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Let $S=S_g$ be the closed orientable surface of genus $g$ and let $\Gamma_3(S)$ be the subgroup of the mapping class group, $Mod(S)$, which acts trivially on $H_1(S;\mathbb{Z/3\mathbb{Z}})$. Define $\Theta(g):=[Mod(S):\Gamma_3(S)]$.

Question: Is it known if $\Theta(g)$ is exponentially large in $g$?

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Yes. $\Gamma_3$ is the kernel of the composition $Mod(S) \to Sp_{2g}(\mathbb Z) \to Sp_{2g}(\mathbb F_3)$. Both of these maps are known to be surjective. The first is a standard fact on the mapping class group, the second follows from strong approximation.

So its index is the cardinality of $Sp_{2g}(\mathbb F_3)$ which is superexponential in $g$ - in fact of size $3^{\Theta(g^2)}$. This follows from the order formula for the symplectic group, but a lower bound can be proved using just a maximal unipotent subgroup.

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  • $\begingroup$ Thanks, @Will . Could you give me a reference please so that I can refer to that in a paper? $\endgroup$
    – Mehdi Yazdi
    Oct 13, 2016 at 20:40
  • $\begingroup$ I looked up the order formula. I can also get the lower bound 3^{g^2} by the order formula, right? $\endgroup$
    – Mehdi Yazdi
    Oct 13, 2016 at 22:06
  • $\begingroup$ @Mehdi - I don't know. You may need to use multiple references for each of the facts. Wikipedia cites the first subjectivity as Theorem 6.4 of A primer on mapping class groups by Farm and Margalit. The reference for strong approximation appears to be ams.org/mathscinet-getitem?mr=0213361, page 188. For the order of the symplectic group, L. C. Grove's Classical Groups and Geometric Algebra, page 27. $\endgroup$
    – Will Sawin
    Oct 13, 2016 at 22:06
  • $\begingroup$ @Mehdi Yeah, that follows directly from the order formula. $\endgroup$
    – Will Sawin
    Oct 13, 2016 at 22:06

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