Pleated Surfaces and Eruptions Seminar

Geodesic Laminations on Surfaces (Bonahon)

Summary: "This article is similarly divided into three parts. The first part is devoted to generalities on geodesic laminations, gives some examples, and discusses dynamically interesting transverse structures for geodesic laminations. This includes the analytic notion of a transverse Holder distribution, and the more combinatorial notion of a transverse cocycle; these two transverse structures are later shown to be equivalent.The second part discusses topological applications of geodesic laminations. In particular, we consider the space of measured geodesic laminations, as a completion of the space of simple closed curves on the surface. We mention the piecewise linear structure MC ( S), and indicate how the combinatorial tangent vectors of this piecewise linear manifold have a geometric interpretation as geodesic laminations with transverse Holder distributions. Finally, the third part is devoted to some geometric applications of geodesic laminations. We 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, we show how the bending of a pleated surface along its pleating locus can be measured by a transverse cocycle. We also connect the bending of pleated surfaces to the rotation angle of closed geodesics in the hyperbolic 3-manifold."

DEFORMING CONVEX REAL PROJECTIVE STRUCTURES (Wienhard, Zhang)

Summary: "In this article we introduce two new flows on the space of convex real projective structures, which are described by explicit deformations of the internal parameters associated to each pair of pants in a pants decomposition. We call these flows the eruption flow and the internal bulging flow associated to the pants. The eruption flows associated to the 2g − 2 pairs of pants commute with the 6g − 6 generalized twist flows associated to the curves in the pants decomposition."

Shearing hyperbolic surfaces, bending pleated surfaces and Thurston’s symplectic form (Bonahon)

Summary: "The article develops a system of local holomorphic coordinates for the space of hyperbolic 3-manifolds with the fundamental group of a surface. These coordinates depend on the choice of a geodesic lamination on the surface, and are a complexified version of Thurston’s shear coordinates for Teichmuller space. The imaginary part of these coordinates measures the bending of a pleated surface realizing the geodesic lamination. We also show how these coordinates are related, via Thurston’s symplectic form on the space of measured geodesic laminations, to the complex length function and to its differential."