On the fundamental groups of compact Sasakian manifolds

TL;DR

This paper investigates the fundamental groups of compact Sasakian manifolds, revealing that all known Kähler groups are Sasaki and establishing restrictions on possible Sasaki groups, with applications to the Abel-Jacobi map.

Contribution

It demonstrates that all known Kähler groups are Sasaki and introduces new restrictions on the fundamental groups of compact Sasakian manifolds.

Findings

All known Kähler groups are Sasaki.

Finite groups are Sasaki.

Restrictions on fundamental groups of Sasakian manifolds.

Abstract

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.