Tree-like is not a transitive relation on paths
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866914402559590400 |
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| author | Brazas, Jeremy Conner, Gregory R. Fabel, Paul Kent, Curtis |
| author_facet | Brazas, Jeremy Conner, Gregory R. Fabel, Paul Kent, Curtis |
| contents | The notions of tree-like loop and Lipschitz tree-like loop were introduced by Hambly and Lyons in their 2010 Annals of Mathematics paper. They showed that the Lipschitz tree-like property determines an equivalence relation on the set of paths of bounded variation in a given metric space and then asked if this notion could be extended to paths without the Lipschitz requirement. We show that after eliminating the Lipschitz requirement, the resulting relation is no longer transitive and thus is not an equivalence relation. The counterexample is obtained by analyzing an explicit fractal construction in the plane. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_16774 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tree-like is not a transitive relation on paths Brazas, Jeremy Conner, Gregory R. Fabel, Paul Kent, Curtis Geometric Topology General Topology Group Theory The notions of tree-like loop and Lipschitz tree-like loop were introduced by Hambly and Lyons in their 2010 Annals of Mathematics paper. They showed that the Lipschitz tree-like property determines an equivalence relation on the set of paths of bounded variation in a given metric space and then asked if this notion could be extended to paths without the Lipschitz requirement. We show that after eliminating the Lipschitz requirement, the resulting relation is no longer transitive and thus is not an equivalence relation. The counterexample is obtained by analyzing an explicit fractal construction in the plane. |
| title | Tree-like is not a transitive relation on paths |
| topic | Geometric Topology General Topology Group Theory |
| url | https://arxiv.org/abs/2603.16774 |