Coarse Helly Structures and Equivariant Factor Maps for Boundaries of Hierarchically Hyperbolic Groups
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| Langue: | anglais |
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2025
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| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | <p>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.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17387269 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Coarse Helly Structures and Equivariant Factor Maps for Boundaries of Hierarchically Hyperbolic Groups SÉRGIO DE ANDRADE, PAULO Hierarchically Hyperbolic Groups, Coarse Geometry, Geometric Group Theory, Cannon-Thurston Maps, Group Boundaries, Coarse Helly Property, Rigidity. <p>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.</p> |
| title | Coarse Helly Structures and Equivariant Factor Maps for Boundaries of Hierarchically Hyperbolic Groups |
| topic | Hierarchically Hyperbolic Groups, Coarse Geometry, Geometric Group Theory, Cannon-Thurston Maps, Group Boundaries, Coarse Helly Property, Rigidity. |
| url | https://doi.org/10.5281/zenodo.17387269 |