Teichmüller sequences on trajectories invariant under a Kleinian group

Abstract

We extend the results of [T2] to the situation where there is a compatibility with the action of a Kleinian group. A classical Techmüller sequence is a sequence of quasiconformal mapsf _i with complex dilatations of the form$k_i \bar \varphi /\left| \varphi \right|$, where ϕ is a quadratic differential and 0<-k _i<1 are numbers such thatk _i→1 asi→∞. We proved in [T2] that if τ is a vertical trajectory associated to ϕ, then there is often, for instance if the sequence is normalized so thatf _i fix 3 points, a subsequence such thatf _i tend either toward a constant or an injective map of τ. If there is compatibility with the action of a non-elementary finitely generated Kleinian groupG, we can given a precise characterization which of these cases occurs. Suppose thatf _i induce isomorphisms ϕ_i ofG onto another Kleinian group and that ϕ_i have algebraic limit ϕ. If the quadratic differential is defined on a component of the ordinary set ofG, if there are no parabolic elements, and if τ 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 constantc if the stabilizerG _τ of τ is elementary; and, if it is non-elementary, then the limit is injective. In the first case, ϕ(g) is parabolic with fixpointc wheneverg∈G _τ is of infinite order; and in the latter case, the limitf is an embedding of τ in a natural topology of τ, andf embeds τ into a component of the limit set of ϕG whose stabilizer is ϕG _τ. Various extensions and generalizations are presented.