## 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 $$g$$.

### MSC:

 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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