FOLDING FREE-GROUP AUTOMORPHISMS

Dynamics of free group automorphisms

We present a coarse convexity result for the dynamics of free group automorphisms: Given an automorphism φ of a finitely generated free group F , we show that for all x ∈ F and 0 ≤ i ≤ N , the length of φ(x) is bounded above by a constant multiple of the sum of the lengths of x and φ (x), with the constant depending only on φ.

Folding by Automorphisms

Let Φ be a simply-laced root system with simple roots ∆ = {αi : i ∈ I} embedded in the real vector space V with inner product 〈·,·〉. For convenience, we may assume that the roots have been normalized so that 〈α, α〉 = 2 for all α ∈ Φ. As a second convenience, we may assume that V has been enlarged, if necessary, so that 〈·,·〉 is nondegenerate on V . Now let σ be a diagram automorphism of (Φ,∆). ...

Complexes with free actions by the group of automorphisms of a free group

We define two finite-dimensional complexes on which the group Aut(Fn) acts freely, and study their homotopy groups. The definitions of these complexes were motivated by lowdimensional pseudo-isotopy theory. So not surprisingly, the complexes are related to a number of spaces arising in algebraic K-theory, and to another in low-dimensional topology. The complexes are denoted by Xn and X alg n . ...

Normal Forms for Basis-Conjugating Automorphisms of a Free Group

Free-group Automorphisms, Train Tracks, and the Beaded Decomposition

We study the automorphisms φ of a finitely generated free group F . Building on the train-track technology of Bestvina, Feighn and Handel, we provide a topological representative f : G→ G of a power of φ that behaves very much like the realization on the rose of a positive automorphism. This resemblance is encapsulated in the Beaded Decomposition Theorem which describes the structure of paths i...