Intersections of analytically and geometrically finite subgroups of Kleinian groups

Abstract

We consider the intersection of pairs of subgroups of a Kleinian group of the second kind K whose limit sets intersect, where one subgroup G is analytically finite and the other J is geometrically finite, possibly infinite cyclic. In the case that J is infinite cyclic generated by M, we show that either some power of M lies in G or there is a doubly cusped parabolic element Q of G with the same fixed point as M. In the case that J is nonelementary, we show that the intersection of the limit sets of G and J is equal to the limit set of the intersection G ∩ J G \cap J union with a subset of the rank 2 parabolic fixed points of K. Hence, in both cases, the limit set of the intersection is essentially equal to the intersection of the limit sets. The main facts used in the proof are results of Beardon and Pommerenke [4] and Canary [6], which yield that the Poincaré metric on the ordinary set of an analytically finite Kleinian group G is comparable to the Euclidean distance to the limit set of G.