Coarse Helly Structures and Equivariant Factor Maps for Boundaries of Hierarchically Hyperbolic Groups
Year: 2025
Type: article
Abstract: 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.
Author: SÉRGIO DE ANDRADE, PAULO
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7092360038
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