Journal of the
Korean Mathematical Society JKMS

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)