Minimal index torsion-free subgroups of Kleinian groups

Abstract

A Kleinian group Γ is a discrete subgroup of PSL(2, C), the full group of orientation-preserving isometries of 3-dimensional hyperbolic space. In the language of [T1] Q = H3/Γ is a hyperbolic 3-orbifold; that is a metric 3-orbifold in which all sectional curvatures are -1, and for which Γ is the orbifold fundamental group (see [T1] for further details). A Fuchsian group is a discrete subgroup of PSL(2, R) and as such acts discontinuously on the hyperbolic plane. We define Γ to be of finite co-volume (resp. co-area) if the volume (resp. area) of the quotient orbifold Q is finite. By Selberg’s Lemma, if Γ is a Kleinian (resp. Fuchsian) group of finite covolume (resp. co-area) it contains a torsion-free subgroup of finite index. By definition any torsion-free subgroup cannot contain any finite subgroups of Γ , so the index must be a multiple of the lowest common multiple of all orders of finite subgroups of Γ (see [CFJR] for a proof). Γ being of finite co-volume implies there are only finitely many conjugacy classes of finite subgroups. Thus, we make the following