Stacks of Hyperbolic Spaces and Ends of 3-manifolds

In this paper we introduce the notion of a “stack” of geodesic spaces. Loosely speaking, this consists of a geodesic space decomposed into a sequence of “sheets” indexed by a set of consecutive integers. A stack is said to be “hyperbolic” if it is Gromov hyperbolic and its sheets are uniformly Gromov hyperbolic. We define a Cannon-Thurston map for such a stack, and show that the boundary of a one-sided proper hyperbolic stack is a dendrite. If the stack arises from a sequence of closed hyperbolic surfaces with a lower bound on injectivity radius, then this allows us to define an “ending lamination” on the surface. We show that the ending lamination has a certain dynamical property that implies unique ergodicity. We also show that such a sequence is a bounded distance from a Teichmüller ray — a result obtained independently by Mosher. This can be reinterpreted in terms of the Bestvina-Feighn flaring condition, and gives a coarse geometrical characterisation of Teichmüller rays. Applying this to a simply degenerate end of a hyperbolic 3-manifold with bounded geometry, we recover Thurston’s ending lamination conjecture, proven by Minsky, in this case. Various related issues are discussed. 2010 Mathematics Subject Classification: 57M50, 20F65