Homogeneous length functions on groups

A pseudo-length function defined on an arbitrary group $G = (G,\cdot,e, (,)^{-1})$isamap$\ell: G \to [0,+\infty)$obeying$\ell(e)=0$, the symmetry property $\mathrm{\ell }(x^{-1})=\mathrm{\ell }(x)$, and the triangle inequality $\mathrm{\ell }(xy)⩽\mathrm{\ell }(x)+\mathrm{\ell }(y)$ 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 $\ell(x^n) = n,\ell(x)$forall$n\in$\mathbb{N}$$\mathrm{.}Weshowthatthisimpliesthat$\ell([x,y])=0$forall$x,y \in G$. 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.