On the fundamental groups of compact Sasakian manifolds

We study the fundamental groups of compact Sasakian manifolds, which we call Sasaki groups. It is shown that all known K$\ddot{a}$hler groups are Sasaki, in particular, all finite groups are Sasaki. On the other hand, we show there exists many restrictions on the fundamental groups of compact Sasakian manifolds. We also study the Abel-Jacobi map of a compact Sasakian manifold and its applications to Sasaki groups.