Comparison of volumes of Siegel sets and fundamental domains for \(\mathrm {SL}_n (\pmb {\mathbb {Z}})\)

Original Paper
  • 57 Downloads

Abstract

The purpose of this paper is to calculate explicitly the volumes of Siegel sets which are coarse fundamental domains for the action of \({\mathrm {SL}} _n (\mathbb {Z})\) in \(\mathrm {SL} _n (\mathbb {R})\), so that we can compare these volumes with those of the fundamental domains of \({\mathrm {SL}} _n (\mathbb {Z})\) in \(\mathrm {SL} _n (\mathbb {R})\), which are also computed here, for any \(n\ge 2\). An important feature of this computation is that it requires keeping track of normalization constants of the Haar measures. We conclude that the ratio between volumes of fundamental domains and volumes of Siegel sets grows super-exponentially fast as n goes to infinity. As a corollary, we obtained that this ratio gives a super-exponencial lower bound, depending only on n, for the number of intersecting Siegel sets. We were also able to give an upper bound for this number, by applying some results on the heights of intersecting elements in \( {\mathrm {SL}} _n (\mathbb {Z})\).

Keywords

Arithmetic groups Siegel sets Coarse fundamental domains Volumes 

Mathematics Subject Classification

20G20 20G30 51N30 14L35 

Notes

Acknowledgements

I would like to thank Professor Mikhail Belolipetsky for several suggestions on the development of this paper and also on the text. I also thank Paul Garret and Martin Orr for their very helpful works and for always answering my emails with good suggestions, and Cayo Dória for helping me to understand better some topics. Finally, I also thank the refferee for carefully reading the paper and for giving suggestions that improved the presentation of the results.

References

  1. 1.
    Borel, A.: Introduction aux Groupes Arithmètiques. Hermann, Paris (1969)MATHGoogle Scholar
  2. 2.
    Borel, A., Harish-Chandra, : Arithmetic subgroups of algebraic groups. Ann. Math. 75, 485–535 (1962)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Duke, W., Rudinick, Z., Sarnak, P.: Density of integer points on affine homogeneous varieties. Duke Math. J. 71(1), 143–179 (1993)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Garret, P.: Volume of \({\rm SL}_n (\mathbb{Z})\backslash {\rm SL}_n (\mathbb{R})\) and \(Sp_n(\mathbb{Z})\backslash Sp_n(\mathbb{R})\). Paul Garrett’s homepage (2014). http://www-users.math.umn.edu/~garrett/m/v/volumes.pdf. Accessed 07 Nov 2017
  5. 5.
    Habegger, P., Pila, J.: Some unlikely intersections beyond André–Oort. Compos. Math. 148, 1–27 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Harder, G.: A Gauss–Bonnet formula for discrete arithmetically defined groups. Ann. Sci. Ec. Norm. Supér. \(4^e\) série, tome 4(3), 409–455 (1971)Google Scholar
  7. 7.
    Morris, D.W.: Introduction to Arithmetic Groups. Deductive Press (2015). arXiv:math/0106063
  8. 8.
    Orr, M.: Height bounds and the Siegel property (2016). Preprint arXiv:1609.01315v3
  9. 9.
    Siegel, C.L.: Einführung in die Theorie der Modulfunktionen n-ten Grades. Math. Ann. 116, 617–657 (1939)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Siegel, C.L.: A mean value theorem in geometry of numbers. Ann. Math. 45(2), 340–347 (1945)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Siegel, C.L.: Lectures on the Geometry of Numbers. Springer, Berlin (1989)CrossRefMATHGoogle Scholar
  12. 12.
    Stein, E.M., Weiss, G.L.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series 32. Princeton University Press, Princeton (1971)MATHGoogle Scholar
  13. 13.
    Venkataramana, T.N.: Lattices in Lie groups. In: Workshop on Geometric Group Theory, India (2010)Google Scholar
  14. 14.
    Young, R.: The Dehn function of \({\rm SL} _n (\mathbb{Z})\). Ann. Math. 177, 969–1027 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Instituto Nacional de Matemática Pura e AplicadaRio de JaneiroBrazil

Personalised recommendations