Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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IDEAL RIGHT-ANGLED PENTAGONS IN HYPERBOLIC 4-SPACE

  • Kim, Youngju (Department of Mathematics Education Konkuk University) ;
  • Tan, Ser Peow (Department of Mathematics National University of Singapore)
  • Received : 2018.02.09
  • Accepted : 2019.01.24
  • Published : 2019.07.01
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Abstract

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.

Keywords

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FIGURE 1. An ideal right-angled pentagon

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FIGURE 2. The horocycle/horosphere pair (C0, S0) assigned to the ideal vertex

DBSHBB_2019_v56n4_1131_f0003.png 이미지

FIGURE 3. A normalized augmented ideal right-angled pentagon (L1, Π2, L3, Π4, L5)

DBSHBB_2019_v56n4_1131_f0004.png 이미지

FIGURE 4. An augmented ideal right-angled pentagon (L1, Π2, L3, Π4, L5) with (C0, S0)

DBSHBB_2019_v56n4_1131_f0005.png 이미지

FIGURE 5. PLP-configurations at p01 and p50)

DBSHBB_2019_v56n4_1131_f0006.png 이미지

FIGURE 6. An ideal right-angled pentagon in hyperbolic 2-space

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FIGURE 7. A normalized augmented pentagon (Π1, L2, Π3, L4, Π5)

DBSHBB_2019_v56n4_1131_f0008.png 이미지

FIGURE 8. An augmented ideal right-angled pentagon with three planes and a horocycle

DBSHBB_2019_v56n4_1131_f0009.png 이미지

FIGURE 9. An ideal right-angled pentagon associated to alinked two-generator group in ℍ2

DBSHBB_2019_v56n4_1131_f0010.png 이미지

FIGURE 10. D: the boundary of the fundamental domain of 1, I3, I5> in ∂ℍ4

DBSHBB_2019_v56n4_1131_f0011.png 이미지

FIGURE 11. The fundamental domain D' in ℝ3

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