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 Ideal right-angled pentagons in hyperbolic 4-space J. Korean Math. Soc. 2019 Vol. 56, No. 4, 1131-1158 https://doi.org/10.4134/JKMS.j190388Published 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 Full-Text :