Skinning bounds along thick rays

Abstract

We show that the diameter of the skinning map of an acylindrical hyperbolic 3manifold M is bounded on ε-thick Teichmüller geodesics by a constant depending only on ε and the topological type of ∂ M. Figure 1: At left is the manifold M X t , with surface E t and collar N t in the convex core about the convex core boundary. At right is the geodesic triangle X 0 Y t X t .Theorem 1 (Bound along thick rays). Let S be a closed orientable surface of genus greater than 1 and let ε > 0. Then there is a D such that if M is any compact hyperbolic 3-manifold with totally geodesic boundary X M ∼ = S and G : [0,Specifically, there are constants A and B depending only on S and ε such that Sketch of the proofThe idea of the proof is as follows, see Figure 1.Let X t be the surfaces along the geodesic ray, let Y t be the mirror image of the skinning surface at X t , and let M t = M X t be the interior of M equipped with the hyperbolic metric corresponding to X t . McMullen proved [9] that the skinning map of an acylindrical manifold is uniformly contracting, and this means that the distance between X t and Y t is growing at a definite linear rate. The geodesic [X t ,Y t ] from X t to Y t fellow travels our geodesic G along a thick segment [X t , Z t ] of linearly growing length, thanks to work of Rafi [14]. This implies, using work of Brock-Canary- Minsky [3], the existence of a linearly deep and uniformly thick collar about the convex hull boundary of M t . We establish this in Theorem 3 in Section 3.In Section 4, we use the Geometric Inflexibility Theorem of Brock-Bromberg [1]. This tells us that, in the complement of the thick collars of Theorem 3, the geometry of the manifold is changing, in a C 1 -sense, at a rate exponentially small in t. (Here the metric distortion is measured in terms of the strain field of the family of metrics.)We formulate two consequences of this. Theorem 11 gives the pointwise C 1 estimates in the form that we will use. Theorem 12 uses an additional estimate from [1]