Automorphisms of Groups and a Higher Rank JSJ Decomposition II: The single ended case

Abstract

The JSJ decomposition encodes the automorphisms and the virtually cyclic splittings of a hyperbolic group. For general finitely presented groups, the JSJ decomposition encodes only their splittings. In this sequence of papers we study the automorphisms of a hierarchically hyperbolic group (HHG) that satisfies some weak acylindricity conditions. To study these automorphisms we construct an object that can be viewed as a higher rank JSJ decomposition. This higher rank decomposition encodes the dynamics of individual automorphisms and the structure of the outer automorphism group of an HHG.