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

Let F _k be a free group of rank k ≥ 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 noncommutative generalization of the translation number. We concentrate on the situation where φ: F _k → Aut(X) corresponds to a free action of F _k on a simplicial tree X, in particular, where 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 ≥ 1 and is a rational number with 2kλ ∈ ℤ1/(2k − 1)] for every φ ∈ 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+(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). The second and the third author were supported by the NSF grant DMS#0404991 and the NSA grant DMA#H98230-04-1-0115.