Let be the free group on generators and the surface group of genus . We consider two particular generating sets: the set of all primitive elements in and the set of all simple loops in . 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 that are Aut-invariant. This answers an open problem posed by Miklós Abért.
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Michael Brandenbursky, Michał Marcinkowski, Aut-invariant norms and Aut-invariant quasimorphisms on free and surface groups. Comment. Math. Helv. 94 (2019), no. 4, pp. 661–687DOI 10.4171/CMH/470