On the space of left-orderings of virtually solvable groups

TL;DR

This paper characterizes the topological structure of the space of left-orderings for countable virtually solvable groups, showing it is either finite or a Cantor set, and explicitly describes this space for the group SOL.

Contribution

It provides a complete topological classification of the space of left-orderings for virtually solvable groups and explicitly describes this space for the SOL group.

Findings

The space of left-orderings is either finite or a Cantor set.

Explicit description of the left-orderings space for the SOL group.

Topological classification applies to all countable virtually solvable groups.

Abstract

We show that the space of left-orderings of a countable virtually solvable group is either finite or homeomorphic to a Cantor set. We also provide an explicit description of the space of left-orderings of $SOL=\Z^2\rtimes_T\Z$.