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Connectivity of the space of ending laminations. (English) Zbl 1190.57013
For a surface of genus at least four or a punctured surface of genus at least two, the main result of the present paper states that the space of ending laminations (or of arational measured foliations) on the surface is connected (this is also the space of ending laminations of geometrically infinite Kleinian groups, without accidental parabolics and isomorphic to the fundamental group of the surface). Since this space is known to be homeomophic to the Gromov boundary of the complex of curves of the surface, as a consequence also the Gromov boundary is connected. This in turn implies that quasi-isometry of the complexes of curves of two surfaces implies homeomorphism of the surfaces (rigidity of the complex of curves). As another consequence, also the space of doubly degenerate Kleinian surface groups is connected (by the ending lamination theorem, these Kleinian groups are determined by their two ending laminations).

MSC:
57M50 General geometric structures on low-dimensional manifolds
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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