2022/12/19 by Michael Handel, Lee Mosher, Handel, Michael +1
Mathematics · #Mathematical Dynamics and Fractals #Geometric and Algebraic Topology #Advanced Topology and Set Theory
paper · pdf · doi:10.48550/arxiv.2212.09907
This is the second of a three part study of relative free splitting complexes FS(Γ;\mathscr A), known from Part~I to be Gromov hyperbolic. Here and in~Part III we focus on stable translation lengths τϕ≥ 0 of the simplicial isometries of FS(Γ;\mathscr A) induced by relative outer automorphisms ϕ∈ Out(Γ;\mathscr A), stating and proving quantitative generalizations of earlier theorems for Out(Fn). The main technical result proved here in Part~II is the Two Over All Theorem, which expresses a uniform exponential flaring property along arbitrary Stallings fold paths in FS(Γ;\mathscr A), a new result even for Out(Fn). We give two applications of this theorem. First, the natural map from the relative outer space \mathscr O(Γ;\mathscr A) to the relative free splitting complex FS(Γ;\mathscr A) is coarsely Lipschitz, with respect to the log-Lipschitz semimetric on~\mathscr O(Γ;\mathscr A). Second, if ϕ∈ Out(Γ;\mathscr A) has a filling attracting lamination with expansion factor λ>1 then the stable translation length of ϕ acting on FS(Γ;\mathscr A) has an upper bound of the form~B log(λ).