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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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