Diameters of spherical Alexandrov spaces and curvature one orbifolds (Q2733844)

A spherical orbifold is a quotient of the \(n\)-dimensional sphere by a finite group of isometries (i.e.\ a finite orthogonal group). In this paper, an explicit lower bound (depending only on the dimension) for the diameter of such spherical orbifolds is found. For the particular case of Coxeter groups, this lower bound is independent of the dimension. As the author points out, all these bounds can be used to study Riemannian geometry for orbifolds, because the space of directions of a Riemannian orbifold is an spherical orbifold. NEWLINENEWLINENEWLINEIn the second half of the paper, the author considers non-finite subgroups of the orthogonal group. In this case it is assumed that the subgroup is closed and acts non transitively on the sphere. The quotient has a natural structure of Alexandrov space (i.e. to have a lower curvature bound) and it is called ``spherical Alexandrov space''. In this situation there is also a lower bound on the diameter that depends on the dimension, which in this case is not explicit. NEWLINENEWLINENEWLINEThe paper is self-contained, including basic material and interesting examples. The techniques are based mainly on the study of subgroups of the orthogonal group.