Abstract
AbstractThecomplexityof a branched cover of a Riemann surfaceMto the Riemann sphereS2is defined as its degree times the hyperbolic area of the complement of its branching set inS2. ThecomplexityofMis defined as the infimum of the complexities of all branched covers ofMtoS2. We prove that ifMis a connected, closed, orientable Riemann surface of genusg≥1, then its complexity equals 2π(mmin+2g−2) , wheremminis the minimum total length of a branch datum realisable by a branched coverp:M→S2.
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