Arithmeticity and automorphisms of Riemann surfaces
by
Michael Belolipetsky
Sobolev Instritute of Mathematics
Coauthors: Gareth Jones (University of Southampton)

At the end of nineteenth century Schwarz proved that the automorphism group of a compact Riemann surface of genus g >= 2 is finite, and Hurwitz showed that its order is at most 84(g-1). This bound is sharp, by which we mean that it is attained for infinitely many g, and the least genus of such an extremal surface is 3. However, it is also well known that there are infinitely many genera for which the bound 84(g-1) is not attained. It therefore makes sense to consider the maximal order N(g) of the group of automorphisms of any Riemann surface of genus g. Accola and Maclachlan in 1968 independently proved that N(g) >= 8(g+1). This bound is also sharp, and the least genus attaining it is 23. Thus we have the following sharp bounds for N(g) with g >= 2:

8(g+1) <= N(g) <= 84(g-1).

We consider these bounds from an arithmetic point of view, defining arithmetic Riemann surfaces to be those which are uniformized by arithmetic Fuchsian groups. The motivation for this approach can be found in the works of Borel, Margulis and various others on arithmetic groups. Concerning Riemann surfaces with large groups of automorphisms, the surprising fact, which can easily be seen, is that all the extremal surfaces for Hurwitz's upper bound are arithmetic, whereas all the extremal surfaces for the Accola-Maclachlan lower bound are non-arithmetic. This raises the natural question: ``What can be said about the other two bounds?"

The non-arithmetic analog of Hurwitz's upper bound, obtained in [1], is 156(g-1)/7; this bound is sharp, and the least genus attaining it is 50. The arithmetic analog of the Accola-Maclachlan lower bound was obtained in my joint paper with Gareth Jones [2]. The result is that for each g >= 2 there is an arithmetic surface of genus g with 4(g-1) automorphisms, and that this bound is attained for infinitely many g, starting with 24.

We now collect these results together: defining N_ar(g) and N_na(g) to be the maximal orders of the automorphisms groups of the arithmetic and non-arithmetic surfaces of genus g respectively, we have sharp bounds

4(g-1) <= Nar(g) <= 84(g-1),

8(g+1) <= Nna(g) <=  156 7 (g-1).

References

[1]

M. Belolipetsky. On the number of automorphisms of a nonarithmetic Riemann surface. Siberian Math. J. 38 (1997), 860-867.

[2]

M. Belolipetsky, G. Jones, A bound for the number of automorphisms of an arithmetic Riemann surface, Math. Proc. Camb. Phil. Soc., to appear.

Date received: January 26, 2000