• 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.