# Patrick S. Nairne

# About me

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.

# @me

myfirstnameandsurnamenexttoeachother (without the "S.") @ hotmail . co . uk

# Research

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)

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.Embedding relatively hyperbolic groups into products of binary trees

https://arxiv.org/abs/2404.02730

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 Buyalo, Dranishnikov and Schroeder's analogous result for hyperbolic groups. To prove the main theorem, we build upon the projection complex theory of Bestvina, Bromberg and Fujiwara by proving that you can remove some edges from the "quasi-tree of metric spaces" and be left with a "tree of metric spaces" that is quasiisometric to it. By defining diaries and linear statistics we provide a powerful framework by which you 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.

# Teaching

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

# Talks

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