• 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.