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The key result in the present paper is a direct analogue of the celebratedThurston's Theorem for marked Thurston maps with parabolic orbifolds. Combiningthis result with previously developed techniques, we prove that every Thurstonmap can be constructively geometrized in a canonical fashion. As a consequence,we give a partial resolution of the general problem of decidability of Thurstonequivalence of two postcritically finite branched covers of the two-sphere.

Constructive geometrization of Thurston maps and decidability of Thurston equivalence