Circular orderability of 3-manifold groups

TL;DR

This paper explores the circular orderability of 3-manifold groups, establishing a link with finite cyclic covers and the L-space conjecture, and demonstrating that most graph manifold groups are circularly orderable.

Contribution

It introduces the concept of circular orderability for 3-manifold groups and relates it to existing properties like left-orderability and the L-space conjecture.

Findings

A 3-manifold has a circularly orderable fundamental group iff a finite cyclic cover has a left-orderable fundamental group.

Most graph manifold groups are circularly orderable.

Circular orderability behaves differently from left-orderability under Dehn surgery and cyclic branched covers.

Abstract

This paper initiates the study of circular orderability of $3$-manifold groups, motivated by the L-space conjecture. We show that a compact, connected, $\mathbb{P}^2$-irreducible $3$-manifold has a circularly orderable fundamental group if and only if there exists a finite cyclic cover with left-orderable fundamental group, which naturally leads to a "circular orderability version" of the L-space conjecture. We also show that the fundamental groups of almost all graph manifolds are circularly orderable, and contrast the behaviour of circularly orderability and left-orderability with respect to the operations of Dehn surgery and taking cyclic branched covers.