MATH 7520 - Negative Curvature in Group Theory (Berstein Seminar)

Jason Manning, spring 2017.

Hyperbolic groups have been a central topic of geometric group theory since Gromov's 1987 essay. Gromov noticed that certain ideas from the differential geometry of negatively curved manifolds were vastly simplified and generalized by framing them in terms of "large-scale" metric properties, such as uniform thin-ness of triangles, or the linearity of the isoperimetric inequality. Small cancellation groups (for example) satisfy these properties, though they are not usually fundamental groups of negatively curved manifolds. The success of this project has encouraged people to look for negative curvature phenomena in ever broader classes of groups, including relatively hyperbolic and acylindrically hyperbolic groups.

We'll read and discuss various papers related to these topics, most likely starting with the various definitions of hyperbolicity for groups. As this is a Berstein seminar, most classes will consist of student lectures. I'll try to point out open questions and possible research directions as they arise.