Minicourse in Geometric Group Theory:
"The hitchhiker's guide to hierarchical hyperbolicity"

Speaker: Giorgio Mangioni (Heriot-Watt University)

Schedule:

      Lecture 1: Monday, Apr. 27, 14:15 - 16:00, HCØ 4.4.20
      Lecture 2: Thursday, Apr. 30, 14:15 - 16:00, HCØ 4.4.1
      Lecture 3: Friday, May 1, 14:15 - 16:00, HCØ 4.4.1
      Lecture 4: Friday, May 8, 14:15 - 16:00, HCØ 4.4.1

Abstract:

Hierarchically hyperbolic groups (HHGs) are a new class of non-positively curved groups which provide a common framework for mapping class groups of surfaces, cubulated groups, and several more. This allows one to use mapping class group techniques to study cube complexes, and vice versa. This four-lecture minicourse will give an insight into what an HHG is, what properties it enjoys, and how one can use this machinery to prove cool facts about classical groups.

In the first lecture, we will go through the motivating example of mapping class groups, whose hierarchical structure, developed by Masur and Minsky, was a huge leap forward in our understanding of the geometry of these groups.

It is quite surprising that CAT(0) cube complexes have a similar structure, considering that mapping class groups cannot act "nicely" on them. In the second lecture, we will abstract the common features of both, arriving at the notion of a hierarchically hyperbolic group. By carefully explaining each axiom, we hope to make the definition less intimidating.

The third lecture will be a carousel of nice properties of HHGs. We shall mainly focus on their coarse median structure and its consequences; if time allows, we will run through some of the coolest recent developments of the theory.

In the last lecture, I will shamelessly advertise my work on quotients of HHGs by random walks, and how, in the case of mapping class groups, they inherit most "rigidity" properties of the original group.