The Subadditive Ergodic Theorem and generic stretching factors for free group automorphisms |
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Authors: | Vadim Kaimanovich Ilya Kapovich and Paul Schupp |
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Institution: | (1) School of Engineering and Science, International University-Bremen, P.O. Box 750 561, 28725 Bremen, Germany;(2) Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA |
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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 ∈ 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. |
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Keywords: | |
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