TITLE: Frucht's type theorem for Riemann surface
QUESTION [7 upvotes]: 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.
Thanks in adavanced.

REPLY [10 votes]: 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).