Landing rays and ray Cannon-Thurston maps
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915211122835456 |
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| author | Halder, Rakesh Mj, Mahan Sardar, Pranab |
| author_facet | Halder, Rakesh Mj, Mahan Sardar, Pranab |
| contents | For a hyperbolic subgroup H of a hyperbolic group G, we describe sufficient criteria to guarantee the following.
1) Geodesic rays in H starting at the identity land at a unique point of the boundary of G.
2)The inclusion of H into G does not extend continuously to the boundary.
As a consequence we obtain sufficient conditions that provide a mechanism to guarantee the non-existence of Cannon-Thurston maps. One such criterion we use extensively is an adaptation of a property proven by Jeon, Kapovich, Leininger and Ohshika. As a consequence we describe a number of classes of examples demonstrating the non-existence of Cannon-Thurston maps. We recover, in the process, a simple counter-example lying at the heart of Baker and Riley's examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_17815 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Landing rays and ray Cannon-Thurston maps Halder, Rakesh Mj, Mahan Sardar, Pranab Geometric Topology Group Theory 20F65, 20F67 (Primary) 30C10, 37F10 (Secondary) For a hyperbolic subgroup H of a hyperbolic group G, we describe sufficient criteria to guarantee the following. 1) Geodesic rays in H starting at the identity land at a unique point of the boundary of G. 2)The inclusion of H into G does not extend continuously to the boundary. As a consequence we obtain sufficient conditions that provide a mechanism to guarantee the non-existence of Cannon-Thurston maps. One such criterion we use extensively is an adaptation of a property proven by Jeon, Kapovich, Leininger and Ohshika. As a consequence we describe a number of classes of examples demonstrating the non-existence of Cannon-Thurston maps. We recover, in the process, a simple counter-example lying at the heart of Baker and Riley's examples. |
| title | Landing rays and ray Cannon-Thurston maps |
| topic | Geometric Topology Group Theory 20F65, 20F67 (Primary) 30C10, 37F10 (Secondary) |
| url | https://arxiv.org/abs/2503.17815 |