The Subadditive Ergodic Theorem and generic stretching factors for free group automorphisms

Abstract

Given a free group $F_k$ of rank $k\ge 2$ with a fixed set of free generators we associate to any homomorphism $\phi$ from $F_k$ to a group $G$ with a left-invariant semi-norm a generic stretching factor, $\lambda(\phi)$, which is a non-commutative generalization of the translation number. We concentrate on the situation when $\phi:F_k\to Aut(X)$ corresponds to a free action of $F_k$ on a simplicial tree $X$, in particular, when $\phi$ 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 $\lambda=\lambda(\phi)$. We show that $\lambda \ge 1$ and is a rational number with $2k\lambda\in \mathbb Z[ \frac{1}{2k-1} ]$ for every $\phi\in Aut(F_k)$. We also prove that the set of all $\lambda(\phi)$, where $\phi$ varies over $Aut(F_k)$, has a gap between 1 and $1+\frac{2k-3}{2k^2-k}$, and the value 1 is attained only for ``trivial'' reasons. Furthermore, there is an algorithm which, when given $\phi$, calculates $\lambda(\phi)$.