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

Vadim Kaimanovich, Ilya Kapovich, Paul Schupp

Research output: Contribution to journalArticlepeer-review

Abstract

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 ℤ[1/(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).

Original languageEnglish (US)
Pages (from-to)1-46
Number of pages46
JournalIsrael Journal of Mathematics
Volume157
DOIs
StatePublished - Jan 2007

ASJC Scopus subject areas

  • Mathematics(all)

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