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.
@article{10_4171_ggd_780,
author = {Vaibhav Gadre and Sebastian Hensel},
title = {Linear progress in fibres},
journal = {Groups, geometry, and dynamics},
pages = {1099--1129},
year = {2024},
volume = {18},
number = {3},
doi = {10.4171/ggd/780},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/780/}
}
TY - JOUR
AU - Vaibhav Gadre
AU - Sebastian Hensel
TI - Linear progress in fibres
JO - Groups, geometry, and dynamics
PY - 2024
SP - 1099
EP - 1129
VL - 18
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/780/
DO - 10.4171/ggd/780
ID - 10_4171_ggd_780
ER -
%0 Journal Article
%A Vaibhav Gadre
%A Sebastian Hensel
%T Linear progress in fibres
%J Groups, geometry, and dynamics
%D 2024
%P 1099-1129
%V 18
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/780/
%R 10.4171/ggd/780
%F 10_4171_ggd_780
