Homogeneous length functions on groups

A pseudo-length function defined on an arbitrary group $G = (G,\cdot,e, (\,)^{-1})$ is a map $\ell: G \to [0,+\infty)$ obeying $\ell(e)=0$, the symmetry property $\ell(x^{-1}) = \ell(x)$, and the triangle inequality $\ell(xy) \leqslant \ell(x) + \ell(y)$ for all $x,y \in G$. We consider pseudo-length functions which saturate the triangle inequality whenever $x=y$, or equivalently those that are homogeneous in the sense that $\ell(x^n) = n\,\ell(x)$ for all $n\in\mathbb{N}$. We show that this implies that $\ell([x,y])=0$ for all $x,y \in G$. This leads to a classification of such pseudo-length 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.