Structure invariant properties of the hierarchically hyperbolic boundary

TL;DR

This paper demonstrates that key topological and dynamical properties of the boundary of a hierarchically hyperbolic group are invariant under different structures, using a maximization procedure.

Contribution

It establishes the invariance of boundary properties under a specific maximization process in hierarchically hyperbolic groups.

Findings

Boundary properties are independent of the chosen hierarchically hyperbolic structure.

Invariance under the maximization procedure is proven.

Topological and dynamical features of the boundary are preserved.

Abstract

We prove several topological and dynamical properties of the boundary of a hierarchically hyperbolic group are independent of the specific hierarchically hyperbolic structure. This is accomplished by proving that the boundary is invariant under a "maximization" procedure introduced by the first two authors and Durham.