Appears in collection : 2022 - T2 - WS3 - Hyperbolic groups and their generalisations

The JSJ (for groups) was originally constructed to study the automorphisms and the cyclic splittings of a (torsion-free) hyperbolic group. Later, the JSJ was generalized to all finitely presented groups. In this generality it encodes the splittings but not all the automorphisms. We further generalize the JSJ decomposition to study automorphisms of groups that act on products of hyperbolic spaces, and more generally to study automorphisms of (some) hierarchically hyperbolic groups. The object that we construct can be viewed as a higher rank JSJ decomposition.