Connections between the topology of the Morse boundary, the Morse local-to-global property and acylindrical hyperbolicity
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914555506982912 |
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| author | Abbott, Carolyn Zbinden, Stefanie |
| author_facet | Abbott, Carolyn Zbinden, Stefanie |
| contents | We relate the topology of the Morse boundary of a group to geometric and algorithmic properties of the group. In particular, we show that a group has $σ$-compact Morse boundary if and only if it is Morse local-to-global. We also provide tools such as the geodesic Morse local-to-global property to show that groups are (not) Morse local-to-global. Our strategy generalizes tools from small cancellation theory, such as the intersection of relators, to arbitrary finitely generated groups. Further, we introduce a class of groups akin to graded small-cancellation groups and show that, for groups in this class, a geodesic is Morse if and only if its intersection with relators grows sublinearly in the length of the relators.
We use this to construct the first example of a non-virtually cyclic Morse local-to-global group with an infinite-order Morse element that is not acylindrically hyperbolic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_11126 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Connections between the topology of the Morse boundary, the Morse local-to-global property and acylindrical hyperbolicity Abbott, Carolyn Zbinden, Stefanie Group Theory We relate the topology of the Morse boundary of a group to geometric and algorithmic properties of the group. In particular, we show that a group has $σ$-compact Morse boundary if and only if it is Morse local-to-global. We also provide tools such as the geodesic Morse local-to-global property to show that groups are (not) Morse local-to-global. Our strategy generalizes tools from small cancellation theory, such as the intersection of relators, to arbitrary finitely generated groups. Further, we introduce a class of groups akin to graded small-cancellation groups and show that, for groups in this class, a geodesic is Morse if and only if its intersection with relators grows sublinearly in the length of the relators. We use this to construct the first example of a non-virtually cyclic Morse local-to-global group with an infinite-order Morse element that is not acylindrically hyperbolic. |
| title | Connections between the topology of the Morse boundary, the Morse local-to-global property and acylindrical hyperbolicity |
| topic | Group Theory |
| url | https://arxiv.org/abs/2605.11126 |