Regular polytopes from twisted Coxeter groups and unitary reflexion groups
In several papers the authors studied the following ``amalgamation problem''. Given two abstract regular d-polytopes \(P_ 1\) and \(P_ 2\) such that the vertex figures of \(P_ 1\) are isomorphic to the facets of \(P_ 2\), does there exist a regular \((d+1)\)-polytope P with facets isomorphic to \(P_ 1\) and vertex-figures isomorphic to \(P_ 2?\) The universal member among all solutions (if it exists) is denoted by \(\{P_ 1,P_ 2\}\). Several techniques have been developed to decide for some interesting cases the existence and finiteness or non-finiteness of \(\{P_ 1,P_ 2\}\). Twisting operations on Coxeter groups have been introduced by the authors in Math. Z. 201, 209-226 (1989; Zbl 0646.51023), as a method to solve the amalgamation problem. In the present paper the particular case is studied where the polytopes are obtained from unitary reflexion groups generated by reflexions of period 2.
scientific article; zbMATH DE number 4169552
scientific article