### Research Focus

Geometric group theory

I work in geometric group theory. I study infinite discrete groups via associated metric spaces, Riemannian manifolds, graphs and cell complexes, and I focus on geometric features such as curvature, the shapes of balls, and characteristics of discs spanning loops (which can be called soap-film geometry). The setting for much of my research to date has been the geometry of the word problem for groups — a meeting-point of algebra, algorithmic complexity, geometry, topology and formal languages. My work has also touched on graph theory and cryptography.

### Publications

(Selected)

- Soficity and variations on Higman's group (with Martin Kassabov and Vivian Kuperberg)
- Computing area in presentations of the trivial group, Proc. Amer. Math. Soc. 145 (2017), 5059-5069
- Hyperbolic hydra (with Noel Brady and Will Dison), Groups, Geometry and Dynamics, pages 961–976, 7 (4), 2013
- Cannon-Thurston maps do not always exist (with Owen Baker), Forum of Mathematics, Sigma, 1, e3 (11 pages), 2013
- Hydra groups (with Will Dison), Commentarii Mathematici Helvetici, 88 (3), (2013), 507-540,
- The Dehn function of Stallings' group (with Will Dison, Murray Elder, and Robert Young), Geometric and Functional Analysis 19 no. 2 (2009), 406–422.
- Extrinsic versus intrinsic diameter for Riemannian filling-discs and van Kampen diagrams (with Martin Bridson), Journal of Differential Geometry 82 no. 1 (2009), 115–154.
- Filling functions; part II of Geometry of the Word Problem for Finitely Generated Groups, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser-Verlag, 2007.
- Diameters of Cayley graphs of Chevalley groups (with Martin Kassabov), Eur. J. Comb 28 (2007), no. 3, 791–800.
- The absence of efficient dual pairs of spanning trees in planar graphs (with Bill Thurston), Electronic J. Comb. 13 (2006), N13.
- Higher connectedness of asymptotic cones, Topology 42 (2003), 1289–1352.
- Isoperimetric inequalities for nilpotent groups (with Steve Gersten and Derek Holt), Geometric and Functional Analysis 13 (2003), 795–814.