One-relator Sasakian groups

TL;DR

This paper characterizes which one-relator groups can be fundamental groups of compact Sasakian manifolds, showing they are either finite cyclic or related to Riemann surface groups, and classifies high-deficiency groups with this property.

Contribution

It provides a complete classification of one-relator and high-deficiency groups that are fundamental groups of compact Sasakian manifolds.

Findings

One-relator groups are Sasakian iff they are finite cyclic or surface groups with orbifold points.

Classifies all high-deficiency groups that are Sasakian.

Establishes a clear criterion linking group properties to Sasakian geometry.

Abstract

We prove that any one-relator group $G$ is the fundamental group of a compact Sasakian manifold if and only if $G$ is either finite cyclic or isomorphic to the fundamental group of a compact Riemann surface of genus g > 0 with at most one orbifold point of order $n \geq 1$. We also classify all groups of deficiency at least two that are also the fundamental group of some compact Sasakian manifold.