Parabolic Elements in Kleinian Groups

Abstract

Let G be a Kleinian group, and T(G) its deformation space. We are concerned here with the question of which loxodromic (including hyperbolic) elements of G can be made parabolic on the boundary of T(G). For some geometrically finite boundary groups (see [8]), one obtains necessary conditions in terms of primitive elements of G represented by simple disjoint loops on U(G)/G. In this paper we restrict our attention to geometrically finite function groups (including groups with torsion), and show that these necessary conditions are sufficient. The restriction to function groups is to some extent a matter of convenience; our techniques can be applied in certain more general situations. The groups we obtain as boundary groups are all geometrically finite. Nothing is known about which elements can be made parabolic at boundary groups which are not geometrically finite. If G is Fuchsian acting on the upper half plane U, then every element of G representing a simple loop on U/G is primitive. Abikoff [1] and Marden, in the torsion-free case (unpublished), showed that a set of elements representing simple disjoint loops on U/G can be made parabolic on the boundary of the deformation space of G, where the deformations are all supported on U. Similar results along the lines of this paper have also been obtained by Thurston (unpublished). The author wishes to thank T. Jorgensen, L. Keen and I. Kra for informative conversations.