# index of the subgroup of the mapping class group acting trivially on Z/3Z homology

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 $g$?

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.