FIGURE 1. An ideal right-angled pentagon
FIGURE 2. The horocycle/horosphere pair (C0, S0) assigned to the ideal vertex
FIGURE 3. A normalized augmented ideal right-angled penta-gon (L1, Π2, L3, Π4, L5)
FIGURE 4. An augmented ideal right-angled pentagon (L1, Π2, L3, Π4, L5) with (C0, S0)
FIGURE 5. PLP-configurations at p01 and p50
Figure 6. An ideal right-angled pentagon in hyperbolic 2-space
FIGURE 7. A normalized augmented pentagon (Π1, L2, Π3, L4, Π5)
FIGURE 8. An augmented ideal right-angled pentagon with three planes and a horocycle
FIGURE 9. An ideal right-angled pentagon associated to a linked two-generator group in ℍ2
FIGURE 10. D: the boundary of the fundamental domain of < I1, I3, I5> in ∂ℍ4
FIGURE 11. The fundamental domain D' in ℝ3
참고문헌
- L. V. Ahlfors, Old and new in Mobius groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 93-105. https://doi.org/10.5186/aasfm.1984.0901
- L. V. Ahlfor, Mobius transformations and Clifford numbers, in Differential geometry and complex analysis, 65-73, Springer, Berlin, 1985.
-
L. V. Ahlfor, On the fixed points of Mobius transformations in
${\mathbb{R}}^n$ , Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 15-27. https://doi.org/10.5186/aasfm.1985.1005 - A. Basmajian, Constructing pairs of pants, Ann. Acad. Sci. Fenn. Ser. A I Math. 15 (1990), no. 1, 65-74. https://doi.org/10.5186/aasfm.1990.1505
- A. Basmajian, Generalizing the hyperbolic collar lemma, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 154-158. https://doi.org/10.1090/S0273-0979-1992-00298-7
- A. Basmajian and B. Maskit, Space form isometries as commutators and products of involutions, Trans. Amer. Math. Soc. 364 (2012), no. 9, 5015-5033. https://doi.org/10.1090/S0002-9947-2012-05639-X
- A. F. Beardon, The Geometry of Discrete Groups, Graduate Texts in Mathematics, 91, Springer-Verlag, New York, 1983.
- R. Benedetti and C. Petronio, Lectures on Hyperbolic Geometry, Universitext, Springer-Verlag, Berlin, 1992.
- 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.
- W. Fenchel, Elementary Geometry in Hyperbolic Space, De Gruyter Studies in Mathematics, 11, Walter de Gruyter & Co., Berlin, 1989.
-
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. - 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.
- Y. Kim, Quasiconformal stability for isometry groups in hyperbolic 4-space, Bull. Lond. Math. Soc. 43 (2011), no. 1, 175-187. https://doi.org/10.1112/blms/bdq092
- Y. Kim, Geometric classification of isometries acting on hyperbolic 4-space, J. Korean Math. Soc. 54 (2017), no. 1, 303-317. https://doi.org/10.4134/JKMS.j150734
- C. Kourouniotis, Complex length coordinates for quasi-Fuchsian groups, Mathematika 41 (1994), no. 1, 173-188. https://doi.org/10.1112/S0025579300007270
- B. Maskit, Kleinian Groups, Grundlehren der Mathematischen Wissenschaften, 287, Springer-Verlag, Berlin, 1988.
- J. R. Parker and I. D. Platis, Complex hyperbolic Fenchel-Nielsen coordinates, Topology 47 (2008), no. 2, 101-135. https://doi.org/10.1016/j.top.2007.08.001
- J. G. Ratcliffe, Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics, 149, Springer-Verlag, New York, 1994.
- C. Series, An extension of Wolpert's derivative formula, Pacific J. Math. 197 (2001), no. 1, 223-239. https://doi.org/10.2140/pjm.2001.197.223
- S. P. Tan, Complex Fenchel-Nielsen coordinates for quasi-Fuchsian structures, Internat. J. Math. 5 (1994), no. 2, 239-251. https://doi.org/10.1142/S0129167X94000140
- 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. https://doi.org/10.1016/j.aim.2012.03.009
- M. Wada, Conjugacy invariants of Mobius transformations, Complex Variables Theory Appl. 15 (1990), no. 2, 125-133. https://doi.org/10.1080/17476939008814442
- P. L.Waterman, Mobius transformations in several dimensions, Adv. Math. 101 (1993), no. 1, 87-113. https://doi.org/10.1006/aima.1993.1043