2005/04/06 by Vadim Kaimanovich, Vadim A. Kaimanovich, Ilya Kapovich +5
Mathematics · #20F65 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #math.GR #math.GT #msc:20F65
paper · pdf · doi:10.48550/arxiv.math/0504105
arxiv created 2005/04/06 · openalex publication_date 2005/04/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a free group Fk of rank k≥ 2 with a fixed set of free generators we associate to any homomorphism ϕ from Fk 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:Fk→ Aut(X) corresponds to a free action of Fk on a simplicial tree X, in particular, when ϕ corresponds to the action of Fk on its Cayley graph via an automorphism of Fk. 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∈ \mathbb Z[ (1)/(2k-1) ] for every ϕ∈ Aut(Fk). We also prove that the set of all λ(ϕ), where ϕ varies over Aut(Fk), has a gap between 1 and 1+(2k-3)/(2k2-k), and the value 1 is attained only for ``trivial'' reasons. Furthermore, there is an algorithm which, when given ϕ, calculates λ(ϕ).