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arXiv · 2307.10760

Length functions on groups and actions on graphs

Matthew Collins·Armando Martino
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Abstract

We study generalisations of Chiswell's Theorem that $0$-hyperbolic Lyndon length functions on groups always arise as based length functions of the the group acting isometrically on a tree. We produce counter-examples to show that this Theorem fails if one replaces $0$-hyperbolicity with $\delta$-hyperbolicity. We then propose a set of axioms for the length function on a finitely generated group that ensures the function is bi-Lipschitz equivalent to a (or any) length function of the group acting on its Cayley graph.

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Matthew Collins, Armando Martino. 2023-07-20. Length functions on groups and actions on graphs. https://arxiv.org/abs/2307.10760

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