Linear progress in fibres

Gadre, V. and Hensel, S. (2024) Linear progress in fibres. Groups, Geometry and Dynamics, (doi: 10.4171/GGD/780) (Early Online Publication)

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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.

Item Type:Articles
Additional Information:The second author is supported in part by the DFG grant HE 7523/1-1 within the SPP 2026 “Geometry at Infinity”.
Status:Early Online Publication
Refereed:Yes
Glasgow Author(s) Enlighten ID:Gadre, Dr Vaibhav
Authors: Gadre, V., and Hensel, S.
Subjects:Q Science > QA Mathematics
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Groups, Geometry and Dynamics
Publisher:European Mathematical Society
ISSN:1661-7207
ISSN (Online):1661-7215
Published Online:29 February 2024
Copyright Holders:Copyright © 2024 European Mathematical Society
First Published:First published in Groups, Geometry and Dynamics 2024
Publisher Policy:Reproduced under a Creative Commons license
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