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The subadditive ergodic theorem and generic stretching factors for free group automorphisms. (English) Zbl 1173.20031
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 \(\lambda(\varphi)\).

MSC:
20F65 Geometric group theory
20E05 Free nonabelian groups
20E36 Automorphisms of infinite groups
37A30 Ergodic theorems, spectral theory, Markov operators
60G50 Sums of independent random variables; random walks
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