Abstract
In this paper we introduce the notion of a of geodesic spaces. Loosely speaking, this consists of a geodesic space decomposed into a sequence of indexed by a set of consecutive integers. A stack is said to be if it is Gromov hyperbolic and its sheets are uniformly Gromov hyper- bolic. 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 on the surface. We show that the ending lamination has a certain dynamical property that im- plies unique ergodicity. We also show that such a sequence is a bounded distance from a Teichmuller ray — a result obtained in- dependently by Mosher. This can be reinterpreted in terms of the Bestvina-Feighn flaring condition, and gives a coarse geometrical characterisation of Teichmuller rays. Applying this to a simply degenerate end of a hyperbolic 3-manifold with bounded geome- try, we recover Thurston's ending lamination conjecture, proven by Minsky, in this case. Various related issues are discussed.
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