Isometry groups and mapping class groups of spherical 3-orbifolds

Abstract

We study the isometry group of compact spherical orientable 3-orbifolds S 3 /G, where G is a finite subgroup of SO(4), by determining its isomorphism type and, when S 3 /G is a Seifert fibrered orbifold, by describing the action on the Seifert fibrations induced by isometric copies of the Hopf fibration of S 3 . Moreover, we prove that the inclusion of Isom(S 3 /G) into Diff(S 3 /G) induces an isomorphism of the π 0 groups, thus proving the π 0 -part of the natural generalization of the Smale Conjecture to spherical 3-orbifolds.

On the diffeomorphism type of Seifert fibered spherical 3-orbifolds

“…For this reason, the classification of compact spherical orientable three-orbifolds S 3 /G up to orientation-preserving diffeomorphisms corresponds to the algebraic classification of finite subgroups of SO(4) up to conjugation in SO(4), originally due to Seifert and Threlfall ([TS31] and [TS33]), which we shall now briefly recall. For more detail, see [DV64], which we essentially follow although it must be mentioned that in Du Val's list of finite subgroups of SO(4) there are three missing cases, see also [CS03,MS15,MS19].…”

“…The methods we use in the paper are related to the fact that compact spherical 3-orbifolds are globally the quotient of the 3-sphere S 3 by the action of a finite group G of isometries. In [MS15,MS19] we have analyzed different aspects of the classification of finite subgroups of SO(4) up to conjugacy. Here we continue this kind of analysis but the additional difficulty consists in considering a classification up to "fibration-preserving conjugacy".…”

Classification of non-free Kleinian groups generated by two parabolic transformations

“…The isometry groups of the dihedral spherical orbifolds obtained as the π-orbifolds associated with 2-bridge links are calculated by [57,28]. Moreover, in the recent papers [40,41], Mecchia and Seppi classified the Seifert fibered spherical 3-orbifolds and calculated the isometry groups of such orbifolds. Since every spherical dihedral orbifold is Seifert fibered, the results in this section are implicitly contained in [40,41].…”

“…Moreover, in the recent papers [40,41], Mecchia and Seppi classified the Seifert fibered spherical 3-orbifolds and calculated the isometry groups of such orbifolds. Since every spherical dihedral orbifold is Seifert fibered, the results in this section are implicitly contained in [40,41]. However, we give a self-contained proof, because it is not a simple task to translate their results into the form we need.…”