On the growth of Lie algebras and Jordan algebras

We prove that every increasing, polynomially bounded function is realizable, up to bi-Lipschitz equivalence, as the growth function of some finitely generated Lie algebra. This does not extend to general increasing subexponential functions. We further prove that the classes of growth functions of Jordan algebras and associative algebras coincide, answering in the affirmative a question of Martínez and Zelmanov.