Density of tube packings in hyperbolic space (Q706068)

The motivation of this topic has been the Euclidean result of \textit{A. Bezdek} and \textit{W. Kuperberg} [Mathematika 37, No. 1, 74--80 (1990; Zbl 0715.52006)] proving the same optimal density \(\pi/\sqrt {12}\) as that of Euclidean plane packing of equal circles. In hyperbolic space \(H^3\), there is a result of \textit{T. H. Marshall} and \textit{G. J. Martin} [Ann. Acad. Sci. Fenn., Math. 30, No. 1, 3--48 (2005; Zbl 1079.52011)], providing an upper bound for very large radius tubes which is asymptotically sharp. The author's results refer to moderate radius tubes where the packing has a symmetry group of \(H^3\). Thus the density can be considered in a hyperbolic manifold of finite volume. Seemingly, the author intensively works in this field which promises further results.