# Work

## Publications

### Formal language convexity in left-orderable groups

Preprint - submitted, May 2019.

Poster - YGGT, Bilbao, July 2019.

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

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 $F_2 \times \mathbb{Z}$.

## Pre-PhD Stuff

Warning: extreme juvenilia!

### 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

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.

### Expository Papers

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.

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

### 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.

Work - Hang Lu Su