The finite subgroups of maximal arithmetic kleinian groups

Ted Chinburg; Eduardo Friedman

Annales de l'institut Fourier (2000)

  • Volume: 50, Issue: 6, page 1765-1798
  • ISSN: 0373-0956

Abstract

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Given a maximal arithmetic Kleinian group Γ ⊂ PGL ( 2 , ℂ ) , we compute its finite subgroups in terms of the arithmetic data associated to Γ by Borel. This has applications to the study of arithmetic hyperbolic 3-manifolds.

How to cite

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Chinburg, Ted, and Friedman, Eduardo. "The finite subgroups of maximal arithmetic kleinian groups." Annales de l'institut Fourier 50.6 (2000): 1765-1798. <http://eudml.org/doc/75471>.

@article{Chinburg2000,
abstract = {Given a maximal arithmetic Kleinian group $\Gamma \subset \{\rm PGL\}(2,\{\Bbb C\})$, we compute its finite subgroups in terms of the arithmetic data associated to $\Gamma $ by Borel. This has applications to the study of arithmetic hyperbolic 3-manifolds.},
author = {Chinburg, Ted, Friedman, Eduardo},
journal = {Annales de l'institut Fourier},
keywords = {finite subgroups; Kleinian groups; minimal arithmetic hyperbolic 3-orbifolds; quaternion algebras over number fields},
language = {eng},
number = {6},
pages = {1765-1798},
publisher = {Association des Annales de l'Institut Fourier},
title = {The finite subgroups of maximal arithmetic kleinian groups},
url = {http://eudml.org/doc/75471},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Chinburg, Ted
AU - Friedman, Eduardo
TI - The finite subgroups of maximal arithmetic kleinian groups
JO - Annales de l'institut Fourier
PY - 2000
PB - Association des Annales de l'Institut Fourier
VL - 50
IS - 6
SP - 1765
EP - 1798
AB - Given a maximal arithmetic Kleinian group $\Gamma \subset {\rm PGL}(2,{\Bbb C})$, we compute its finite subgroups in terms of the arithmetic data associated to $\Gamma $ by Borel. This has applications to the study of arithmetic hyperbolic 3-manifolds.
LA - eng
KW - finite subgroups; Kleinian groups; minimal arithmetic hyperbolic 3-orbifolds; quaternion algebras over number fields
UR - http://eudml.org/doc/75471
ER -

References

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  1. [1] A. BOREL, Commensurability classes and volumes of hyperbolic 3-manifolds, Ann. Scuola Norm. Sup. Pisa, 8 (1981), 1-33. Also in Borel's Oeuvres, Berlin, Springer, 1983. Zbl0473.57003MR82j:22008
  2. [2] T. CHINBURG and E. FRIEDMAN, An embedding theorem for quaternion algebras J. London Math. Soc., (2) 60 (1999), 33-44. Zbl0940.11053MR2000j:11173
  3. [3] T. CHINBURG and E. FRIEDMAN, The smallest arithmetic hyperbolic 3-orbifold, Invent. Math., 86 (1986), 507-527. Zbl0643.57011MR88a:22022
  4. [4] T. CHINBURG, E. FRIEDMAN, K. JONES and A. W. REID, The arithmetic hyperbolic 3-manifold of smallest volume, Ann. Scuola Norm. Sup. Pisa (to appear). Zbl1008.11015
  5. [5] M. DEURING, Algebren, Springer Verlag, Berlin, 1935. Zbl0011.19801JFM61.0118.01
  6. [6] W. FEIT, Exceptional subgroups of GL2, Appendix to chapter XI of Introduction to Modular Forms by S. Lang, Berlin, Springer Verlag, 1976. 
  7. [7] F. W. GEHRING and G. J. MARTIN, 6-torsion and hyperbolic volume, Proc. Amer. Math. Soc., 117 (1993), 727-735. Zbl0790.30032MR93d:30056
  8. [8] F. W. GEHRING, C. MACLACHLAN, G. J. MARTIN and A. W. REID, Arithmeticity, discreteness and volume, Trans. Amer. Math. Soc., 349 (1997), 3611-3643. Zbl0889.30031
  9. [9] K. N. JONES and A. W. REID, Minimal index torsion-free subgroups of Kleinian groups, Math. Ann., 310 (1998), 235-250. Zbl0890.57016MR99a:57010
  10. [10] J. MARTINET, Petits discriminants des corps de nombres. In J. Armitage, editor, Number Theory Days, 1980 (Exeter 1980). London Math. Soc. Lecture Notes Ser. 56, Cambridge: Cambridge Univ. Press, 1982. Zbl0491.12005
  11. [11] J.-P. SERRE, Trees, Springer Verlag, Berlin, 1980. Zbl0548.20018
  12. [12] L. WASHINGTON, Introduction to Cyclotomic Fields, Springer Verlag, Berlin, 1982. Zbl0484.12001MR85g:11001
  13. [13] M.-F. VIGNÉRAS, Arithmétique des algèbres de Quaternions, Lecture Notes in Math. 800, Springer Verlag, Berlin, 1980. Zbl0422.12008MR82i:12016

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