A note on the connectivity of components of Kleinian groups

Abstract

1. We wish to call attention to an error in a statement on Kleinian groups that says a component of a Kleinian group has connectivity 1, 2, or infinite. This statement appears seemingly for the first time in Appell and Goursat [2, pp. 168169] and subsequently in Lehner [4, Theorem 54, p. 109]. Using a simple construction we will exhibit Kleinian groups with components of arbitrary connectivity (in fact it is possible to construct Kleinian groups with a countable sequence of nonconjugate components, each with a prescribed finite or infinite connectivity), and note under what conditions the original statement does hold. We will observe that the exceptional cases occur only when the subgroup that leaves the component invariant is finite, and we will restrict ourselves to this case. For the sake of completeness we will classify all the possible relations between the connectivity of a component and the order of the subgroup that leaves the component invariant, and finally we will observe that for finitely generated groups the original statement is valid. The construction used here is standard although the contention that the statement is valid for finitely generated Kleinian groups rests on fairly recent results of Ahlfors [1] and Selberg [5].