Maps between Boundaries of Relatively Hyperbolic Groups
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910030465335296 |
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| author | Pal, Abhijit Sardar, Rana |
| author_facet | Pal, Abhijit Sardar, Rana |
| contents | F. Paulin proved that if the Gromov boundaries of two hyperbolic groups are quasi-Mobius equivalent, then the groups themselves are quasi-isometric. The goal of this article is to extend Paulin's result to the setting of relatively hyperbolic groups by introducing the notion of relative quasi-Mobius maps between the Bowditch boundaries of relatively hyperbolic groups. We show that any coarsely cusp-preserving quasi-isometry between two relatively hyperbolic groups induces a homeomorphism between their Bowditch boundaries, and that this induced homeomorphism is relative quasi-Mobius and linearly distorts the exit points of bi-infinite geodesics into combinatorial horoballs. Conversely, we prove that if a homeomorphism between the Bowditch boundaries of two relatively hyperbolic groups preserves parabolic fixed points and is either relative quasi-Mobius or linearly distorts the exit points of bi-infinite geodesics into combinatorial horoballs, then it arises from a coarsely cusp-preserving quasi-isometry between the groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_14863 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maps between Boundaries of Relatively Hyperbolic Groups Pal, Abhijit Sardar, Rana Geometric Topology 20F65, 20F67, 20E08 F. Paulin proved that if the Gromov boundaries of two hyperbolic groups are quasi-Mobius equivalent, then the groups themselves are quasi-isometric. The goal of this article is to extend Paulin's result to the setting of relatively hyperbolic groups by introducing the notion of relative quasi-Mobius maps between the Bowditch boundaries of relatively hyperbolic groups. We show that any coarsely cusp-preserving quasi-isometry between two relatively hyperbolic groups induces a homeomorphism between their Bowditch boundaries, and that this induced homeomorphism is relative quasi-Mobius and linearly distorts the exit points of bi-infinite geodesics into combinatorial horoballs. Conversely, we prove that if a homeomorphism between the Bowditch boundaries of two relatively hyperbolic groups preserves parabolic fixed points and is either relative quasi-Mobius or linearly distorts the exit points of bi-infinite geodesics into combinatorial horoballs, then it arises from a coarsely cusp-preserving quasi-isometry between the groups. |
| title | Maps between Boundaries of Relatively Hyperbolic Groups |
| topic | Geometric Topology 20F65, 20F67, 20E08 |
| url | https://arxiv.org/abs/2401.14863 |