Regular left-orders on groups

joint with Yago Antolín and Cristóbal Rivas

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 whose left-orders are all 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.

Preprint, April 2021

Youtube video of talk - November 2020.

Formal language convexity in left-orderable groups

Abstract: We propose a criterion for the regularity of a formal language representation when passing 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’ question 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 $F_2 \times \mathbb{Z}$.

Preprint - accepted in IJAC, May 2020 (submitted May 2019).

Youtube video of talk - March 2020.

Slides - UPV, Bilbao, February 2020; Heriot-Watt, Edinburgh, March 2020.

Poster - YGGT, Bilbao, July 2019.

Lightning talk slides - Of Coarse!, Ventotene, September 2019.



Introduction to Growth in Groups - Winter 2017

Adapted into blog post

The growth function of a group is a large-scale geometric property that is directly connected to two out of three of Dehn’s decision problems: the word problem and the isomorphism problem.

Computation in the Completion of the Free Group Algebra - Winter 2015

Slides; Complete abstract

It is known (e.g., due to independent results of Malcev and B.H. Neumann) that $\mathbb{Q}[F_n]$, the (rational) group algebra of the free group of rank $n$, can be embedded in a division algebra $D$. We consider the problem of making this embedding algorithmic.


Heisenberg Group Sphere Count

Github repository

This algorithms generate the sphere counts for the three-dimensional Heisenberg group over the integers (nilpotent of step 2), and gives the sphere count in terms of its Malcev coordinates.

Research Logs

These are my research notes from old projects which did not turn into other material, in case it is of use to someone starting out a similar project. Warning: by the nature of these documents, some statements or ideas in there may be completely wrong.

Growth in the Heisenberg Group - Fall 2016

Keywords: growth in groups, nilpotent groups, Malcev normal form, Carnot groups, Gromov, growth in the polynomial range, rational growth, Duchin, Shapiro, geodesics, Cayley graph, CC metric, complexity, polynomial-time algorithm.

Expository Papers

Introduction to $\ell^2$-Betti numbers - Spring 2016

The goal of this presentation is to set up the framework for $\ell^2$ - Betti Numbers from the point of view of von Neumann Algebras.

Overview of the classification of tripartite entanglement under SLOCC - Summer 2014, warning: extreme juvenilia!

The goal of this document is to give the undergraduate reader an overview of tripartite quantum entanglement under SLOCC, with no background assumed.

Last updated: 2021-04-21

Work - Hang Lu Su