# Formal language convexity in left-orderable groups

## TL;DR

This paper investigates the regularity of positive cone languages in left-orderable groups, establishing criteria for preservation under subgroups and providing examples and counterexamples related to regular language representations.

## Contribution

It introduces a criterion for regularity preservation in subgroup positive cones and constructs specific examples, including positive cones generated by a fixed number of elements.

## Key findings

- Regularity of positive cone languages passes to finite index subgroups.
- No quasi-geodesic regular language represents a positive cone in certain hyperbolic groups.
- Constructed groups with positive cones generated by exactly k elements for all k ≥ 3.

## 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 $k$ generators, for every $k \geq 3$. As a special case of our construction, we obtain a finitely generated positive cone for $F_2 \times \mathbb{Z}$.