# Frucht's type theorem for Riemann surface

Frucht's theorem is a theorem in algebraic graph theory conjectured by Dénes Kőnig in 1936 and proved by Robert Frucht in 1939. It states that every finite group is the group of symmetries of a finite undirected graph. More strongly, for any finite group $G$ there exist infinitely many non-isomorphic simple connected graphs such that the automorphism group of each of them is isomorphic to $G$,You can see here.

I am not familiar with Riemann geometry, but I think my question has some meaning.

Do we have such Frucht's type theorem for Riemann surface? Precisely, for a finite group $G$, is there Riemann surface $M$, which its fundamental group is isomorphic to $G$?

I am interested in the special case, where $G=D_{2n}$ is the dihedral group. Also, when $G=D_{\infty}$ is interesting for me.

• Unlikely if you stick to surfaces. The only nontrivial finite fundamental group of a surface is $\mathbb Z/2=\pi_1(\mathbb R P^2)$, but $\mathbb R P^2$ is strictly speaking not a Riemann surface as it admits no complex structure. On the other hand, every finitely presented group can occur as a fundamental group of a four-manifold. – Sebastian Goette Nov 20 '15 at 20:54
You seem to be asking about the group of isometries, not the fundamental group. If so, for every $n$ and every finite group $G$ there is a compact hyperbolic manifold of dimension $n$ whose isometry group is $G.$ See Belolipetsky and Lubotzky. They actually do $n\geq 4,$ but the paper has references to work in lower dimensions (by Greenberg and Kojima).