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Page 203 of Farb and Margalit's Primer on Mapping Class Groups contains the result:

Let $g ≥ 2$. The order of any finite subgroup of $MCG(S_g)$ is at most $84(g − 1)$.

I've been told by my supervisor that a version exists in terms of the Euler characteristic of a surface $S$; that is, the following result is true:

Let $\chi(S) < 0$. The order of any finite subgroup of $MCG(S)$ is at most $42\left|\chi(S)\right|$.

However, I can't find this anywhere in the literature! Does anyone know of a reference for this, or is anyone able to nudge me towards a proof (or counterexample)? Many thanks.

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    $\begingroup$ @PhilTosteson They mean the case when the surface isn't closed. $\endgroup$
    – Autumn Kent
    Commented Mar 3, 2020 at 17:49
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    $\begingroup$ The proof using the Gauss Bonnet theorem will give this to you in the finite type case. (en.wikipedia.org/wiki/Hurwitz%27s_automorphisms_theorem) $\endgroup$
    – Autumn Kent
    Commented Mar 3, 2020 at 17:50
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    $\begingroup$ @Autumn Kent: Ahhhhh I see, perfect - thanks! $\endgroup$
    – Alex Townsend-Teague
    Commented Mar 3, 2020 at 19:30

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