Linear progress in fibres

  • Vaibhav Gadre

    University of Glasgow, UK
  • Sebastian Hensel

    Mathematisches Institut der Universität München, Germany
Linear progress in fibres cover
Download PDF

This article is published open access under our Subscribe to Open model.

Abstract

A fibred 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 Teichmüller disc, and the extension induced by the Birman exact sequence.

Cite this article

Vaibhav Gadre, Sebastian Hensel, Linear progress in fibres. Groups Geom. Dyn. 18 (2024), no. 3, pp. 1099–1129

DOI 10.4171/GGD/780