Non-fundamental trunc-simplex tilings and their optimal hyperball packings and coverings in hyperbolic space, I. For families F1-F4

Abstract

Supergroups of some hyperbolic space groups are classified as a continuation of our former works. Fundamental domains will be integer parts of
truncated tetrahedra belonging to families F1 - F4, for a while, by the
notation of E. Moln\'{a}r et al. in $2006$. Our paper relies basically on this work.
As an application, optimal congruent hyperball packings and coverings to the truncation base planes with their very good densities are computed.
This covering density is better than the conjecture of L.~Fejes~T\'oth for balls and horoballs in $1964$.