Regular left-orders on groups

TL;DR

This paper investigates regular left-orders on finitely generated groups, exploring their stability under group operations, classifying groups with exclusively regular left-orders, and examining specific examples like Baumslag-Solitar groups.

Contribution

It introduces the concept of regular left-orders, analyzes their stability under extensions and wreath products, and classifies groups with all left-orders being regular, including specific cases like Baumslag-Solitar groups.

Findings

Regular left-orders are stable under extensions and wreath products.

All left-orders are regular on certain groups, including some Baumslag-Solitar groups.

The group $(A*B) imes extbf{Z}$ admits a regular left-order if $A$ and $B$ do.

Abstract

A regular left-order on 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 give a classification of the groups all whose left-orders are regular left-orders. In addition, we prove that solvable Baumslag-Solitar groups $B(1,n)$ admits a regular left-order if and only if $n\geq -1$. Finally, Hermiller and Sunic showed that no free product admits a regular left-order, however we show that if $A$ and $B$ are groups with regular left-orders, then $(A*B)\times \mathbb{Z}$ admits a regular left-order.