Submission date: 26 September 2022

Abstract:

preliminary version
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 that satisfies some weak acylindricity conditions. To study these automorphisms we construct objects that can be viewed as a higher rank JSJ decomposition. In the first paper of the sequence we constructed a higher rank Makanin-Razborov diagram that encodes the automorphisms of an HHG. In the second paper we use this diagram, and techniques from the solution to Tarski's problem on the elementary theory of a free group, to associate two groupoids with this automorphism group. The groupoids generalize the structure of the automorphism group of a hyperbolic group that follows from the existence of its JSJ decomposition. In this paper we construct the groupoids for HHGs with single ended higher rank MR diagrams. In the next paper of the sequence we generalize the construction to HHGs with general higher rank diagrams.

Mathematics Subject Classification:

Keywords and phrases:

Full text arXiv 2209.12891: pdf, ps.