# Automorphism groups of cocompact Fuchsian groups as mapping class groups

Let $$\Gamma$$ be a cocompact Fuchsian group. So it has presentation $$\langle x_1,y_1, \dots, x_g,y_g,z_1, \ldots, z_r \mid [x_1,y_1] \cdots [x_g,y_g]z_1 \cdots z_r=1, \ z_i^{m_i}=1 \rangle$$ for some $$g \geq 2$$, $$r \geq 0$$ and $$m_i \geq 2$$.

Question: Does there exist some $$n \geq 1$$ such that $$\mathrm{Aut}(\Gamma)$$ contains a finite-index subgroup isomorphic to $$\mathrm{Mod}(S_{g,n})$$?

The case $$r=0$$ can be found in Farb and Margalit's book A primer on mapping class groups (page 235). There, it is explained how to deduce from the Dehn-Nielsen-Baer theorem and the Birman exact sequence that $$\mathrm{Aut}(\pi_1(S_g))$$ is isomorphic to $$\mathrm{Mod}^{\pm}(S_{g,1})$$.

On the other hand, a Dehn-Nielsen-Baer theorem for Fuchsian groups can be found in MacLachlan and Harvey's article On mapping class groups and Teichmüller spaces, where it is proved that $$\mathrm{Out}(\Gamma)$$ is virtually isomorphic to some $$\mathrm{Mod}(S_{g,m})$$.

A natural guess is that $$\mathrm{Aut}(\Gamma)$$ should be virtually isomorphic to $$\mathrm{Mod}(S_{g,m+1})$$, with essentially the same arguments but replacing surfaces with 2-orbifolds.

It is probably pretty standard, but I would need a reference to cite in a paper I am writing.

• The Out of some cocompact Fuchsian groups such as triangle groups is finite, isn't it? – YCor Oct 18 '18 at 21:06
• Right. (One possibility to justify that the Out of a triangle group is finite is to notice that any triangle group satisfies Serre's property FA and next to apply Paulin's theorem.) In this case, the Out should be virtually isomorphic to $\mathrm{Mod}(S_{0,3})$, but $\mathrm{Mod}(S_{0,4})$ seems to be a virtually free group, so definitely not commensurable to a triangle group. I will try to understand what happens in this case. – AGenevois Oct 18 '18 at 22:27
• Triangle groups (and other groups such as polygon reflection groups) do not really fit into this question, for a few reasons. First, a reflection isometry of $\mathbb H^2$ reverses orientation, whereas Fuchsian groups are discrete subgroups of $\text{PSL}(2,\mathbb R)$ acting as the group of orientation preserving isometries of $\mathbb H^2$. Second, while one can formulate this question using the full group of isometries, the presentations have a different form from those in the question: they need relations associated to finite dihedral subgroups where two reflection lines cross at a point. – Lee Mosher Oct 19 '18 at 15:04
• Having said all of that, I don't know of a reference. One place I might look is in work of Zieschang, who cleaned up lots of little nooks in surface theory over the years. But I cannot say whether this search will be fruitful. – Lee Mosher Oct 19 '18 at 15:09
• In the book Surfaces and Planar Discontinuous Groups, written by Heiner Zieschang, Elmar Vogt and Hans-Dieter Coldewey, I found an entire chapter dedicated to automorphism groups of planar groups. The result I am looking for does not seem to be written there, but it seems to be good starting point. – AGenevois Oct 26 '18 at 14:44