Asymptotic equivalence of group actions on surfaces and Riemann–Hurwitz solutions
The topological data of a group action on a compact Riemann surface can be encoded using a tuple (h; m 1, ..., m s ) called its signature. There are two number theoretic conditions on a tuple necessary for it to be a signature: the Riemann–Hurwitz formula is satisfied and each m i equals the order of a non-trivial group element. We show on the genus spectrum of a group that asymptotically, satisfaction of these conditions is in fact sufficient. We also describe the order of growth for the number of tuples which satisfy these conditions but are not signatures in the case of cyclic groups.
Mathematics Subject Classification (2010)Primary 14H37 Secondary 14E22
KeywordsAutomorphism Compact Riemann surface Signature
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