Hierarchically hyperbolic groups

A rigorous mathematical way to describe and study the shape of any object is via its symmetries. Two of the fundamental properties of symmetries are that any symmetry can be reversed, and any two symmetries can be composed to obtain a new symmetry. Abstracting these two properties, we arrive at the mathematical notion of a group. Just as numbers are the mathematical objects that measure size, groups are the mathematical objects that measure symmetry. Moving this symmetry intuition into the background and abstracting the notion, one quickly realises that groups appear in many different ways and under different guises all over mathematics and other areas of sciences, and thus one has group theory: the study of abstract groups. This research project, entitled Hierarchically hyperbolic groups, deals with a newly identified family of groups, which are called hierarchically hyperbolic. Hierarchically hyperbolic groups, as the name suggests, are constructed in a hierarchical way starting from hyperbolic pieces. This is a typical strategy in mathematics: to approximate a difficult situation (read a group we do not quite understand) by easier cases (in this case, easier groups). Hierarchically hyperbolic groups are those groups that can be recovered in this fashion using hyperbolic groups. What is a hyperbolic group then? Hyperbolic groups were introduced in the 80s by Gromov in a very successful attempt to use geometric methods in group theory (more precisely, tools from hyperbolic geometry: here is where the name comes from). This geometric approach, as it turned out, imposes strong algebraic, asymptotic and growth properties that hyperbolic groups must satisfy, and it was the cornerstone over which geometric group theory developed. As the intuition deriving from the names suggests, hyperbolic groups are hierarchically hyperbolic, but the converse is not true: hyperbolic pieces can be assembled in a complicated way so that the resulting group is not itself hyperbolic. Even so, it turns out that hierarchically hyperbolic groups too must satisfy strong algebraic, asymptotic and growth properties. Hierarchically hyperbolic groups were introduced recently, at the end of 2014, and therefore with this research proposal I aim to better the understanding of this family of groups. More precisely, on one hand, I will be interested in the structural properties of the whole class - that is, in finding out what operations on groups preserve hierarchical hyperbolicity; and, on the other hand, in producing many concretes examples as well as establishing ways to discern groups that are hierarchically hyperbolic from those that are not.