Geometriae Dedicata

, Volume 151, Issue 1, pp 245–258 | Cite as

Conjugacy classes in Möbius groups

  • Krishnendu Gongopadhyay
Original Paper


Let \({\mathbb H^{n+1}}\) denote the n + 1-dimensional (real) hyperbolic space. Let \({\mathbb {S}^{n}}\) denote the conformal boundary of the hyperbolic space. The group of conformal diffeomorphisms of \({\mathbb {S}^{n}}\) is denoted by M(n). Let M o (n) be its identity component which consists of all orientation-preserving elements in M(n). The conjugacy classification of isometries in M o (n) depends on the conjugacy of T and T −1 in M o (n). For an element T in M(n), T and T −1 are conjugate in M(n), but they may not be conjugate in M o (n). In the literature, T is called real if T is conjugate in M o (n) to T −1. In this paper we classify real elements in M o (n). Let T be an element in M o (n). Corresponding to T there is an associated element T o in SO(n + 1). If the complex conjugate eigenvalues of T o are given by \({\{e^{i\theta_j}, e^{-i\theta_j}\}, 0 < \theta_j \leq \pi, j=1,\ldots,k}\) , then {θ1, . . . , θ k } are called the rotation angles of T. If the rotation angles of T are distinct from each-other, then T is called a regular element. After classifying the real elements in M o (n) we have parametrized the conjugacy classes of regular elements in M o (n). In the parametrization, when T is not conjugate to T −1 , we have enlarged the group and have considered the conjugacy class of T in M(n). We prove that each such conjugacy class can be induced with a fibration structure.


Hyperbolic space Möbius groups Conjugacy classes Real elements 

Mathematics Subject Classification (2000)

Primary: 51M10 Secondary: 20E45 58D99 


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  1. 1.
    Ahlfors L.V.: Möbius Transformations and Clifford Numbers. Differential Geometry and Complex Analysis, pp. 65–73. Springer, Berlin (1985)Google Scholar
  2. 2.
    Birkhoff G.D.: The restricted problem of three bodies. Rend. Circ. Mat. Palermo 39, 265–334 (1915)CrossRefGoogle Scholar
  3. 3.
    Cao C., Waterman P.L.: Conjugacy Invariants of Möbius Groups. Quasiconformal Mappings and Analysis (Ann Arbor, MI, 1995), pp. 109–139. Springer, New York (1998)Google Scholar
  4. 4.
    Chen S.S., Greenberg L.: Hyperbolic Spaces,Contributions to Analysis (A Collection of Papers Dedicated to Lipman Bers), pp. 49–87. Academic Press, New York (1974)Google Scholar
  5. 5.
    Devaney R.L.: Reversible diffeomorphisms and flows. Trans. Am. Math. Soc. 218, 89–113 (1976)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Falbel E., Wentworth R.: On products of isometries of hyperbolic space. Topol. Appl. 156(13), 2257–2263 (2009)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Feit, W., Zuckerman, G.J.: Reality properties of conjugacy classes in spin groups and symplectic groups. Algebraists’ homage: papers in ring theory and related topics (New Haven, Conn., 1981). Contemp. Math., 13, Am. Math. Soc., Providence, R.I. pp. 239–253 (1982)Google Scholar
  8. 8.
    Gongopadhyay K., Kulkarni R.S.: z-Classes of isometries of the hyperbolic space. Conform Geom. Dyn. 13, 91–109 (2009)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Greenberg L.: Discrete subgroups of the Lorentz group. Math. Scand. 10, 85–107 (1962)MathSciNetMATHGoogle Scholar
  10. 10.
    Knüppel F., Nielsen K.: Products of involutions in O +(V). Linear Algebra Appl. 94, 217–222 (1987)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Kulkarni, R.S.: Conjugacy classes in M(n). Conformal Geometry (Bonn, 1985/1986), 41–64, Aspects Math., E12, Vieweg, Braunschweig,(1988)Google Scholar
  12. 12.
    Kulkarni R.S., Raymond F.: 3-dimensional Lorentz space-forms and Seifert fiber spaces. J. Differ. Geom. 21(2), 231–268 (1985)MathSciNetMATHGoogle Scholar
  13. 13.
    Moeglin C., Vignéras M.-F., Waldspurger J.-L.: Correspondences de Howe sur un corps p-adique, Lecture Notes in Mathematics 1291, Springer, Berlin (1987)Google Scholar
  14. 14.
    Moser J.K., Webster S.M.: Normal forms for real surfaces in \({\mathbb {C}^{2}}\) near complex tangents and hyperbolic surface transformations. Acta Math. 150(3–4), 255–296 (1983)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Ratcliffe J.G.: Foundation of Hyperbolic Manifolds, Graduate Texts in Mathematics 149. Springer, Berlin (1994)Google Scholar
  16. 16.
    Short I.: Reversible maps in isometry groups of spherical, Euclidean and hyperbolic space. Math. Proc. R. Ir. Acad. 108(1), 33–46 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Short I., O’Farrell A.G., Lávička R.: Reversible maps in the group of quaternionic Möbius transformations. Math. Proc. Cambridge Philos. Soc. 143(1), 57–69 (2007)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Singh A., Thakur M.: Reality properties of conjugacy classes in algebraic groups. Israel J. Math. 165, 1–27 (2008)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Singh A., Thakur M.: Reality properties of conjugacy classes in G 2. Israel J. Math. 145, 157–192 (2005)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Tiep P.H., Zalesski A.E.: Real conjugacy classes in algebraic groups and finite groups of Lie type. J. Group Theory 8, 291–315 (2005)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Wada M.: Conjugacy invariants of Möbius transformations. Complex Var. Theory Appl. 15(2), 125–133 (1990)MathSciNetMATHGoogle Scholar
  22. 22.
    Wonenburger M.J.: Transformations which are products of two involutions. J. Math. Mech. 16, 327–338 (1966)MathSciNetMATHGoogle Scholar

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© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Mathematics, Indian Institute of Science Education and Research (IISER) MohaliTransit Campus: MGSIPAP ComplexChandigarhIndia

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