Geometric orbifolds with torsion free derived subgroup
The paper under review deals with orientable geometric three-orbifolds with underlying space the three-sphere. It gives a criterion to decide whether the maximal abelian covering is a manifold. Being a manifold is equivalent to saying that its branching locus is nonempty. In addition, when the geometry is not spherical, this is also equivalent to saying that the orbifold fundamental group of the covering has no torsion. By Selberg's lemma, every geometric 3-orbifold has always a finite sheeted covering which is a manifold, however the proof is not constructive and it can be quite difficult to determine the precise covering. This paper gives a precise statement that is straightforward to check. Notice that since we assume that the underlying space of the orbifold is the three-sphere, the orbifold fundamental group is generated by torsion elements, in particular the maximal abelian covering is finite. The criterion is stated in terms of the branching locus (which is a link or trivalent graph in the sphere, as we assume that the orbifold is orientable) and the isotropy groups. The criterion proved in this paper says the following: the maximal abelian covering is a manifold if, and only if, the following two conditions hold: (1) the isotropy group of each singular vertex is the abelian dihedral group of four elements, and (2) the branching locus has no separating edge. An edge of the branching locus is called separating if the number of components of the singular graph increases when we remove this edge. In the paper, to have a separating edge is called 1-connected, which should no be confused with simply connected. In particular, when the branching locus of the orbifold has no vertices, then the maximal abelian covering is always a manifold.
scientific article; zbMATH DE number 5816218
scientific article