In this paper we introduce the notion of a ``stack'' of path-metric spaces. Loosely speaking, this consists of a path-metric space decomposed into a sequence of ``sheets'' indexed by a set of consecutive integers. A stack is a ``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 compact 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 dynamical property that implies unique ergodicity. We also show that such a sequence is a bounded distance from a Teichm\"uller 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 Teichm\"uller rays. Applying this to a simply degenerate end of a hyperbolic 3-manifold with bounded geometry, we recover this case of Thurston's ending lamination conjecture, proven by Minsky. Various related issues are discussed.
2000 Mathematics Subject Classification : 57M50, 30F60, 20F65.
: in ``Geometry and Topology Down Under'', Contemporary Mathematics Vol. 597, (eds. C.D.Hodgson, W.H.Jaco, M.G.Scharlemann, S.Tillman) American Mathematical Society (2013) 65--138.