Intermediate subgroups of braid groups are not bi-orderable

TL;DR

This paper proves that for certain surfaces, any subgroup of the braid group between the pure braid group and the full braid group is not bi-orderable, extending known results about the bi-orderability of these groups.

Contribution

It demonstrates that all intermediate subgroups of braid groups on surfaces (excluding some special cases) are not bi-orderable, generalizing previous results on full and pure braid groups.

Findings

Pure braid groups are bi-orderable.

Full braid groups are not bi-orderable for n ≥ 3.

Intermediate subgroups are not bi-orderable.

Abstract

Let $M$ be the disk or a compact, connected surface without boundary different from the sphere $S^2$ and the real projective plane $\mathbb{R}P^2$, and let $N$ be a compact, connected surface (possibly with boundary). It is known that the pure braid groups $P_n(M)$ of $M$ are bi-orderable, and, for $n\geq 3$, that the full braid groups $B_n(M)$ of $M$ are not bi-orderable. The main purpose of this article is to show that for all $n \geq 3$, any subgroup $H$ of $B_n(N)$ that satisfies $P_n(N) \subsetneq H \subset B_n(N)$ is not bi-orderable.