Regular polyhedra with translational symmetries (Q1375070)

The notion of a regular polyhedron in 3-space refers to the flag-transitive action of the symmetry group. These polyhedra have been classified in papers by \textit{B. Grünbaum} [Aequationes Math. 16, 1-20 (1977; Zbl 0381.51012)], and \textit{A. Dress} [Aequationes Math. 23, 252-265 (1981; Zbl 0506.51010), and ibid. 29, 222-243 (1985; Zbl 0588.51022)]. In the present paper the author looks at them from the purely algebraic point of view. In particular, the case of subgroups of translations in quotients of Coxeter groups is studied. The main example here is the Coxeter group. The groups of translations are found in purely algebraic terms, as so-called hyperbolic translations. The connection with the work of P. McMullen and E. Schulte on abstract regular polytopes is also discussed.