Mathematics Colloquia and Seminars

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Given a topological branched covering f of the sphere S^2 over itself, with branch values contained in a finite subset P in S^2, can f be realized as a rational map on the Riemann sphere? William Thurston gave a criterion in 1982: If the orbifold type of f is hyperbolic, then it can be realized as a rational map if and only if there is no invariant multi-curve satisfying certain conditions. This condition is hard to apply in practice, since it involves checking infinitely many multi-curves.

We give a complete positive combinatorial condition, using the notion of domination of graphs, a stricter condition than being 1-Lipschitz. We also give a physical interpretation of this condition in terms of elasticity of rubber-band spines .