A toolbox for left-orders of low complexity

TL;DR

This thesis applies formal language theory to study left-orderable groups, analyzing the complexity of positive cones and their interactions with group constructions, providing new classifications and constructions in the field.

Contribution

It introduces novel classifications and constructions of positive cones in groups using AFLs, extending previous results and offering new insights into group orderability and language complexity.

Findings

Classified the complexity of positive cones in ^2

Constructed regular positive cones for ,q groups extending prior work

Developed new methods for positive cone constructions in specific groups

Abstract

This thesis explores how concepts of formal language theory can be used to study left-orderable groups. It analyses the languages formed by their positive cones and demonstrates how the abstract families of languages (AFLs) in the Chomsky hierarchy (in particular regular and context-free languages) interact with core group-theoretic constructions under subgroups, extensions, finite generation and taking direct products with $\mathbb{Z}$. These investigations yield new insights into the interplay between decidability and geometry in group theory. Some results which may be improvements to the existing literature are included in the thesis. There is a classification of the complexity of positive cones of $\mathbb{Z}^2$, a more constructive proof on finding regular positive cone languages of language-convex subgroups compared to a result of Su (2020), a construction of countably infinite many regular positive cones of $\mathrm{BS}(1,q)$ for $q \geq -1$ which are all automorphic to each other extending a result of Antol\'in, Rivas, and Su (2022), and a construction of positive cones with finite generating set for groups of the form $F_{2n} \times \mathbb{Z}$ extending a result of Malicet, Mann, Rivas, and Triestino (2019).