J. Korean Math. Soc. 2019; 56(4): 1131-1158
Online first article June 17, 2019 Printed July 1, 2019
https://doi.org/10.4134/JKMS.j190388
Copyright © The Korean Mathematical Society.
Youngju Kim, Ser Peow Tan
Konkuk University; National University of Singapore
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
MSC numbers: Primary 52C15; Secondary 30F99, 57M50
Supported by: This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. NRF-2017R1A2B1002908).
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