Ideal right-angled pentagons in hyperbolic 4-space

J. Korean Math. Soc. Published online 2019 Apr 09

Youngju Kim, and Ser Peow Tan
Konkuk University, National University of Singapore

Abstract : An ideal right-angled pentagon in hyperbolic 4-space is a sequence of oriented geodesics (L_1, ..., L_5)
such that L_i intersects L_{i+1}, i=1, ... , 4, perpendicularly in hyperbolic 4-space and the initial point
of L_1 coincides with the endpoint of L_5 in the boundary at infinity.
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 < A, B > of isometries acting on hyperbolic 4-space such that A is parabolic, while B$ and AB are loxodromic.