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THE COMPLEXITY OF RIEMANN SURFACES AND THE HURWITZ EXISTENCE PROBLEM

Part of: Riemann surfaces

Published online by Cambridge University Press:  02 August 2012

ALDO-HILARIO CRUZ-COTA  and
TERESITA RAMIREZ-ROSAS
ALDO-HILARIO CRUZ-COTA*
Affiliation:
Department of Mathematics, Grand Valley State University, Allendale, MI 49401-9401, USA (email: [email protected])
TERESITA RAMIREZ-ROSAS
Affiliation:
Department of Mathematics, Grand Valley State University, Allendale, MI 49401-9401, USA (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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The complexity of a branched cover of a Riemann surface M to the Riemann sphere S2 is defined as its degree times the hyperbolic area of the complement of its branching set in S2. The complexity of M is defined as the infimum of the complexities of all branched covers of M to S2. We prove that if M is a connected, closed, orientable Riemann surface of genus g≥1, then its complexity equals 2π(mmin+2g−2) , where mmin is the minimum total length of a branch datum realisable by a branched cover p:M→S2.

Keywords

complexity of Riemann surfacesbranched covers between Riemann surfacesHurwitz existence problem

MSC classification

Secondary: 30F99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 1 , February 2013 , pp. 131 - 138
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

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