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Summary: Let $$F_k$$ be a free group of rank $$k\geq 2$$ with a fixed set of free generators. We associate to any homomorphism $$\varphi$$ from $$F_k$$ to a group $$G$$ with a left-invariant semi-norm a generic stretching factor, $$\lambda(\varphi)$$, which is a noncommutative generalization of the translation number. We concentrate on the situation where $$\varphi\colon F_k\to\operatorname{Aut}(X)$$ corresponds to a free action of $$F_k$$ on a simplicial tree $$X$$, in particular, where $$\varphi$$ corresponds to the action of $$F_k$$ on its Cayley graph via an automorphism of $$F_k$$. In this case we are able to obtain some detailed “arithmetic” information about the possible values of $$\lambda=\lambda(\varphi)$$. We show that $$\lambda\geq 1$$ and is a rational number with $$2k\lambda\in\mathbb{Z}[1/(2k-1)]$$ for every $$\varphi\in\operatorname{Aut}(F_k)$$. We also prove that the set of all $$\lambda(\varphi)$$, where $$\varphi$$ varies over $$\operatorname{Aut}(F_k)$$, has a gap between 1 and $$1+(2k-3)/(2k^2-k)$$, and the value 1 is attained only for “trivial” reasons. Furthermore, there is an algorithm which, when given $$\varphi$$, calculates $$\lambda(\varphi)$$.
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