  A fibered hyperbolic 3-manifold induces a map from the hyperbolic plane to
hyperbolic 3-space, the respective universal covers of the fibre and the
manifold. The induced map is an embedding that is exponentially distorted in
terms of the individual metrics. In this article, we begin a study of the
distortion along typical rays in the fibre. We verify that a typical ray in the
hyperbolic plane makes linear progress in the ambient metric in hyperbolic
3-space. We formulate the proof in terms of some soft aspects of the geometry
and basic ergodic theory. This enables us to extend the result to analogous
contexts that correspond to certain extensions of closed surface groups. These
include surface group extensions that are Gromov hyperbolic, the universal
curve over a Teichm\"uller disc, and the extension induced by the Birman exact
sequence.
