Tree-like is not a transitive relation on paths
Geometric Topology
2026-03-18 v1 General Topology
Group Theory
Abstract
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.
Keywords
Cite
@article{arxiv.2603.16774,
title = {Tree-like is not a transitive relation on paths},
author = {Jeremy Brazas and Gregory R. Conner and Paul Fabel and Curtis Kent},
journal= {arXiv preprint arXiv:2603.16774},
year = {2026}
}