14.23 (1999)

Let $F_n$ be a free group with basis $\{x_1, \dots, x_n\}$, and let $\lvert \cdot \rvert$ be the length function with respect to this basis. For $\alpha \in \operatorname{Aut} F_n$ we put $\lVert \alpha \rVert = \max\{\lvert \alpha(x_1) \rvert, \dots, \lvert \alpha(x_n) \rvert\}$. Is it true that there is a recursive function $f : \mathbb{N} \to \mathbb{N}$ with the following property: for any $\alpha \in \operatorname{Aut} F_n$ there is a basis $\{y_1, \dots, y_k\}$ of $\operatorname{Fix}(\alpha) = \{x \mid \alpha(x) = x\}$ such that $\lvert y_i \rvert \leqslant f(\lVert \alpha \rVert)$ for all $i = 1, \dots, k$?