Abstract
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.
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The authors would like to express their gratitude to the referee for suggestions on how to improve the introduction and extend the results developed in Section 3.
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Bozlee, S., Wootton, A. Asymptotic equivalence of group actions on surfaces and Riemann–Hurwitz solutions. Arch. Math. 102, 565–573 (2014). https://doi.org/10.1007/s00013-014-0657-x
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DOI: https://doi.org/10.1007/s00013-014-0657-x