Let $\Sigma\_{g,n}$ denote an $n$-punctured surface of genus $g$, with $2g+n-2 > 0$. Let $\Pi\_{g,n}$ be its fundamental group (for some choice of base point), and let $\Gamma\_{g,n}$ denote its pure, orientation preserving mapping class group.


Then, by the Dehn-Nielsen-Baer theorem, the outer action of $\Gamma\_{g,n}$ on $\Pi\_{g,n}$ is faithful and induces an isomorphism between $\Gamma\_{g,n}$ and the subgroup $\text{Out}^\*(\Pi\_{g,n})$ of $\text{Out}(\Pi\_{g,n})$ which fixes the conjugacy class of each simple closed curve surrounding a puncture.



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> **Question.** When does $\text{Out}^\*(\Pi\_{g,n})$ have finite index inside $\text{Out}(\Pi\_{g,n})$?
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If $(g,\,n)=(1,\,1)$, then the index is $2$. Since $\Pi\_{1,1} = \Pi\_{0,3}$ and $\Gamma\_{0,3}$ is trivial, one gets infinite index for $(g, \, n)=(0,\,3)$. What happens in the other cases? I'm particularly interested in the case where $g\ge 2, \, n = 1$.



Index of the mapping class group $\Gamma_{g,n}$ inside $\text{Out}(\Pi_{g,n})$
This is surely not the most direct answer. But $\mathrm{Out}(\Pi\_{g,1}) \cong \mathrm{Out}(F\_{2g})$ surjects onto $\mathrm{GL}(2g,\mathbf Z)$, and the image of $\Gamma\_{g,1}$ lands in $\mathrm{Sp}(2g,\mathbf Z)$. So the index is infinite for $g \geq 2$.




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In fact a more careful version of this argument shows that $(g,n)=(1,1)$ is the unique case where you get a finite index subgroup. Let me flesh out the argument. For all $n>0$ we have similarly that $\mathrm{Out}(\Pi\_{g,n}) \cong \mathrm{Out}(F\_{2g+n-1})$ surjects to $\mathrm{GL}(2g+n-1,\mathbf Z)$. The corresponding representation $H$ of $\Gamma\_{g,n}$ is just the action of the mapping class group on the first homology of your favorite genus $g$ surface $\Sigma$ with $n$ punctures. But the action must be compatible with a lot of extra structure coming from geometry: there is the short exact sequence
$$ 0 \to \mathbf Z^{n-1} \to H\_1(\Sigma,\mathbf Z) \to H\_1(\overline \Sigma,\mathbf Z) \to 0 $$ and $\Gamma\_{g,n}$ preserves it, where $\overline \Sigma$ is the compact surface obtained by filling in the punctures. Moreover, $H\_1(\overline \Sigma,\mathbf Z)$ is of rank $2g$ and carries a symplectic form preserved by $\Gamma\_{g,n}$; the action of $\Gamma\_{g,n}$ on $\mathbf Z^{n-1}$ is trivial. These conditions define an infinite index subgroup of $\mathrm{GL}(2g+n-1,\mathbf Z)$ unless $(g,n)=1$.


