Abstract

In this paper we show that two “interior points” of the limit set can be connected by a quasi-arc in the limit set.

Introduction

Let G be a singly degenerate group, that is, G is a finitely generated Kleinian group with a single simply connected invariant component. The limit set Λ(G) of such a Kleinian group G is a continuum with no interior point, but its topological and geometrical properties are not fully understood. According to beautiful recent work of Y. Minsky [Min94], if G is torsion free and the quotient manifold ℍ3/G has bounded geometry in the sense that its injectivity radius is bounded below by some positive constant within its convex core, Λ(G) has rich topological properties. More precisely, it is locally connected and its interior (i.e. the set of non end points) has the structure of an R-tree whose distance is equivariant with respect to the action of the Kleinian group (see also [Abi88]). Furthermore, Λ(G) is identified with the leaf space of the ending lamination, that is, topologically it is obtained by collapsing all leaves and closures of component of complements leaves of the ending lamination of the group via Moore's theorem (see [Min94] and [Thu82]).

In this paper, we study the limit set of such a Kleinian group G from the geometrical and analytical point of view. In this direction, B.Maskit showed in [Mak75] that deleting the limit set of any Schottky subgroup from Λ(G) does not destroy the connectivity (see also [Mat01]).