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Let $F_n$ be the free group on $n$ generators and $\Gamma_g$ the surfacegroup of genus $g$. We consider two particular generating sets: the set of allprimitive elements in $F_n$ and the set of all simple loops in $\Gamma_g$. Wegive a complete characterization of distorted and undistorted elements in thecorresponding $Aut$-invariant word metrics. In particular, we reprove Stallingstheorem and answer a question of Danny Calegari about the growth of simpleloops. In addition, we construct infinitely many quasimorphisms on $F_2$ thatare $Aut(F_2)$-invariant. This answers an open problem posed by Mikl\'osAb\'ert.

Aut-invariant norms and Aut-invariant quasimorphisms on free and surface groups