On the topology of the space of bi-orderings of a free group on two generators

TL;DR

This paper proves that the space of bi-orderings of a free group on two generators has no isolated points, extending known results about left-orderings to bi-orderings.

Contribution

It establishes that the space of bi-orderings of a free group on two generators is topologically similar to the space of left-orderings, with no isolated points.

Findings

The space of bi-orderings of a free group on two generators has no isolated points.

The topology of bi-orderings mirrors that of left-orderings in free groups.

Bi-orderings form a dense, non-isolated subset in the space of all orderings.

Abstract

Let $G$ be a group. We can topologize the spaces of left-orderings $LO(G)$ and bi-orderings $O(G)$ of $G$ with the product topology. These spaces may or may not have isolated points. It is known that $LO(F_n)$ has no isolated points, where $F_n$ is a free group on $n\geq 2$ generators. In this paper, we show that $O(F_n)$ has no isolated points as well.