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

Abstract

Let $G$ be a group. We can topologize the spaces of left-orderings $\operatorname{LO}(G)$ and bi-orderings $\operatorname{O}(G)$ of $G$ with the product topology. These spaces may or may not have isolated points. It is known that $\operatorname{LO}(F\_n)$ has no isolated points, where $F\_n$ is a free group on $n\geq 2$ generators. In this paper, we show that $\operatorname{O}(F\_n)$ has no isolated points as well, thereby resolving the second part of Conjecture 2.2 by Sikora \[Bull. London Math. Soc. 36 (2004), 519—526].