Ideal right-angled pentagons in hyperbolic 4-space
J. Korean Math. Soc. 2019 Vol. 56, No. 4, 1131-1158
Published online July 1, 2019
Youngju Kim, Ser Peow Tan
Konkuk University; National University of Singapore
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
MSC numbers : Primary 52C15; Secondary 30F99, 57M50
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