Quasi-isometric invariance of Hierarchically Hyperbolic Groups
Davide Spriano (Oxford)
Abstract: Hierarchically hyperbolic groups and spaces (HHG/S) are a large class of groups and spaces that include several examples of interests like mapping class groups, virtually compact special groups, many 3-manifold groups, etc. Started by Behrstock--Hagen--Sisto, the theory of HHS enjoyed spectacular development in recent years. It was used to solve open conjectures about mapping class groups and Teichmuller space as well as providing a unified framework to study the examples above.
It is a straightforward exercise to show that being a hierarchically hyperbolic space is a quasi-isometric invariant property. When considering hierarchically hyperbolic groups, however, it is significantly less clear whether the notion is preserved by quasi-isometry. In joint work with Harry Petyt we give a strong answer to this question, showing that the property of being a hierarchically hyperbolic groups is not even preserved by finite extensions.