In this paper, we introduce two new kinds of structures on a non-compact surface: broken hyperbolic structures and broken measured foliations. The space of broken hyperbolic structures contains the Teichmüller space of the surface as a subspace. The space of broken measured foliations is naturally identified with the space of affine foliations of the surface. We describe a topology on the union of the space of broken hyperbolic structures and of the space of broken measured foliations which generalizes Thurston's compactification of Teichmüller space.
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