# Work

## Publications

### 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.

## Pre-PhD

### Presentations

*Introduction to Growth in Groups* - Winter 2017

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

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.

### Code

*Heisenberg Group Sphere Count*

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*