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IDEAL RIGHT-ANGLED PENTAGONS IN HYPERBOLIC 4-SPACE

Kim, Youngju;Tan, Ser Peow

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

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

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

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Acknowledgement

Supported by : National Research Foundation of Korea(NRF)