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 Miklos Abert.

中文翻译：

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自由和表面群上的自不变范数和自不变拟态

令 $F_n$ 是 $n$ 生成元上的自由群，$\Gamma_g$ 是 $g$ 属的表面群。我们考虑两个特定的生成集：$F_n$ 中所有原始元素的集合和 $\Gamma_g$ 中所有简单循环的集合。我们在相应的 $Aut$-invariant 词度量中给出了失真和未失真元素的完整特征。特别是，我们谴责 Stallings 定理并回答 Danny Calegari 关于简单循环增长的问题。此外，我们在 $F_2$ 上构造了无限多个 $Aut(F_2)$ 不变的拟态。这回答了 Miklos Abert 提出的一个悬而未决的问题。