Cusp cross-sections of hyperbolic orbifolds by Heisenberg nilmanifolds. I
Let \(\pi\) be a \textbf{K}-Heisenberg crystallographic group with group extension \(1\to \Delta\to \pi\to Q\) where \(\Delta\) is a maximal normal nilpotent subgroup and \(Q\) is a finite subgroup. By using the Seifert fibration, the author gives a discrete faithful representation of the \textbf{K}-Heisenberg crystallographic group, which is stated as follows: Theorem. There exists an injective homomorphism \(\mathcal{A}:\pi\rightarrow \Delta\rtimes \Aut\;\Delta\). Then the author uses this theorem to study the realization problem of Riemannian flat manifolds and Heisenberg infranilmanifolds. The author gives another proof of a result of \textit{D. D. Long} and \textit{A. W. Reid} on compact Riemannian flat orbifolds [Algebr. Geom. Topol. 2, 285--296 (2002; Zbl 0998.57038)]. In addition, the author also gives a generalization of \textit{D. B. McReynold}'s result on Heisenberg infranilmanifolds [Algebr. Geom. Topol. 4, 721--755 (2004; Zbl 1058.57012)].
scientific article; zbMATH DE number 5154818
scientific article