Regular polytopes from twisted Coxeter groups and unitary reflexion groups (Q923365)

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.