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Non-maximal cyclic group actions on compact Riemann surfaces. (English) Zbl 0903.20027

A finite group \(G\) of automorphisms of a Riemann surface \(X\) is non-maximal in genus \(g\) if (i) \(G\) acts as a group of automorphisms of some compact Riemann surface \(X_g\) of genus \(g\) and (ii), for all such surfaces \(X_g\), \(|\operatorname{Aut} X_g|>| G|\). The authors investigate the case where \(G\) is a cyclic group \(C_n\) of order \(n\). Then they obtain that if \(C_n\) acts on only finitely many surfaces of genus \(g\), then they completely solve the problem of finding all such pairs \((n,g)\). Also they find a bound for \(n\) as function of genus \(g\) if \(C_n\) acts on infinitely many surfaces of genus \(g\).

MSC:

20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
30F10 Compact Riemann surfaces and uniformization
20F29 Representations of groups as automorphism groups of algebraic systems
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