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In 1982, Thurston gave a necessary and sufficient condition for a critically finite branched covering of the two-sphere to itself to be combinatorially equivalent to a rational map. We discuss extending this result to geometrically finite rational maps. We give an example to show that Thurston's original condition is not sufficient. This example is topologically pathological near accumulation points of the postcritical set. We give two conditions forbidding such pathology, show that they are equivalent, and (in a sequel to the present paper) will show that Thurston's condition together with this tameness is both necessary and sufficient to haracterize geometrically finite rational maps. © 2010 American Mathematical Society.

Geometrically finite and semi-rational branched coverings of the two-sphere