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I don't know about the curve complex, but the Bestvina-Handel 1995 (Train tracks for surface homeos, 1995) give a completely combinatorial/algorithmic proof, which, at a quick glance, never mentions teichmuller space)
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I don't know about the curve complex, but the Bestvina-Handel 1995 (Train tracks for surface homeos) give a completely combinatorial/algorithmic proof, which, at a quick glance, never mentions teichmuller space)