Coarse Helly Structures and Equivariant Factor Maps for Boundaries of Hierarchically Hyperbolic Groups

The rigidity program in geometric group theory seeks to understand the extent to which the algebraic structure of a group is determined by its large-scale geometry. A central theme is the relationship between quasi-isometries of a group and the dynamics on its boundary. This paper introduces the concept of a coarse Helly structure on a hierarchically hyperbolic space (HHS), a combinatorial framework designed to be robust under quasi-isometries. We establish that for a large class of HHSs admitting such a structure, any quasi-isometry between them induces a canonical homeomorphism on their boundaries. This result provides a broad generalization of the classical Cannon-Thurston map phenomenon. When the quasi-isometry arises from a group isomorphism, the induced boundary map is an equivariant homeomorphism, yielding a factor map between the boundary actions. This framework is applied to demonstrate that for certain classes of groups, quasi-isometric rigidity implies a conjugacy of their boundary dynamical systems. Our methods leverage the full power of the HHS axioms, particularly the partial realization axiom, to build a bridge between the coarse geometry of the space and the fine topology of its boundary.