Densely ordered braid subgroups

TL;DR

This paper investigates the ordering properties of subgroups within braid groups, revealing that certain natural subgroups exhibit dense orderings under the Dehornoy ordering, contrasting with the group's overall discrete order.

Contribution

It demonstrates that specific subgroups of braid groups, including the commutator subgroup and Burau kernel, have dense orderings, extending understanding of braid group order structures.

Findings

Subgroups like the commutator subgroup have dense orderings.

The kernel of the Burau representation can have dense orderings.

Characterization of least positive elements in normal subgroups.

Abstract

Dehornoy showed that the Artin braid groups $B_n$ are left-orderable. This ordering is discrete, but we show that, for $n >2$ the Dehornoy ordering, when restricted to certain natural subgroups, becomes a dense ordering. Among subgroups which arise are the commutator subgroup and the kernel of the Burau representation (for those $n$ for which the kernel is nontrivial). These results follow from a characterization of least positive elements of any normal subgroup of $B_n$ which is discretely ordered by the Dehornoy ordering.