  Let $F_n$ be the free group on $n$ generators and $\Gamma_g$ the surface
group of genus $g$. We consider two particular generating sets: the set of all
primitive elements in $F_n$ and the set of all simple loops in $\Gamma_g$. We
give a complete characterization of distorted and undistorted elements in the
corresponding $Aut$-invariant word metrics. In particular, we reprove Stallings
theorem and answer a question of Danny Calegari about the growth of simple
loops. In addition, we construct infinitely many quasimorphisms on $F_2$ that
are $Aut(F_2)$-invariant. This answers an open problem posed by Mikl\'os
Ab\'ert.
