We already know the bound of the order of the finite subgroups of the $Mod(S_g)$. If we take a further step, to find all the finite subgroups, then what is the result for low genus cases? For example, are there some list of all types of the finite subgroups of $Mod(S_g)$ for $g \leq 10$?
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$\begingroup$ possible duplicate of Automorphisms of Riemann Surfaces $\endgroup$– Ian AgolDec 31, 2013 at 19:33
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2$\begingroup$ By the Nielsen realization problem (solved by Kerckhoff), any finite subgroup of the mapping class group of a compact surface is realized by a group of automorphisms of some conformal/hyperbolic/Riemann structure on the surface. Thus, your question is (essentially) a duplicate of the other question pointed out in abx's answer. en.wikipedia.org/wiki/Nielsen_realization_problem $\endgroup$– Ian AgolDec 31, 2013 at 19:33
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As pointed out in the preceding answer, what you are looking for are all possible automorphisms of a compact Riemann surface of genus $g$. You'll find some answers at this question.
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1$\begingroup$ The book "Characters and Automorphism Groups of Compact Riemann Surfaces" promises GAP calculations up to $g\le 48$, but several people also say that the promised website does not exist so far. $\endgroup$ Jan 1, 2014 at 10:32
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Since the maximal order of a finite subgroup of $Mod(S_g)$ is $84(g-1)$ for $g\ge 2$, and this bound is realised for infinitely many $g$'s, listing all finite subgroups might be quite hard for bigger $g$. ( Larsen proved the remarkable result that the frequency of $g$ for which the bound $84(g − 1)$ is attained is the same as the frequency of the perfect cubes in the integers.) Also, all finite groups can be realised as a subgroup of $Mod(S_g)$, for $g\ge 2$.
So it seems to me that such a list is difficult to obtain (I have not seen it), but I am not an expert, and perhaps someone knowns more here. Of course, the bound for finite cyclic subgroups is only $4g+2$, so that a list of finite cyclic subgroups seems much easier to obtain.
answered Dec 31, 2013 at 14:19