Regular left-orders on groups

Abstract

A regular left-order on a finitely generated group $G$ is a total, left-multiplication invariant order on $G$ whose corresponding positive cone is the image of a regular language over the generating set of the group under the evaluation map. We show that admitting regular left-orders is stable under extensions and wreath products and we give a classification of the groups whose left-orders are all regular left-orders. In addition, we prove that a solvable Baumslag–Solitar group $B(1,n)$ admits a regular left-order if and only if $n\ge -1$. Finally, Hermiller and Šunić showed that no free product admits a regular left-order. We show that if $A$ and $B$ are groups with regular left-orders, then $(A\ast B)\times \mathbb{Z}$ admits a regular left-order.