Kim, Youngju;Tan, Ser Peow

  • Received : 2018.02.09
  • Accepted : 2019.01.24
  • Published : 2019.07.01


An ideal right-angled pentagon in hyperbolic 4-space ${\mathbb{H}}^4$ is a sequence of oriented geodesics ($L_1,{\ldots},L_5$) such that $L_i$ intersects $L_{i+1},i=1,{\ldots},4$, perpendicularly in ${\mathbb{H}}^4$ and the initial point of $L_1$ coincides with the endpoint of $L_5$ in the boundary at infinity ${\partial}{\mathbb{H}}^4$. We study the geometry of such pentagons and the various possible augmentations and prove identities for the associated quaternion half side lengths as well as other geometrically defined invariants of the configurations. As applications we look at two-generator groups ${\langle}A,B{\rangle}$ of isometries acting on hyperbolic 4-space such that A is parabolic, while B and AB are loxodromic.


hyperbolic 4-space;right-angled pentagon;Vahlen matrix;Delambre-Gauss formula;two-generator groups;deformation


  1. L. V. Ahlfors, Old and new in Mobius groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 93-105.
  2. L. V. Ahlfors, Mobius transformations and Clifford numbers, in Differential geometry and complex analysis, 65-73, Springer, Berlin, 1985.
  3. L. V. Ahlfors, On the fixed points of Mobius transformations in $\mathbb{R}^n$, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 15-27.
  4. A. Basmajian, Constructing pairs of pants, Ann. Acad. Sci. Fenn. Ser. A I Math. 15 (1990), no. 1, 65-74.
  5. A. Basmajian, Generalizing the hyperbolic collar lemma, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 154-158.
  6. A. Basmajian and B. Maskit, Space form isometries as commutators and products of involutions, Trans. Amer. Math. Soc. 364 (2012), no. 9, 5015-5033.
  7. A. F. Beardon, The Geometry of Discrete Groups, Graduate Texts in Mathematics, 91, Springer-Verlag, New York, 1983.
  8. R. Benedetti and C. Petronio, Lectures on Hyperbolic Geometry, Universitext, Springer-Verlag, Berlin, 1992.
  9. C. Cao and P. L. Waterman, Conjugacy invariants of Mobius groups, in Quasiconformal mappings and analysis (Ann Arbor, MI, 1995), 109-139, Springer, New York, 1998.
  10. W. Fenchel, Elementary Geometry in Hyperbolic Space, De Gruyter Studies in Mathematics, 11, Walter de Gruyter & Co., Berlin, 1989.
  11. W. M. Goldman, The complex-symplectic geometry of SL(2, $\mathbb{C}$)-characters over surfaces, in Algebraic groups and arithmetic, 375-407, Tata Inst. Fund. Res., Mumbai, 2004.
  12. L. Keen, Collars on Riemann surfaces, in Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), 263-268. Ann. of Math. Studies, 79, Princeton Univ. Press, Princeton, NJ, 1974.
  13. Y. Kim, Quasiconformal stability for isometry groups in hyperbolic 4-space, Bull. Lond. Math. Soc. 43 (2011), no. 1, 175-187.
  14. Y. Kim, Geometric classification of isometries acting on hyperbolic 4-space, J. Korean Math. Soc. 54 (2017), no. 1, 303-317.
  15. C. Kourouniotis, Complex length coordinates for quasi-Fuchsian groups, Mathematika 41 (1994), no. 1, 173-188.
  16. B. Maskit, Kleinian Groups, Grundlehren der Mathematischen Wissenschaften, 287, Springer-Verlag, Berlin, 1988.
  17. J. R. Parker and I. D. Platis, Complex hyperbolic Fenchel-Nielsen coordinates, Topology 47 (2008), no. 2, 101-135.
  18. J. G. Ratcliffe, Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics, 149, Springer-Verlag, New York, 1994.
  19. C. Series, An extension of Wolpert's derivative formula, Pacific J. Math. 197 (2001), no. 1, 223-239.
  20. S. P. Tan, Complex Fenchel-Nielsen coordinates for quasi-Fuchsian structures, Internat. J. Math. 5 (1994), no. 2, 239-251.
  21. S. P. Tan, Y. L. Wong, and Y. Zhang, Delambre-Gauss formulas for augmented, rightangled hexagons in hyperbolic 4-space, Adv. Math. 230 (2012), no. 3, 927-956.
  22. M. Wada, Conjugacy invariants of Mobius transformations, Complex Variables Theory Appl. 15 (1990), no. 2, 125-133.
  23. P. L.Waterman, Mobius transformations in several dimensions, Adv. Math. 101 (1993), no. 1, 87-113.


Supported by : National Research Foundation of Korea(NRF)