My research is in geometric group theory, an area of mathematics devoted to studying groups as geometric objects in order to solve algebraic and algorithmic problems, as well as problems in other fields. For example, geometric group theory was used by Agol and Wise to solve Thurston’s virtual Haken conjecture, a major open problem in low-dimensional topology, and by Sela to solve the famed Tarski conjecture in first-order logic.

The methods of geometric group theory have also had extraordinary success outside of mathematics, with Carlsson’s development of persistent homology in topological data analysis, Ghrist’s applications of braid groups in robotics, and recent exciting applications of graph braid groups in topological quantum computing by Maciazek et al.

My research focuses on non-positive curvature in groups. I am especially interested in graph braid groups, as well as hierarchical hyperbolicity, which is a tool to study groups from a geometric point of view by exploiting patterns of hyperbolic behaviour occurring within them. Recently, I have been studying random quotients of hierarchically hyperbolic groups and using cube complexes to study non-positive curvature in graph braid groups. I have also written code to implement algorithms detailed in my paper on graph braid groups. For example, I have written a program that computes free splittings of graph braid groups and, joint with Tomasz Maciazek, a program that computes presentations of graph braid groups.

A copy of my research statement is available here.

2023

Graph of groups decompositions of graph braid groups

Daniel Berlyne

International Journal of Algebra and Computation, 2023

We provide an explicit construction that allows one to easily decompose a graph braid group as a graph of groups. This allows us to compute the braid groups of a wide range of graphs, as well as providing two general criteria for a graph braid group to split as a non-trivial free product, answering two questions of Genevois. We also use this to distinguish certain right-angled Artin groups and graph braid groups. Additionally, we provide an explicit example of a graph braid group that is relatively hyperbolic, but is not hyperbolic relative to braid groups of proper subgraphs. This answers another question of Genevois in the negative.

Random quotients of hierarchically hyperbolic groups

We show that random quotients of acylindrical hierarchically hyperbolic groups (HHGs) are asymptotically almost surely HHGs, where our model of randomness is defined by taking random walks on the Cayley graph. In particular, this applies to right-angled Artin groups that do not split as a direct product, right-angled Coxeter groups that do not split as a direct product with two infinite factors, and mapping class groups of finite-type surfaces.

We show that any graph product of finitely generated groups is hierarchically hyperbolic relative to its vertex groups. We apply this result to answer two questions of Behrstock, Hagen, and Sisto: we show that the syllable metric on any graph product forms a hierarchically hyperbolic space, and that graph products of hierarchically hyperbolic groups are themselves hierarchically hyperbolic groups. This last result is a strengthening of a result of Berlai and Robbio by removing the need for extra hypotheses on the vertex groups. We also answer two questions of Genevois about the geometry of the electrification of a graph product of finite groups.

2021

Appendix to “Largest acylindrical actions and stability in hierarchically hyperbolic groups”

We consider two manifestations of non-positive curvature: acylindrical actions on hyperbolic spaces and quasigeodesic stability. We study these properties for the class of hierarchically hyperbolic groups, which is a general framework for studying many important families of groups, including mapping class groups, right-angled Coxeter and Artin groups, most 3-manifold groups, and many others. A group that admits an acylindrical action on a hyperbolic space may admit many such actions on different hyperbolic spaces, so it is natural to search for a "best" one. The set of all cobounded acylindrical actions on hyperbolic spaces admits a natural poset structure; in this paper we prove that all hierarchically hyperbolic groups admit a unique action which is the largest in this poset. The action we construct is also universal in the sense that every element which acts loxodromically in some acylindrical action on a hyperbolic space does so in this one. Special cases of this result are themselves new and interesting. For instance, this is the first proof that right-angled Coxeter groups admit universal acylindrical actions. The notion of quasigeodesic stability of subgroups provides a natural analogue of quasiconvexity outside the context of hyperbolic groups. We provide a complete classification of stable subgroups of hierarchically hyperbolic groups, generalizing and extending results that are known for mapping class groups and right-angled Artin groups. We also provide a characterization of contracting quasigeodesics; interestingly, in this generality the proof is much simpler than in the special cases where it was already known. In the appendix, it is verified that any space satisfying the a priori weaker property of being an "almost hierarchically hyperbolic space" is actually a hierarchically hyperbolic space. The results of the appendix are used to streamline the proofs in the main text.

Hierarchical hyperbolicity of graph products and graph braid groups

This paper is a translation of the paper "Idealtheorie in Ringbereichen", written by Emmy Noether in 1920, from the original German into English. It in particular brings the language used into the modern world so that it is easily understandable by the mathematicians of today. The paper itself deals with ideal theory, and was revolutionary in its field, that is modern algebra. Topics covered include: the representation of an ideal as the least common multiple of irreducible ideals; the representation of an ideal as the least common multiple of maximal primary ideals; the association of prime ideals with primary ideals; the representation of an ideal as the least common multiple of relatively prime irreducible ideals; isolated ideals; the representation of an ideal as the product of coprime irreducible ideals; equivalent concepts regarding modules.