Hyperbolic 3-manifolds and geodesics in Teichmuller space

We are interested in studying the geometry of hyperbolic 3-manifolds homotopy equivalent to a given compact 3-manifold M. When M is a surface bundle over a circle, or when M is an interval bundle over a surface, one can associate to such a hyperbolic 3-manifold a geodesic in $T$(S), the Teichmüller space of the surface S. The interplay between the geometry of hyperbolic 3-manifolds and geodesics in the Teichmüller space is the basic subject of our study. More specifically, our goal is to predict the behavior of geodesics in $T$(S) based on their end points, then apply the results in studying the geometry of hyperbolic 3-manifolds. Our results are, in summary: (1) A hyperbolic 3-manifold has bounded geometry if and only if the corresponding Teichmüller geodesic stays in the thick part of $T$(S). (2) In general, every curve that is short in a hyperbolic 3-manifold is also short for some metric in the corresponding Teichmüller geodesic. (3) The converse of the second statement above is not true.