Surface group actions and length bounds

Brian H. Bowditch

We give a generalisation of Minsky's a-priori bounds theorem to Gromov hyperbolic spaces. Let Sigma be a compact orientable surface whose fundamental group acts on a Gromov hyperbolic space. To any curve one can associate its stable length in this action. Suppose we have sequence of simple closed curves that form a tight geodesic in the curve complex of Sigma. Under certain assumptions on the action, we show that one can bound the stable length of each curve in the sequence in terms of the lengths of terminal curves and the topological type of the Sigma. This can be applied to give a description of models of hyperbolic 3-manifolds in term of Gromov hyperbolicity.

September 2010.

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