Linear progress in fibres

TL;DR

This paper investigates how typical rays in fibers of hyperbolic 3-manifolds progress linearly in the ambient space, using ergodic theory and geometric analysis, and extends the results to various group extension contexts.

Contribution

It establishes linear progress of typical rays in fibered hyperbolic 3-manifolds and extends the analysis to related group extension scenarios.

Findings

Typical rays make linear progress in hyperbolic 3-space.

The method applies to Gromov hyperbolic surface group extensions.

Results connect geometric distortion with ergodic theory.

Abstract

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.