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](https://en.wikipedia.org/wiki/Frucht%27s_theorem).


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.


Thanks in adavanced.



Frucht's type theorem for Riemann surface
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.](http://arxiv.org/abs/math/0406607) They actually do $n\geq 4,$ but the paper has references to work in lower dimensions (by Greenberg and Kojima).


