Some generalized Coxeter groups and their orbifolds (Q1382019)

Classes of geometric 3-orbifolds are constructed whose (orbifold-) fundamental groups are generalized Coxeter groups (of the form \(\langle a_1, \dots, a_n\mid a_i^{u_i}= (a_{i+1} a_i^{-1})^{v_i} =1\rangle\); for \(u_i=v_i=2\) these are Coxeter groups), certain \(\mathbb{Z}\)-extensions of such generalized Coxeter groups and also some generalized triangle groups. The singular sets of these orbifolds are obtained by connecting two concentric circles by segments (rungs) resp. by cyclic quotients of such orbifolds. It is shown that these orbifolds belong to hyperbolic geometry, and also to some others of Thurston's eight 3-dimensional geometries (depending on the branching orders associated to the singular sets), by constructing explicitly fundamental polyhedra for the universal covering groups and applying Poincaré's theorem.