I was a PhD student studying maths at the University of Oxford under the supervision of Professor Cornelia Druțu from 2020-2024. I was interested in
the quasiisometric rigidity of groups and metric spaces; in particular quasiisometric embeddings and other forms of coarse embeddings between groups and spaces.
algorithmic properties of groups; the relationship between the geometry of a group and its computational properties.
I successfully defended my thesis on 21st May 2024. It can be found here: https://ora.ox.ac.uk/objects/uuid:4dc87a3f-66a8-4072-a029-949924a2c476
myfirstnameandsurnamenexttoeachother (without the "S.") @ hotmail . co . uk
Embeddings of Trees, Cantor Sets and Solvable Baumslag-Solitar Groups [Geometriae Dedicata]
https://link.springer.com/article/10.1007/s10711-022-00745-z
This paper studies quasiisometric embeddings between the solvable Baumslag-Solitar groups, extending the work of Farb and Mosher on quasiisometries between them. It turns out that the existence of such an embedding is determined by the boundedness of a fascinating integer sequence.
Regularity of Quasigeodesics Characterises Hyperbolicity (with Sam Hughes and Davide Spriano) [to appear in Proceedings of the Royal Society of Edinburgh Section A: Mathematics]
https://arxiv.org/abs/2205.08573
We show that a group is hyperbolic if and only if its quasigeodesics form regular languages. One direction of the above equivalence was proved by Rees and Holt; we prove the other. We also prove that a geodesic metric space is non-hyperbolic if and only if for any L >0 you can find an L-locally (K,0)-quasigeodesic loop of length at most ML where K and M are uniformly bounded constants.
Two new constructions in the theory of projection complexes
https://arxiv.org/abs/2404.02730
In this paper we provide two new constructions that are useful for the theory of projection complexes developed by Bestvina, Bromberg, Fujiwara and Sisto. We prove that there exists a subtree of the projection complex which is quasiisometric to the projection complex. We use this subtree to form a tree of metric spaces, which is a subgraph of the quasi-tree of metric spaces and quasiisometric to it. These constructions simplify the metric structure (up to quasiisometry) of the projection complex and the quasi-tree of metric spaces. As an application, we use these constructions to provide a shorter proof of Hume's theorem that the mapping class group admits a quasiisometric embedding into a finite product of trees.
Embedding relatively hyperbolic groups into products of binary trees
https://arxiv.org/abs/2412.16029
We prove that if a group G is relatively hyperbolic with respect to virtually abelian peripheral subgroups then G quasiisometrically embeds into a product of binary trees. This extends the result of Buyalo, Dranishnikov and Schroeder in which they prove that a hyperbolic group quasiisometrically embeds into a product of binary trees. Inspired by Buyalo, Dranishnikov and Schroeder's Alice's Diary, we develop a general theory of diaries and linear statistics. The notions provide a framework by which one can take a quasiisometric embedding of a metric space into a product of infinite-valence trees and upgrade it to a quasiisometric embedding into a product of binary trees.
M1 Linear Algebra (Tutor) - 2022
M1 Groups and Group Actions (Tutor) - 2023
Algebraic Topology (TA) - 2021
Group Theory (Tutor) - 2021
Topology and Groups (TA) - 2020, 2023
Cone types of geodesics and quasigeodesics in groups - Cambridge Junior Geometry Seminar - 1st March 2024
Quasiisometric embeddings of groups into products of binary trees - GAGTA 2024 - Luminy - 5th February 2024
Regular languages of geodesics and quasigeodesics in groups - PGTC conference - Heriot-Watt University - 13th July 2023
Obstructions to embeddings between trees - University of Bristol, Geometry and Topology Seminar - 11th October 2022
Quasiisometric Embeddings of Solvable Baumslag-Solitar Groups - Graduate Student Concentration Week on Metric Geometry at Texas A&M University - July 28th 2022
QI embeddings of Baumslag-Solitar Groups - Institut de Mathématiques de Jussieu - Séminaire d'Algèbres d'Opérateurs - 2nd June 2022
Embeddings of Trees and Solvable Baumslag-Solitar Groups - Oxford Junior Topology and Group Theory Seminar - 27th April 2022
Quasiisometric Embeddings of Treebolic Spaces - TU Graz, Strukturtheorie Seminar - 4th November 2021
Asymptotic Cones and the Filling Order of a Metric Space - Oxford Junior Topology and Group Theory Seminar - 3rd February 2021
Open set