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Given a closed, oriented surface $\Sigma$ of genus greater than 1, let $Mod(\Sigma)$ denote the mapping class group of orientation preserving diffeomorphisms of $\Sigma$ up to isotopy. Given any finite subgroup $F\subset Mod(\Sigma),$ the affirmative solution to the Nielsen Realization problem states that there exists a hyperbolic metric $h$ on $\Sigma$ such that $F$ is isomorphic to the isometry group of $h.$ Now, by Hurwitz's Theorem, the order of $F$ is bounded above by $84(g-1).$ I've always wondered if there was a topological/algebraic method to deduce a uniform bound on the order of finite subgroups of the mapping class group without going through the above mentioned argument. If $Mod(\Sigma)$ were a linear group over a field of characteristic zero, I suppose this would follow immediately from Selberg's Lemma, but I vaguely recall hearing that $Mod(\Sigma)$ has no faithful, finite dimensional linear representations.

So, my question is: is there a way to see that finite subgroups of the mapping class group have a uniform bound on their order without using the above argument, and if so how sharp can one make the bounds in this manner.

I've long been curious about this question and finally remembered to post it on mathoverflow. As always, thanks for your input.

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    $\begingroup$ "I vaguely recall hearing that Mod(Σ) has no faithful, finite dimensional linear representations." --- Actually, it's a major open problem whether or not mapping class groups are linear. $\endgroup$
    – Dan Petersen
    Commented Aug 31, 2014 at 20:38
  • $\begingroup$ Thank you Dan, a google search drew up some notes of Benson Farb stating exactly this. $\endgroup$
    – Andy Sanders
    Commented Aug 31, 2014 at 21:22
  • $\begingroup$ @AndySanders: presumably the context there is that there are no known faithful representations, when the genus is 3 or larger. They may exist, but likely we have not seen such representations, yet. $\endgroup$
    – Ryan Budney
    Commented Feb 26, 2023 at 2:29

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You can also use Serre's theorem which says that kernel of the natural homomorphism from the mapping class group of $\Sigma$ to $\text{Sp}(2g;\mathbb{Z}/3\mathbb{Z})$ is torsion free, and therefore every finite subgroup injects to $\text{Sp}(2g;\mathbb{Z}/3\mathbb{Z})$. But that gives a polynomial bound of degree $g^2$ compared to $84(g-1)$.

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  • $\begingroup$ Thank you Lee. This is certainly the style of observation I had in mind. $\endgroup$
    – Andy Sanders
    Commented Aug 31, 2014 at 15:52
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    $\begingroup$ @AndySanders: My pleasure. And I even fixed the typos. $\endgroup$
    – Lee Mosher
    Commented Aug 31, 2014 at 15:54
  • $\begingroup$ I thought that Serre's result still uses Nielsen's realization for cyclic groups (due to Nielsen-Fenchel). Am I wrong ? $\endgroup$
    – BS.
    Commented Aug 31, 2014 at 16:39
  • $\begingroup$ Well, I'm willing to give Nielsen realization for cyclic groups, there's no hyperbolic geometry there. $\endgroup$
    – Andy Sanders
    Commented Aug 31, 2014 at 16:40

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