Length functions on groups and actions on graphs

TL;DR

This paper explores the relationship between length functions on groups and their actions on graphs, extending known theorems and providing counterexamples for hyperbolic cases, along with axioms ensuring bi-Lipschitz equivalence.

Contribution

It generalizes Chiswell's Theorem to hyperbolic groups and introduces axioms for length functions that guarantee bi-Lipschitz equivalence with Cayley graph actions.

Findings

Chiswell's Theorem holds for 0-hyperbolic Lyndon length functions.

Counterexamples show the theorem fails for δ-hyperbolic functions.

Proposed axioms ensure bi-Lipschitz equivalence to group actions on Cayley graphs.

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.