# Patrick Nairne

# About me

I'm a third year PhD student studying maths at the University of Oxford under the supervision of Professor Cornelia Druțu. I am 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.

More recently, I have been studying the world of symmetric spaces and spherical/Euclidean buildings.

# @me

nairne (at) maths.ox.ac.uk

N3.13 - Andrew Wiles Building - Mathematical Institute - University of Oxford

# 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.

# Teaching

M1 Linear Algebra - 2022

Algebraic Topology (TA) - 2021Group Theory (Tutor) - 2021

Topology and Groups (TA) - 2020

# Talks

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