# Timothy Riley

D.Phil. (2002) Oxford University

### Research Area

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.

### Selected Publications

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