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Commensurable Arithmetic Groups and Volumes

  • Colin Maclachlan
  • Alan W. Reid
Part of the Graduate Texts in Mathematics book series (GTM, volume 219)

Abstract

In this chapter, we return to considering arithmetic Kleinian and Fuchsian groups and the related quaternion algebras. Recall that the wide commensurability classes of arithmetic Kleinian groups are in one-to-one correspondence with the isomorphism classes of quaternion algebras over a number field with one complex place which are ramified at all the real places. There is a similar one-to-one correspondence for arithmetic Fuchsian groups. Thus, for a suitable quaternion algebra A, let C(A) denote the (narrow) commensurability class of associated arithmetic Kleinian or Fuchsian groups. In this chapter, we investigate how the elements of C(A)are distributed and, in particular, determine the maximal elements of C(A) of which there are infinitely many. Since these groups are all of finite covolume, their volumes are, of course, commensurable. As a starting point to determining these volumes, a formula for the groups Pρ(O 1), where O is a maximal order in A, is obtained in terms of the number-theoretic data defining the number field and the quaternion algebra. This relies critically on the fact that the Tamagawa number of the quotient A A 1 /A k 1 of the idèle group A A 1, is 1, as discussed in Chapter 7. From this formula, one can determine the covolumes of the maximal elements of C(A)and show that all of these volumes are integral multiples of a single number. Much of this chapter is based on work of Borel.

Keywords

Prime Ideal Maximal Order Fuchsian Group Quaternion Algebra Triangle Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Further Reading

