MATH 7390: Topics in Algebra: Non-positively Curved Groups (Fall 2009)

Instructor: Tim Riley

This course will be an introduction to classes of infinite discrete groups regarded as non-positively curved — in particular, Gromov-hyperbolic, CAT(–1), CAT(0), automatic, semi-hyperbolic, and systolic groups.

Arguably, the subject has three points of origin. The oldest lies with Max Dehn and concerns combinatorial techniques to solve problems in low-dimensional topology. More recently, Misha Gromov carried ideas from Riemannian geometry into the coarse word of discrete groups and Jim Cannon studied algorithmic and combinatorial aspects of Cayley graphs related to group actions on hyperbolic space.

I will explain some of the implications of non-positive curvature assumptions on groups in contexts such as subgroup structure, boundaries, grammatical complexity of normal forms, the Baum-Connes Conjecture, and the Novikov Conjecture. I will survey some of the open problems of the subject.