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 ≥ −1. Finally, Hermiller and S̆unić 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 ∗B)× Z admits a regular left-order. MSC 2020 classification: 06F15, 20F60, 68Q45