  1. Borel, A. (1981). Commensurability classes and volumes of hyperbolic three-manifolds. Ann. Scuola Norm. Sup. Pisa, 8: 1–33.MathSciNetzbMATHGoogle Scholar
  2. Chinburg, T. and Friedman, E. (1999). An embedding theorem for quaternion algebras. J. London Math. Soc, 60: 33–44.MathSciNetCrossRefGoogle Scholar
  3. Vignéras, M.-F. (1980a). Arithmétique des Algèbres de Quaternions.Lecture Notes in Mathematics No. 800. Springer-Verlag, Berlin.Google Scholar
  4. Shimizu, H. (1965). On zeta functions of quaternion algebras. Annals of Math, 81: 166–193.zbMATHCrossRefGoogle Scholar
  5. Thurston, W. (1979). The geometry and topology of three-manifolds. Notes from Princeton University.Google Scholar
  6. Janusz, G. (1996). Algebraic Number Fields, 2nd ed. Graduate Studies in Mathematics Vol. 7. American Mathematical Society, Providence, RI.Google Scholar
  7. Gromov, M. (1981). Hyperbolic manifolds according to Thurston and Jorgensen. Sémin. Bourbaki, 554: 40–53.MathSciNetCrossRefGoogle Scholar
  8. Odlyzko, A. (1975). Some analytic estimates of class numbers and discriminants. Invent. Math, 29: 275–286.MathSciNetzbMATHCrossRefGoogle Scholar
  9. Martinet, J. (1982). Petits descriminants des corps de nombres, pages 151–193. L.M.S. Lecture Note Series Vol. 56. Cambridge University Press, Cambridge.Google Scholar
  10. Takeuchi, K. (1983). Arithmetic Fuchsian groups of signature (1; e). J. Math. Soc. Japan, 35: 381–407.zbMATHCrossRefGoogle Scholar
  11. Maclachlan, C. and Rosenberger, G. (1983). Two-generator arithmetic Fuchsian groups. Math. Proc. Cambridge Phil. Soc, 93: 383–391.MathSciNetzbMATHCrossRefGoogle Scholar
  12. Maclachlan, C. (1996). Triangle subgroups of hyperbolic tetrahedral groups. Pacific J. Math, 176: 195–203.MathSciNetGoogle Scholar
  13. Maclachlan, C. and Martin, G. (1999). 2-generator arithmetic Kleinian groups. J. Reine Angew. Math, 511: 95–117.Google Scholar
  14. Reid, A. (1995). A non-Haken hyperbolic 3-manifold covered by a surface bundle. Pacific J. Math, 167: 163–182.MathSciNetzbMATHCrossRefGoogle Scholar
  15. Maclachlan, C. and Reid, A. (1989). The arithmetic structure of tetrahedral groups of hyperbolic isometries. Mathematika, 36: 221–240.MathSciNetzbMATHCrossRefGoogle Scholar
  16. Gehring, F., Maclachlan, C., Martin, G., and Reid, A. (1997). Arithmet- icity, discreteness and volume. Trans. Am. Math. Soc, 349: 3611–3643.MathSciNetzbMATHCrossRefGoogle Scholar
  17. Siegel, C. (1969). Berechnung von Zetafunktion an ganzzahligen Stellen. Nach. Akad. Wiss. Göttingen, 1969: 87–102.Google Scholar
  18. Klingen, H. (1961). Über die Werte der Dedekindschen Zetafunktion. Math. Annalen, 145: 265–277.MathSciNetCrossRefGoogle Scholar
  19. Takeuchi, K. (1977b). Commensurability classes of arithmetic triangle groups. J. Fac. Sci. Univ. Tokyo, 24: 201–222.zbMATHGoogle Scholar
  20. Zagier, D. (1986). Hyperbolic 3-manifolds and special values of Dedekind zeta-functions. Invent. Math, 83: 285–301.MathSciNetzbMATHCrossRefGoogle Scholar
  21. Takeuchi, K. (1977a). Arithmetic triangle groups. J. Math. Soc. Japan, 29: 91–106.MathSciNetzbMATHCrossRefGoogle Scholar
  22. Maclachlan, C. and Rosenberger, G. (1992b). Two-generator arithmetic Fuchsian groups II. Math. Proc. Cambridge Phil. Soc, 111: 7–24.MathSciNetzbMATHCrossRefGoogle Scholar
  23. Sunaga, J. (1997a). Some arithmetic Fuchsian groups with signature (0; el, e2, e3, e4). Tokyo J. Math, 20: 435–451.MathSciNetzbMATHCrossRefGoogle Scholar
  24. Sunaga, J. (1997b). Some arithmetic Fuchsian groups with signature (0; el, e2, e3, e4) II. Saitama Math. J, 15: 15–46.MathSciNetzbMATHGoogle Scholar
  25. Nakinishi, T., Näätänen, M., and Rosenberger, G. (1999). Arithmetic Fuchsian groups of signature. Contemp. Math, 240: 269–277.CrossRefGoogle Scholar
  26. Maclachlan, C. and Rosenberger, G. (1992a). Commmensurability classes of two generator Fuchsian groups, pages 171–189. L. M. S. Lecture Note Series Vol 173. Cambridge University Press, Cambridge.Google Scholar
  27. Vulakh, L. Y. (1994). Reflections in extended Bianchi groups. Math. Proc. Cambridge Phil. Soc, 115: 13–25.MathSciNetzbMATHCrossRefGoogle Scholar
  28. Vinberg, E. (1990). Reflective subgroups in Bianchi groups. Sel. Soviet Math, 9: 4: 309–314.MathSciNetzbMATHGoogle Scholar
  29. Shaiheev, M. (1990). Reflective subgroups in Bianchi groups. Sel. Soviet Math, 9: 4: 315–322.MathSciNetGoogle Scholar
  30. Elstrodt, J., Grunewald, F., and Mennicke, J. (1983). Discontinuous groups on three-dimensional hyperbolic space: analytic theory and arithmetic applications. Russian Math. Surveys, 38: 137–168.zbMATHCrossRefGoogle Scholar
  31. Chinburg, T. and Friedman, E. (1986). The smallest arithmetic hyperbolic three-orbifold. Invent. Math, 86: 507–527.MathSciNetzbMATHCrossRefGoogle Scholar
  32. Lang, S. (1970). Algebraic Number Theory. Addison-Wesley, Reading, MA.Google Scholar
  33. Chinburg, T., Friedman, E., Jones, K., and Reid, A. (2001). The arithmetic hyperbolic 3-manifold of smallest volume. Ann. Scuola Norm. Sup. Pisa, 30: 1–40.MathSciNetzbMATHGoogle Scholar
  34. Cohen, H. (1993). A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics Vol. 138. Springer-Verlag, New York.Google Scholar
  35. Goodman, O. (2001). Snap: A computer program for studying arithmetic invariants of hyperbolic 3-manifolds. http://www.ms.unimelb.edu.au/snap.
  36. Maclachlan, C. (1986). Fuchsian Subgroups of the Groups PSL2(Od), pages 305–311. L.M.S. Lecture Note Series Vol. 112. Cambridge University Press, Cambridge.Google Scholar
  37. Gabai, D., Meyerhoff, R., and Thurston, N. (2002). Homotopy hyperbolic 3-manifolds are hyperbolic. Annals of Math.Google Scholar
  38. Culler, M. and Shalen, P. (1992). Paradoxical decompositions, 2-generator Kleinian groups and volumes of hyperbolic 3-manifolds. J. Am. Math. Soc, 5: 231–288.MathSciNetzbMATHGoogle Scholar
  39. Culler, M., Hersonsky, S., and Shalen, P. (1998). The first Betti number of the smallest hyperbolic 3-manifold. Topology, 37: 807–849.MathSciNetCrossRefGoogle Scholar
  40. Gehring, F., Maclachlan, C., and Martin, G. (1998). Two-generator arithmetic Kleinian groups II. Bull. London Math. Soc, 30: 258–266.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Colin Maclachlan
    • 1
  • Alan W. Reid
    • 2
  1. 1.Department of Mathematical Sciences, Kings CollegeUniversity of AberdeenAberdeenUK
  2. 2.Department of MathematicsUniversity of Texas at AustinAustinUSA

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