Intermediate subgroups of braid groups are not bi-orderable

2025/10/28 by Cruz, R. M. de A.
#06F15 #20F36 #20F60 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2510.24947

Abstract

Let M be the disk or a compact, connected surface without boundary different from the sphere S² and the real projective plane ℝP², 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≥ 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 ≥ 3, any subgroup H of B_n(N) that satisfies P_n(N) \subsetneq H ⊂ B_n(N) is not bi-orderable.

Citations

• Ordered groups and topology
• A survey of surface braid groups and the lower algebraic K-theory of their group rings
• Orderable groups
• Basic results on braid groups
• Describing all bi-orderings on Thompson's group F
• On the dynamics of (left) orderable groups
• Bi-orderings on pure braided Thompson's groups
• Diagram groups are totally orderable
• Orderable 3-manifold groups
• Completing Artin's braid group on infinitely many strands
• Braid Groups are Linear

Related

• Ordering pure braid groups on closed surfaces
• On finite quotients of surface braid groups having order at most 127
• Remark on ordered braid groups
• The lower central and derived series of the braid groups of compact surfaces
• Braids, orderings and minimal volume cusped hyperbolic 3-manifolds