My research focuses on non-positive curvature in groups. I am especially interested in groups acting on cube complexes, 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. This tool applies to a wide range of groups, including:
- right-angled Artin groups, right-angled Coxeter groups, and fundamental groups of special cube complexes;
- mapping class groups and Teichmüller space;
- 3-manifold groups containing no Nil or Sol components;
- graph products of hyperbolic groups;
- braid groups.
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 am also working on writing code to implement algorithms detailed in my paper on graph braid groups. For example, I have written a programme that computes free splittings of graph braid groups and, joint with Tomasz Maciazek, a programme that computes presentations of graph braid groups.
A copy of my research statement is available here.
Graph of groups decompositions of graph braid groupsInternational Journal of Algebra and Computation, 2023(To appear)
Random quotients of hierarchically hyperbolic groupsIn preparation, 2023
Hierarchical hyperbolicity of graph productsGroups, Geometry, and Dynamics, 2022
Appendix to “Largest acylindrical actions and stability in hierarchically hyperbolic groups”Transactions of the American Mathematical Society, Series B, 2021Primary article by C. Abbott, J. Behrstock, and M. G. Durham
Hierarchical hyperbolicity of graph products and graph braid groupsCUNY Academic Works, 2021Ph.D. thesis, City University of New York
Teichmüller’s theorem and its applicationsUniversity of Warwick, 2015Master’s thesis
Ideal Theory in Rings (Translation of “Idealtheorie in Ringbereichen” by Emmy Noether)arXiv, 2014