Intersections of topologically tame subgroups of Kleinian groups

Abstract

In this paper, we continue the investigation of intersections of pairs of finitely generated subgroups Γ1 and Γ2 of a Kleinian group Γ, a question which has been examined by a number of authors. We give a brief survey of this work at the end of the introduction. We consider here intersections of topologically tame subgroups of Γ. We are interested primarily in determining the connection between the limit set of the intersection Γ1 ∩ Γ2 and the intersection of the limit sets of Γ1 and Γ2; as such, we will always assume that the intersection of the limit sets in nonempty. The major application of this sort of result is to detect a nontrivial element in the intersection of a pair of subgroups by looking at points in the intersection of their limit sets. What we show is that, modulo some well understood exceptional cases, the limit set of the intersection of two topologically tame subgroups of a Kleinian group is essentially the intersection of the limit sets. In order to state the results more precisely, we look at particular cases. As the intersection of two elementary subgroups is uninteresting, we will always assume that Γ2 is nonelementary. The case that one subgroup Γ1 = 〈γ〉 is cyclic splits, depending on whether γ is loxodromic or parabolic. Theorem A concerns the case that γ is loxodromic, and states that, modulo one exceptional case, some power of γ lies in Γ2. The exceptional case is, up to finite index subgroups, that of a finite volume hyperbolic 3-manifold which fibers over the circle and has fundamental group Γ, where the fiber has fundamental group Γ2 and Γ is generated by Γ2 and γ. Theorem A: Let Γ be a Kleinian group, let Γ2 be a nonelementary, topologically tame subgroup of Γ, and let γ be a loxodromic element of Γ fixing a point of Λ(Γ2). Then, either Γ is virtually fibered over Γ2 or γn ∈ Γ2 for some n > 0.