Sasaki versus K\"ahler groups

TL;DR

This paper investigates the fundamental groups of compact Sasaki manifolds, revealing their distinct properties from K"ahler groups, including limitations on closure under products and bounds on realizable dimensions, especially in dimension 5.

Contribution

It characterizes the differences between Sasaki and K"ahler groups, highlighting unique properties and constraints of Sasaki groups, particularly in low dimensions.

Findings

Sasaki groups are not closed under direct products.

There is an upper bound on the dimension of Sasaki manifolds for a given group.

The most diverse Sasaki groups occur in dimension 5.

Abstract

We study fundamental groups of compact Sasaki manifolds and show that compared to K\"ahler 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.