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Congruence classes of points in quaternionic hyperbolic space

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Abstract

An important problem in quaternionic hyperbolic geometry is to classify ordered m-tuples of pairwise distinct points in the closure of quaternionic hyperbolic n-space, \(\overline{\mathbf{H}_{\mathbb H}^n}\), up to congruence in the holomorphic isometry group \(\mathrm{PSp}(n,1)\) of \(\mathbf{H}_{\mathbb H}^n\). In this paper we concentrate on two cases: \(m=3\) in \(\overline{\mathbf{H}_{\mathbb H}^n}\) and \(m=4\) on \(\partial \mathbf{H}_{\mathbb H}^n\) for \(n\ge 2\). New geometric invariants and several distance formulas in quaternionic hyperbolic geometry are introduced and studied for this problem. The congruence classes are completely described by quaternionic Cartan’s angular invariants and the distances between some geometric objects for the first case. The moduli space is constructed for the second case.

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References

  1. Apanasov, B.N., Kim, I.: Cartan angular invariant and deformations of rank 1 symmetric spaces. Sbornik Math. 198(2), 147–169 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beardon, A.F.: The Geometry of Discrete Groups. Spring, New York (1983)

    Book  MATH  Google Scholar 

  3. Bisi, C., Gentili, G.: Möbius transformations and the Poincare distance in the quaternionic setting. Indiana Univ. Math. J. 58, 2729–2764 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brehm, U.: The shape invariant of triangles and trigonometry in two-point homogeneous spaces. Geom. Dedicata. 33, 59–76 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cao, C., Waterman, P.L.: Conjugacy invariants of Möbius Groups. Quasiconformal Mappings and Analysis. Springer, New York (1998)

    Google Scholar 

  6. Cao, W.S., Parker, J.R.: Jørgensen’s inequalities and collars in n-dimensional quaternionic hyperbolic space. Q. J. Math. 62, 523–543 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cartan, E.: Sur le groupe de la g\(\acute{e}\)om\(\acute{e}\)trie hypersph\(\acute{e}\)rique. Comment. Math. Helv. 4, 158–171 (1932)

    Article  MathSciNet  Google Scholar 

  8. Chen, S.S., Greenberg, L.: Hyperbolic Spaces, Contributions to Analysis. Academic Press, New York (1974)

    Google Scholar 

  9. Cunha, H., Gusevskii, N.: On the moduli space of quadruples of points in the boundary of complex hyperbolic space. Transform. Groups 15(2), 261–283 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cunha, H., Gusevskii, N.: The moduli space of points in the boundary of complex hyperbolic space. J. Geom. Anal. 22, 1–11 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Falbel, E., Platis, I.D.: The \({\rm {PU}}(2,1)\) confguration space of four points in \(S^3\) and the cross-ratio variety. Math. Ann. 340(4), 935–962 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Falbel, E.: A spherical CR structure on the complement of the fgure eight knot with discrete holonomy. J. Difer. Geom. 79(1), 69–110 (2008)

    MathSciNet  MATH  Google Scholar 

  13. Goldman, W.M.: Complex hyperbolic geometry. Oxford University Press, New York (1999)

    MATH  Google Scholar 

  14. Grossi, C.: PhD Thesis, Universidade Estadual de Campinas, (2006)

  15. Kim, I., Parker, J.R.: Geometry of quaternionic hyperbolic manifolds. Math. Proc. Camb. Philos. Soc. 135, 291–320 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Korányi, A., Reimann, H.M.: The complex cross-ratio on the Heisenberg group. Enseign. Math. 33, 291–300 (1987)

    MathSciNet  MATH  Google Scholar 

  17. Parker, J.R.: Complex hyperbolic Kleinian groups. Cambridge University Press (to appear)

  18. Parker, J.R., Platis, I.D.: Complex hyperbolic Fenchel–Nielsen coordinates. Topology 47(2), 101–135 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Parker, J.R., Platis, I.D.: Global geometrical coordinates on Falbel’s cross-ratio variety. Can. Math. Bull. 52, 285–294 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Platis, I.D.: Cross-ratios and the Ptolemaean inequality in boundaries of symmetric spaces of rank 1. Geometr. Dedicata. 169, 187–208 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported by National Natural Science Foundation of China and Educational Commission of Guangdong Province. The authors would like to thank Prof. John R. Parker and the referee for their useful suggestions, which totally reshaped and enhanced this paper.

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  1. School of Mathematics and Computational Science, Wuyi University, Jiangmen, 529020, Guangdong, People’s Republic of China

    Wensheng Cao

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  1. Wensheng Cao
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Correspondence to Wensheng Cao.

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Cao, W. Congruence classes of points in quaternionic hyperbolic space. Geom Dedicata 180, 203–228 (2016). https://doi.org/10.1007/s10711-015-0099-z

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  • DOI: https://doi.org/10.1007/s10711-015-0099-z

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