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

Vadim Kaimanovich·Ilya Kapovich·Paul Schupp
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Abstract

Given a free group $F_k$ of rank $k\ge 2$ with a fixed set of free generators we associate to any homomorphism $ϕ$ from $F_k$ to a group $G$ with a left-invariant semi-norm a generic stretching factor, $λ(ϕ)$, which is a non-commutative generalization of the translation number. We concentrate on the situation when $ϕ:F_k\to Aut(X)$ corresponds to a free action of $F_k$ on a simplicial tree $X$, in particular, when $ϕ$ 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 $λ=λ(ϕ)$. We show that $λ\ge 1$ and is a rational number with $2kλ\in \mathbb Z[ \frac{1}{2k-1} ]$ for every $ϕ\in Aut(F_k)$. We also prove that the set of all $λ(ϕ)$, where $ϕ$ 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 $ϕ$, calculates $λ(ϕ)$.

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Vadim Kaimanovich, Ilya Kapovich, Paul Schupp. 2005-04-06. The Subadditive Ergodic Theorem and generic stretching factors for free group automorphisms. https://arxiv.org/abs/math/0504105

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