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.

Riemann surfaces, or aboutRiemannian manifolds? These are quite different things. $\endgroup$ – Peter Mueller Nov 20 '15 at 21:46