Hierarchically Hyperbolic Groups (HHGs) were introduced by Behrstock—Hagen—Sisto to provide a common framework to study several groups of interest in geometric group theory, and have been an object of great interest in the area ever since. The goal of the talk is to provide an introduction to the theory of HHGs and discuss the advantages of the unified approach that they provide. If time permits, we will conclude with applications to growth and asymptotic cones of groups.

I will discuss joint work with S. Galatius and A. Kupers in which we investigate the homology of general linear groups over a ring $A$ by considering the collection of all their classifying spaces as a graded $E_\infty$-algebra. I will first explain diverse results that we obtained in this investigation, which can be understood without reference to $E_\infty$-algebras but which seem unrelated to each other: I will then explain how the point of view of cellular $E_\infty$-algebras unites them.

I will introduce the Friedl-Tillmann polytope for one-relator groups, and then discuss how it can be generalised to the Friedl-Lück polytope, how it connects to the Thurston polytope, and how we can view it as a convenient source of intuition and ideas.