Geodesic laminations on surfaces. (English) Zbl 0996.53029

Bonahon, Francis

Lyubich, M. (ed.) et al., Laminations and foliations in dynamics, geometry and topology. Proceedings of the conference held at SUNY at Stony Brook, USA, May 18-24, 1998. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 269, 1-37 (2001).

From the introduction: “Geodesic laminations on surfaces were introduced by W. P. Thurston a little over 20 years ago, and have since been a very powerful tool in hyperbolic geometry, low-dimensional topology and dynamical systems. In particular, there are several different contexts where geodesic laminations now routinely occur. Geodesic lamination can be considered as:
– topological objects, occuring as generalizations of simple closed curves on surfaces;
– geometric objects, such as bending laminations of hyperbolic convex cores, shearing loci of earthquakes on hyperbolic surfaces, stable laminations of pseudo-Anosov diffeomorphisms, maximal stretch laminations between hyperbolic surfaces;
– interesting dynamical objects, in particular because of their connection with interval exchange maps.
This versatility can be somewhat confusing for the non-expert. To the expert, it provides interesting challenges when these very different points of view need to interact with each other. The minicourse given at the Workshop was devoted to illustrations of the above three lectures focused on one of these viewpoints. This article is similarly divided into three parts…We made the deliberate effort of closely following the style and structure of the minicourse”.
Part I. “The dynamical viewpoint”, is devoted to generalities on geodesic laminations. Some examples are given, and discusses dynamically interesting transverse structures for geodesic laminations. This part includes the analytic notion of a transverse Hölder distribution, and the more combinatorial notion of a transverse cocycle (these two transverse structures are later shown to be equivalent).
Part II “The topological viewpoint” discusses topological applications of geodesic laminations. In particular, the author considers the space of measured geodesic laminations, as a completion of the space of simple closed curves on the surface. The piecewise linear structure \({\mathcal M}{\mathcal L} (S)\) is mentioned and it is indicate how the combinatorial tangent vectors of this piecewise linear manifold have a geometric interpretation as geodesic laminations with transverse Hölder distributions.
Part III “The geometric viewpoint” is devoted to some geometric applications of geodesic laminations. The author chose to focus on geodesic laminations as bending loci of boundaries of convex cores of hyperbolic 3-dimensional manifolds, and as pleating loci of pleated surfaces. In particular, it is shown how the bending of a pleated surface along its pleating locus can be measured by a transverse cocycle. Also, the bending of pleated surfaces is related to the rotation angle of closed geodesics in hyperbolic 3-manifolds.
For the entire collection see [Zbl 0959.00033].

Reviewer: Ioan Pop (Iaşi)

MSC:

53C22	Geodesics in global differential geometry
30F40	Kleinian groups (aspects of compact Riemann surfaces and uniformization)
57N05	Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)

Keywords:

geodesic laminations; surfaces; hyperbolic geometry; transverse cocycle