Length functions on groups and actions on graphs

Abstract

We study generalizations of Chiswell’s theorem that 0-hyperbolic Lyndon length functions on groups always arise as based length functions of the group acting isometrically on a tree. We produce counter-examples to show that this Theorem fails if one replaces 0-hyperbolicity with δ-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 with respect to some finite generating set.