A bound for the number of automorphisms of an arithmetic Riemann surface. (English) Zbl 1064.30041

A classical theorem of Hurwitz states that the automorphism group of a compact Riemann surface of genus \(g\) is at most \(84(g-1)\). In the opposite direction in 1968 Accola and Maclachlan showed that for a given genus \(g\) one could always find a compact Riemann surface with at least \(8(g+1)\) automorphisms. This, like Hurwitz’ theorem, is sharp; in contrast to Hurwitz’ theorem the extremal groups are all non-arithmetic. In this paper the authors show that for arithmetic surfaces the automorphism group is at least \(4(g-1)\) and that this bound is sharp for infinitely many \(g\).


30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
14H37 Automorphisms of curves
14H55 Riemann surfaces; Weierstrass points; gap sequences
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