403 Malott Hall
D.Phil. (2002) Oxford University
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.
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.
The absence of efficient dual pairs of spanning trees in planar graphs (with Bill Thurston), Electronic J. Comb. 13, N13 (2006).
Diameters of Cayley graphs of Chevalley groups (with Martin Kassabov), Eur. J. Comb 28 no. 3 (2007), 791–800.
Filling functions; part II of Geometry of the Word Problem for Finitely Generated Groups, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser-Verlag, 2007.
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.
The Dehn function of Stallings' group (with Will Dison, Murray Elder, and Robert Young), Geometric and Functional Analysis 19 no. 2 (2009), 406–422.
Hydra groups (with Will Dison), Commentarii Mathematici Helvetici, 88 (3), pages 507-540, 2013
Hyperbolic hydra (with Noel Brady and Will Dison), to appear in Groups, Geometry and Dynamics
Cannon-Thurston maps do not always exist (with Owen Baker), Forum of Mathematics, Sigma, 1, e3 (11 pages), 2013