Sasaki versus Kähler groups

TL;DR

The paper analyzes fundamental groups of compact Sasaki manifolds and contrasts them with Kähler and projective groups. It develops a framework where $\pi_1(M)$ sits in a central extension $0\to C\to\pi_1(M)\to\pi_1^{\text{orb}}(X)\to0$ arising from a quasi-regular Boothby–Wang fibration over a projective orbifold base $X$, enabling a comparison with projective groups and application of Lefschetz-type results. Key contributions include: (i) every projective group embeds as a Sasaki group in dimension $\ge5$ (and orbifold versions), (ii) there exist Sasaki groups that are not Kähler, (iii) Sasaki groups in dimension $5$ form the richest class with distinct dimension-down/downward inclusions, and (iv) a set of restrictions (e.g., no direct products in general, strong constraints for 3-manifold and rank-one nonpositively curved cases) that separate Sasaki groups from the Kähler projective picture. Collectively, the results clarify the limits of the Kähler–Sasaki analogy, reveal dimension-specific phenomena, and provide constructive methods to realize or obstruct Sasaki group realizations.

Abstract

We study fundamental groups of compact Sasaki manifolds and show that compared to Kähler groups, they exhibit rather different behaviour. This class of groups is not closed under taking direct products, and there is often an upper bound on the dimension of a Sasaki manifold realising a given group. The richest class of Sasaki groups arises in dimension 5.

Figures (2)

• Figure 1: Kähler and projective groups
• Figure 2: Sasaki and projective groups

Theorems & Definitions (31)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6: Structure Theorem
• Theorem 7
• Remark 8
• Lemma 9
• Lemma 10