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Uniform convergence in the mapping class group. (English) Zbl 1153.57013
In conversation at the 2005 Ahlfors-Bers colloquium, Ed Taylor asked the authors whether there is a formulation of convex cocompactness for the mapping class group, analogous to the following notion for Kleinian groups: a non-elementary Kleinian group \(\Gamma\) is convex cocompact if and only if the action of \(\Gamma\) on the limit set \(\Lambda_{\Gamma}\) is a uniform convergence action. Recall that an action of a group \(G\) on a perfect compact metrizable space \(X\) is a (discrete) convergence action if the diagonal action on the space of distinct triples in \(X\) is properly discontinuous, and that it is uniform if this associated action is cocompact.
Theorem 1.3. Let \(G\) be a non-elementary subgroup of \(Mod(S).\) Then \(G\) is convex cocompact if and only if \(G\) acts as a uniform convergence group on \(Z\Lambda_{\Gamma}.\) Theorem 1.4. Suppose that \(G < Mod(S)\) is a non-elementary group. Then \(G\) is a convex cocompact if and only if \(\Omega_{G}\neq\varnothing\) and \(G\) acts cocompactly on it.

MSC:
57M50 General geometric structures on low-dimensional manifolds
57M60 Group actions on manifolds and cell complexes in low dimensions
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