Negative Curvature in Group Theory

This course will begin with a rapid introduction to hyperbolic groups. 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 (or ARE THEY??!). 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. As this is a Berstein seminar, most classes will consist of student lectures, and the content will be chosen collectively by the participants. I'll try to point out open questions and possible research directions as they arise.


(links should work on campus, or off-campus, using PassKey)


  Date   Speaker   Topic
1/25/17 Jason Intro to hyperbolicity
1/27/17 Jason Central problems in hyperbolic groups
1/30/17 - 2/1/17 Teddy Quasi-geodesic stability [BH, III.H.1]
2/3/17 Oliver Isometries of hyperbolic spaces [CDP, Ch. 9]
2/6/17 - 2/8/17 Tim Baumslag-Solitar subgroups (see here)
2/10/17 Florian Rips complex of a hyperbolic group [BH, III.Γ.3]
2/13/17 Carolyn Abbott (special guest!) Acylindrically hyperbolic groups
2/15/17 Yu-chan Linear isoperimetric function. See Notes On Hyperbolic and Automatic Groups, by Patty & Papasoglu, Section 3.4 (cf. [BH, III.H.2])
2/17/17 Jason CAT(0) geometry and hyperbolicity
2/22/17 - 2/24/17 Drew Random groups. See: Sharp phase transition theorems for hyperbolicity of random groups, by Yann Ollivier.
2/27/17 Pallavi Rips construction. See: Incoherent negatively curved groups, by Daniel Wise.
3/1/17 Teddy CAT(0) spaces with isolated flats. See Geometric invariants of spaces with isolated flats by Hruska, and Hadamard spaces with isolated flats by Hruska and Kleiner.
3/3/17 Jason Cone types and automata. [BH, III.Γ.2]
3/6/17 Teddy CAT(0) spaces with isolated flats, continued.
3/8/17 - 3/10/17 Oliver Boundaries of hyperbolic groups. Reference M. Bestvina and G. Mess, The boundary of negatively curved groups.
3/13/17 Drew Surface subgroups of random groups. Reference: D. Calegari and A. Walker Random groups contain surface subgroups.
3/20/17 - 3/22/17 Yu-chan Cannon's conjecture and the surface subgroup problem. Reference: V. Markovic, Criterion for Cannon's conjecture.
Jason Some aspects of Agol's theorem
3/29/17 Corey Bregman Gromov norm and bounded cohomology
4/10/17 Oliver Relatively hyperbolic groups
4/12/17 Pallavi Classifying one-dimensional boundaries of hyperbolic groups. Reference: M. Kapovich and B. Kleiner, Hyperbolic groups with low-dimensional boundary.
4/14/17-4/17/17 Teddy Bestvina-Feighn combination theorem. Reference: M. Bestvina and M. Feighn, A combination theorem for negatively curved groups.
4/21/17 Drew More on Markovic's paper Criterion for Cannon's conjecture.
4/24/17 James Farre Mineyev's theorem on boundedness of cohomology for hyperbolic groups. Reference: I. Mineyev, Straightening and bounded cohomology of hyperbolic groups.
Jason Manning's home page.

Last Updated 2017-04-21