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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.22994 |
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| _version_ | 1866912300410077184 |
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| author | Nguyen, Hoang Thanh Qing, Yulan |
| author_facet | Nguyen, Hoang Thanh Qing, Yulan |
| contents | The quasi-redirecting (QR) boundary is a close generalization of the Gromov boundary to all finitely generated groups. In this paper, we establish that the QR boundary exists as a topological space for several well-studied classes of groups. These include fundamental groups of irreducible non-geometric 3-manifolds, groups that are hyperbolic relative to subgroups with well-defined QR boundaries, right-angled Artin groups whose defining graphs are trees, and right-angled Coxeter groups whose defining flag complexes are planar. This result significantly broadens the known existence of QR boundaries. Additionally, we give a complete characterization of the QR boundaries of Croke-Kleiner admissible groups that act geometrically on CAT(0) spaces. We show that these boundaries are non-Hausdorff and can be understood as one-point compactifications of the Morse-like directions. Finally, we prove that if G is hyperbolic relative to subgroups with well-defined QR boundaries, then the QR boundary of G maps surjectively onto its Bowditch boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22994 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasi-redirecting boundaries of non-positively curved groups Nguyen, Hoang Thanh Qing, Yulan Group Theory 20f65, 20f67 The quasi-redirecting (QR) boundary is a close generalization of the Gromov boundary to all finitely generated groups. In this paper, we establish that the QR boundary exists as a topological space for several well-studied classes of groups. These include fundamental groups of irreducible non-geometric 3-manifolds, groups that are hyperbolic relative to subgroups with well-defined QR boundaries, right-angled Artin groups whose defining graphs are trees, and right-angled Coxeter groups whose defining flag complexes are planar. This result significantly broadens the known existence of QR boundaries. Additionally, we give a complete characterization of the QR boundaries of Croke-Kleiner admissible groups that act geometrically on CAT(0) spaces. We show that these boundaries are non-Hausdorff and can be understood as one-point compactifications of the Morse-like directions. Finally, we prove that if G is hyperbolic relative to subgroups with well-defined QR boundaries, then the QR boundary of G maps surjectively onto its Bowditch boundary. |
| title | Quasi-redirecting boundaries of non-positively curved groups |
| topic | Group Theory 20f65, 20f67 |
| url | https://arxiv.org/abs/2503.22994 |