Formal language convexity in left-orderable groups

Abstract

We propose a criterion for preserving the regularity of a formal language representation when passing from groups to subgroups. We use this criterion to show that the regularity of a positive cone language in a left-orderable group passes to its finite index subgroups, and to show that there exists no left order on a finitely generated acylindrically hyperbolic group such that the corresponding positive cone is represented by a quasi-geodesic regular language. We also answer one of Navas’ questions by giving an example of an infinite family of groups which admit a positive cone that is generated by exactly [Formula: see text] generators, for every [Formula: see text]. As a special case of our construction, we obtain a finitely generated positive cone for [Formula: see text].

Regular left-orders on groups

“…Then (A * B) × Z admits a regular left-order. This sort of phenomenon was already observed in [30], where H.L. Su found a finitely generated positive cone for F 2 × Z (and hence a regular positive cone).…”

“…This is a subtle question. For example F 2 × Z has a Reg-left-order (Theorem 1.5 or [30]), however F 2 does not have Reg-left-orders [16].…”

“…The inheritance under passing to finite index subgroups is due to Su [30]. The other two closure properties are proved in Section 3 where we study left-orders associated to group extensions.…”

“…In previous works, the authors have given examples of groups not admitting regular left-orders [1,30]. In this paper instead we focus on constructing regular left-orders.…”

“…Hermiller and Sunić [16] proved that no non-abelian free groups admit a regular positive cone. Su [30] proved that acylindrically hyperbolic groups do not have positive cones that are quasi-geodesic and regular. This formalized and strengthened an argument sketched by Calegari [8].…”