Free lattice ordered groups and the topology on the space of left orderings of a group

TL;DR

This paper explores the topology of the space of left orderings of groups, establishing new connections with free lattice ordered groups, and proving properties like the absence of isolated points and the uncountability of such spaces.

Contribution

It introduces a novel link between kernels in free lattice ordered groups and the topology of left orderings, providing new proofs of key properties for free groups.

Findings

No left orderable group has countably infinite left orderings.

The space of left orderings of a free group has no isolated points.

Existence of a dense orbit in the space of left orderings of a free group.

Abstract

For any left orderable group G, we recall from work of McCleary that isolated points in the space of left orderings correspond to basic elements in the free lattice ordered group over G. We then establish a new connection between the kernels of certain maps in the free lattice ordered group over G, and the topology on the space of left orderings of G. This connection yields a simple proof that no left orderable group has countably infinitely many left orderings. When we take G to be the free group of rank n, this connection sheds new light on the space of left orderings of the free group: by applying a result of Kopytov, we show that there exists a left ordering of the free group whose orbit is dense in the space of left orderings. From this, we obtain a new proof that the space of left orderings of a free group contains no isolated points.