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

TL;DR

This paper develops a higher rank JSJ decomposition for hierarchically hyperbolic groups, capturing automorphism dynamics and group structure, extending classical concepts from hyperbolic groups to a broader class.

Contribution

It introduces a novel higher rank JSJ decomposition for HHGs that encodes automorphism dynamics and group structure, advancing understanding of their automorphism groups.

Findings

Constructed a higher rank JSJ decomposition for HHGs.

The decomposition encodes automorphism dynamics.

Provides new insights into the outer automorphism 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 (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.