Exotic left orderings of the free groups from the Dehornoy ordering

TL;DR

This paper demonstrates that certain free subgroups of the three-strand braid group can be ordered in a way that has no convex subgroups, revealing new properties of left orderings in free groups.

Contribution

It introduces a novel class of left orderings of free groups derived from the Dehornoy ordering with no convex subgroups.

Findings

Constructed explicit left orderings of free groups from braid groups.

Proved these orderings have no convex subgroups.

Extended understanding of order structures in free groups.

Abstract

We show that the restriction of the Dehornoy ordering to an appropriate free subgroup of the three-strand braid group defines a left ordering of the free group on k generators, k>1, that has no convex subgroups.