On Uniformly Perfect Morse Boundaries
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866915779096608768 |
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| author | Han, Suzhen Liu, Qing |
| author_facet | Han, Suzhen Liu, Qing |
| contents | We introduce and geometrically characterize the notion of uniformly perfect Morse boundary for proper geodesic metric spaces. As a unifying result, we prove that the Morse boundary of any finitely generated, non-elementary group is uniformly perfect whenever it is nonempty. This theorem applies to a broad class of groups, including all acylindrically hyperbolic groups, Artin groups, and hierarchically hyperbolic groups.
Furthermore, we establish a rigidity theorem for homeomorphisms between such boundaries: for any two spaces with uniformly perfect Morse boundaries, a homeomorphism is induced by a quasi-isometry if and only if it satisfies any one of several natural geometric conditions. These conditions include being bi-Hölder, quasi-conformal, quasi-symmetric, or $2$-stable and quasi-Möbius. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06160 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Uniformly Perfect Morse Boundaries Han, Suzhen Liu, Qing Group Theory Geometric Topology 20F65, 20F67, 57M07 We introduce and geometrically characterize the notion of uniformly perfect Morse boundary for proper geodesic metric spaces. As a unifying result, we prove that the Morse boundary of any finitely generated, non-elementary group is uniformly perfect whenever it is nonempty. This theorem applies to a broad class of groups, including all acylindrically hyperbolic groups, Artin groups, and hierarchically hyperbolic groups. Furthermore, we establish a rigidity theorem for homeomorphisms between such boundaries: for any two spaces with uniformly perfect Morse boundaries, a homeomorphism is induced by a quasi-isometry if and only if it satisfies any one of several natural geometric conditions. These conditions include being bi-Hölder, quasi-conformal, quasi-symmetric, or $2$-stable and quasi-Möbius. |
| title | On Uniformly Perfect Morse Boundaries |
| topic | Group Theory Geometric Topology 20F65, 20F67, 57M07 |
| url | https://arxiv.org/abs/2602.06160 |