Relative Free Splitting Complexes II: Stable Translation Lengths and the Two Over All Theorem

TL;DR

The paper develops a quantitative dynamical framework for relative outer automorphism groups acting on relative free splitting complexes FS(Γ;𝔄). Central to the work is the Two Over All Theorem, a uniform exponential flaring property for Stallings fold paths, which yields uniform control of edge-crossing growth and underpins coarse Lipschitz projections from relative outer space 𝒪(Γ;𝔄) to FS(Γ;𝔄. The authors derive an upper bound τ_φ ≤ B log(λ_φ) for stable translation lengths when φ has a filling attracting lamination with expansion factor λ_φ, and show a coarsely Lipschitz Lipschitz projection from 𝒪(Γ;𝔄) to FS(Γ;𝔄). They also develop a robust relative train track technology, including fold axes, EG-aperiodic strata, and bounded cancellation, to connect laminations with growth factors and to control dynamical behavior of Out(Γ;𝔄) elements. Overall, the results supply strong, uniform, quantitative parallels to classic results for Out(F_n) and map class groups, with potential applications to Dehn function lower bounds and hierarchical geometric structures beyond hyperbolic-type groups.

Abstract

This is the second of a three part study of relative free splitting complexes $\mathcal{FS}(Γ;\mathscr A)$, known from Part~I to be Gromov hyperbolic. Here and in~Part III we focus on stable translation lengths $τ_φ\ge 0$ of the simplicial isometries of $\mathcal{FS}(Γ;\mathscr A)$ induced by relative outer automorphisms $φ\in \text{Out}(Γ;\mathscr A)$, stating and proving quantitative generalizations of earlier theorems for $\text{Out}(F_n)$. The main technical result proved here in Part~II is the \emph{Two Over All Theorem}, which expresses a uniform exponential flaring property along arbitrary Stallings fold paths in $\mathcal{FS}(Γ;\mathscr A)$, a new result even for $\text{Out}(F_n)$. 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 $\mathcal{FS}(Γ;\mathscr A)$ is coarsely Lipschitz, with respect to the log-Lipschitz semimetric on~${\mathscr O}(Γ;\mathscr A)$. Second, if $φ\in \text{Out}(Γ;\mathscr A)$ has a filling attracting lamination with expansion factor $λ>1$ then the stable translation length of $φ$ acting on $\mathcal{FS}(Γ;\mathscr A)$ has an upper bound of the form~$B \log(λ)$.

Figures (2)

• Figure 1: Sewing needle folds $f \colon S \to T$, with $v_n = \gamma^n \cdot v_0$, and $v'_n = f(v_n) = \gamma^n \cdot v'_0$, and $E_n = \gamma^n E_0$. The upper portion of the diagram depicts the axis of $\gamma$ in $S$ with fundamental domain $E=E_0$ having initial vertex $v_0$ and terminal vertex $v_1 = \gamma \cdot v_0$. The "lower left" version of $f$ and $T$ shows a nondegenerate sewing needle fold, in which $f(E_0)$ is expressed as the concatenated path $\overline{v'_0 \, z_0 \, z_1 \, v'_1}$, and the action of $\gamma$ of $T$ has an axis with fundamental domain $E'_0 = \overline{z_0 z_1}$. The "lower right" version of $f$ and $T$ shows a degenerate sewing needle fold, in which $f(E_0)$ is $\overline{v'_0 \, z \, v'_1}$, and the subgroup $\langle\gamma\rangle$ is the stabilizer of the vertex $z$.
• Figure 2: Various commutative diagrams involving canonical boundaries $\partial(\Gamma;\mathscr A)$ and $\partial_{\infty}(\Gamma;\mathscr A)$. For each Grushko free splitting $T$ of $\Gamma$ rel $\mathscr A$, a squiggly arrow labelled $I_T$ denotes an equivariant homeomorphism ("identification") of a canonical boundary with the corresponding boundary of $T$. On the left are diagrams associated to each $\Gamma$-equivariant simplicial map $f \colon S \to T$ between Grushko free splittings of $\Gamma$ rel $\mathscr A$. On the right are diagrams associated to each choice of $\Phi \in \mathop{\mathrm{Aut}}\nolimits(\Gamma;\mathscr A)$ and of a $\Phi$-twisted equivariant simplicial map $g \colon S \to T$ between free splittings of $\Gamma$ rel $\mathscr A$; the maps $\partial\Phi$ and $\partial_{\infty}\Phi$ are $\Phi$-twisted equivariant self-homeomorphims. The two diagrams on the left fit into a single commutative diagram with upward pointing "inclusion" arrows (not shown) from the lower diagram to the upper diagram, and similarly for the two on the right.

Theorems & Definitions (80)

• Theorem : Lipschitz Projection Theorem
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• Definition 2.4: Free factor support of an invariant subforest.
• Definition 2.5: Visibility of a free factor system in a free splitting
• Definition 2.6: Relative free factors
• proof
• Definition 2.8: Kurosh rank, following CollinsTurner:efficient