Teichmuller sequences on trajectories invariant under a Kleinian group

We extend the results of [T2] to the situation where there is a compatibility with the action of a Kleinian group. A classical Teichmuller sequence is a sequence of quasiconformal maps f(i) with complex dilatations of the form k(i) (phi) over bar/vertical bar phi vertical bar, where phi is a quadratic differential and 0 < k(i) < 1 are numbers such that k(i) -> 1 as i -> infinity. We proved in [T2] that if tau is a vertical trajectory associated to W, then there is often, for instance if the sequence is normalized so that f(i) fix 3 points, a subsequence such that f(i) tend either toward a constant or an injective map of tau. If there is compatibility with the action of a non-elementary finitely generated Kleinian group G, we can give a precise characterization which of these cases occurs. Suppose that f(i) induce isomorphisms phi(i) of G onto another Kleinian group and that phi(i) have algebraic limit phi. If the quadratic differential is defined on a component of the ordinary set of G, if there are no parabolic elements, and if tau is extended maximally so that all branches coming together at a singular point are included, then we can state the main result as follows. The limit is a constant c if the stabilizer G, Of T is elementary; and, if it is non-elementary, then the limit is injective. In the first case, phi(g) is parabolic with fixpoint c whenever g is an element of G(tau) is of infinite order; and in the latter case, the limit f is an embedding of tau in a natural topology of tau, and f embeds tau into a component of the limit set of phi G whose stabilizer is phi G(tau). Various extensions and generalizations are presented.