 1936. MR 0344200 (49:8940)
  • [LW] E. Lindenstrauss and B. Weiss.
    On sets invariant under the action of the diagonal group.
    Ergodic Theory Dynam. Systems 21(2001), 1481-1500. MR 1855843 (2002j:22009)
  • [Mg] G. A. Margulis.
    Problems and conjectures in rigidity theory.
    In Mathematics: Frontiers and Perspectives, pages 161-174. Amer. Math. Soc., 2000. MR 1754775 (2001d:22008)
  • [Min] H. Minkowski.
    Diophantische Approximationen.
    Chelsea, 1957. MR 0086102 (19:124f)
  • [Oh] H. Oh.
    Finiteness of compact maximal flats of bounded volume.
    Ergodic Theory Dynam. Systems 24(2004), 217-225.MR 2041269
  • [Pan] P. Pansu.
    Introduction to $L\sp 2$ Betti numbers.
    In Riemannian Geometry, pages 53-86. Amer. Math. Soc., 1996.MR 1377309 (97c:58148)
  • [Rag] M. S. Raghunathan.
    Discrete Subgroups of Lie Groups.
    Springer-Verlag, 1972. MR 0507234 (58:22394a)
  • [Re] R. Remak.
    Verallgemeinerung eines Minkowskischen Satzes.
    Math. Zeitschr. 17-18(1928), 1-34; 173-200.
  • [Sk1] B. F. Skubenko.
    A new variant of the proof of the inhomogeneous Minkowski conjecture for $n=5$a.
    Trudy Mat. Inst. Steklov. 142(1976), 240-253, 271. MR 0563094 (58:27803)
  • [Sk2] B. F. Skubenko.
    A proof of Minkowski's conjecture on the product of $n$ linear inhomogeneous forms in $n$ variables for $n\leq 5$.
    J. Soviet Math. 6(1976), 627-650.
  • [So] C. Soulé.
    The cohomology of ${\rm SL}\sb{3}( Z)$.
    Topology 17(1978), 1-22. MR 0470141 (57:9908)
  • [TW] G. Tomanov and B. Weiss.
    Closed orbits for actions of maximal tori on homogeneous spaces.
    Duke Math. J. 119(2003), 367-392. MR 1997950 (2004g:22006)
  • [Va] P. Vámos.
    The missing axiom of matroid theory is lost forever.
    J. London Math. Soc. 18(1978), 403-408.MR 0518224 (80a:05062)
  • [Wd] A. C. Woods.
    Covering six space with spheres.
    J. Number Theory 4(1972), 157-180. MR 0302570 (46:1714)

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Additional Information

Curtis T. McMullen
Affiliation: Department of Mathematics, Harvard University, 1 Oxford St, Cambridge, Massachusetts 02138-2901

DOI: http://dx.doi.org/10.1090/S0894-0347-05-00483-2
PII: S 0894-0347(05)00483-2
Received by editor(s): August 27, 2004
Published electronically: March 24, 2005
Additional Notes: Research partially supported by the NSF and the Guggenheim Foundation.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.