Automorphisms of Groups and a Higher Rank JSJ Decomposition I: RAAGs and a Higher Rank Makanin-Razborov Diagram

TL;DR

This paper introduces a higher rank JSJ decomposition and Makanin-Razborov diagram for hierarchically hyperbolic groups, focusing on right-angled Artin groups, to better understand their automorphisms and splittings.

Contribution

It constructs a higher rank JSJ decomposition and Makanin-Razborov diagram for hierarchically hyperbolic groups, extending the classical theory to more complex group structures.

Findings

Constructed a higher rank JSJ decomposition for RAAGs.

Developed a higher rank Makanin-Razborov diagram for HHGs.

Demonstrated the construction in the case of right-angled Artin groups.

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 that satisfies some weak acylindricity conditions. To study these automorphisms we construct an object that can be viewed as a higher rank JSJ decomposition. In the first paper we demonstrate our construction in the case of a right angled Artin group. For studying automorphisms of a general HHG we construct what we view as a higher rank Makanin-Razborov diagram, which is the first step in the construction of the higher rank JSJ.