Locally unitary groups and regular polytopes (Q1865250)

The authors study the relation between abstract regular polytopes and complex groups generated by reflections. They attach a diagram, similar to a Coxeter diagram, to each group generated by affine reflections on a hermitian complex space \({\mathbb C}^n.\) For groups generated by three reflections in \({\mathbb C}^3\) the diagram is a triangle. These groups, tetrahedral diagrams and their geometric groups in \({\mathbb C}^4,\) and many related questions are being investigated. After introducing abstract polytopes, the authors determine regular polyhedra with triangular faces and their groups, as well as \(4\)-polytopes, and they give conditions under which these polytopes are finite.