Homogeneous length functions on groups

Abstract

A pseudolength function defined on an arbitrary group $G=\left(G,\cdot ,e,{\left(\right)}^{-1}\right)$ is a map $\ell :G\to \left[0,+\infty \right)$ obeying $\ell \left(e\right)=0$, the symmetry property $\ell \left(x^{-1}\right)=\ell \left(x\right)$, and the triangle inequality $\ell \left(xy\right)\le \ell \left(x\right)+\ell \left(y\right)$ for all $x,y\in G$. We consider pseudolength functions which saturate the triangle inequality whenever $x=y$, or equivalently those that are homogeneous in the sense that $\ell \left(x^n\right)=n\ell \left(x\right)$ for all $n\in \mathbb{N}$. We show that this implies that $\ell \left(\left[x,y\right]\right)=0$ for all $x,y\in G$. This leads to a classification of such pseudolength functions as pullbacks from embeddings into a Banach space. We also obtain a quantitative version of our main result which allows for defects in the triangle inequality or the homogeneity property.