On the homeomorphisms of the space of geodesic laminations on a hyperbolic surface

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We prove that for any orientable connected surfaceSSof finite type which is not a sphere with at most four punctures or a torus with at most two punctures, the natural homomorphism from the extended mapping class group ofSSto the group of homeomorphisms of the space of geodesic laminations onSS, equipped with the Thurston topology, is an isomorphism.