"Hic etiam homines magna cornua habentes longitudine quatuor pedum, et sunt etiam serpentes tante magnitudinis, ut unum bovem comedant integrum." Borgia map (c. 1430 AD), Vatican Library.

We will survey the subgroup structure of groups that are non–positively curved in senses such as CAT(0), CAT(-1), and hyperbolic — apparently benign classes of groups that can harbour geometric and computational wildness in their subgroups.

Likely topics include Dehn functions, distortion, the Rips Construction, Mihailova's Construction, Bestvina–Brady groups, Cannon–Thurston maps, and the apparent ubiquity of surface subgroups.

We will emphasise grasping the definitions and basic examples, and setting the main theorems and outstanding open problems in context.

This is a seminar in which the participants will take turns to present. There are no prerequisites for attendance other than a willingness to participate. We will work on improving presentation skills and will disseminate our findings through a blog.