The free splitting complex of a free group, II: Loxodromic outer automorphisms

Abstract

We study the loxodromic elements for the action of O u t ( F n ) \mathsf {Out}(F_n) on the free splitting complex of the rank n n free group F n F_n . Each outer automorphism is either loxodromic or has bounded orbits without any periodic point, or has a periodic point; all three possibilities can occur. Two loxodromic elements are either coaxial or independent, meaning that their attracting-repelling fixed point pairs on the Gromov boundary of the free splitting complex are either equal or disjoint as sets. Each alternative is characterized in terms of attracting laminations; in particular, an outer automorphism is loxodromic if and only if it has a filling attracting lamination. As an application, each attracting lamination determines its corresponding repelling lamination independent of the outer automorphism. As part of this study, we describe the structure of the subgroup of O u t ( F n ) \mathsf {Out}(F_n) that stabilizes the fixed point pair of a given loxodromic outer automorphism, and we give examples which show that this subgroup need not be virtually cyclic. As an application, the action of O u t ( F n ) \mathsf {Out}(F_n) on the free splitting complex is not acylindrical, and its loxodromic elements do not all satisfy the weak proper discontinuity (WPD) property of Bestvina and Fujiwara.

Distortion for abelian subgroups of Out(Fn)

“…Proof. First, Corollary 1.3(2) of [HM14] gives that Stab(Λ + ) = Stab(Λ − ) (which we will henceforth refer to as Stab(Λ ± )), so the ratio in the statement is always well defined. Now P F Λ + and P F Λ − each determine a homomorphism from Stab(Λ ± ) to R and it suffices to show that these homomorphisms have the same kernel.…”

“…Now ψ has a paired lamination Λ − ψ ∈ L(ψ −1 ) which a priori could be different from Λ − . But Corollary 1.3(1) of [HM14] says that in fact Λ − ψ = Λ − and therefore that Λ − ∈ L(ψ −1 ). A final application of [BFH00, Corollary 3.3.1] gives that ψ / ∈ ker P F Λ − .…”

Relatively irreducible free subroups in Out($\mathbb{F}$)

“…From Handel-Mosher's work on loxodromic elements of the free splitting complex [15] one defines the concept of a co-edge number for a free factor system F : it is an integer ≥ 1 which is the minimum , over all subgraphs H of a marked graph G such that H realizes F , of the number of edges in G −H . Lemma 4.8 in [15] gives an explicit formula for computing the co-edge number for a given free factor system.…”

The minimally displaced set of an irreducible automorphism is locally finite

“…Attaching these "collapsed" points, leads one to define the simplicial bordification of the deformation space. If one starts with Culler-Vogtmann space, the result is the free splitting complex, which is related to the free factor complex (see for instance [3,5,6,18,19,21,25]).…”