1.1 Group systems and actions

Nearly all statements and results about groups and connected spaces that we are interested in still hold when ‘connectivity’ is relaxed and we work with ‘systems’ componentwise. In general (almost categorical) terms, a system of [?-objects] is a disjoint union $\mathcal O = \bigsqcup _{j \in J} O_j$ of [?-objects] $O_j$ indexed by some set J. An [?-isomorphism] of systems $\psi \colon \mathcal O \to \mathcal O'$ is a bijection $\sigma \colon J \to J'$ of the corresponding indexing sets and a union of [?-isomorphisms] $\psi _j \colon O_j \to O_{\sigma \cdot j}'$ . The calligraphic font is reserved for systems.

In more concrete terms, here are some basic concepts that will show up in the paper:

1. 1. an isomorphism of group systems $\psi \colon {\mathcal {G}} \to {\mathcal {G}}'$ is a bijection whose restriction to any component ${G_j \subset {\mathcal {G}}}$ is a group isomorphism of components; for group systems, we always assume (for convenience) components are nontrivial if the system is nonempty.

2. 2. two isomorphisms of group systems $\psi ,\psi '\colon {\mathcal {G}} \to {\mathcal {G}}'$ are in the same outer class $[\psi ]$ if the component isomorphisms $\psi _j,\psi _j' \colon G_j \to G_{\sigma \cdot j}'$ differ only by post-composition with an inner automorphism of $G_{\sigma \cdot j}'$ for all $j \in J$ .

3. 3. a metric on a set system $\mathcal X$ is a disjoint union of metrics $d_j\colon X_j \times X_j \to \mathbb R_{\ge 0}$ on the components $X_j \subset \mathcal X$ .

4. 4. for a group system ${\mathcal {G}}$ indexed by J and object system $\mathcal O$ indexed by $J'$ , a $\underline{\mathcal{G}-\mathrm{action}}$ on $\mathcal O$ (or $\underline{\mathcal{G}-\text{object} \text{system} \mathcal{O}}$ ) is a union of component $G_j$ -actions on $O_{\beta \cdot j}$ for some bijection $\beta \colon J \to J'$ .

5. 5. for an automorphism of a group system $\psi \colon {\mathcal {G}} \to {\mathcal {G}}$ and a ${\mathcal {G}}$ -object system $\mathcal O$ , the ψ-twisted ${\mathcal {G}}$ -object system $\mathcal O \psi $ is given by precomposing the component $G_{\sigma \cdot j}$ -action on $O_{\beta \sigma \cdot j}$ with the component isomorphism $\psi _j\colon G_j \to G_{\sigma \cdot j}$ to get a $G_j$ -object $O_{\beta \sigma \cdot j}$ .

1.2 Pretrees, trees and hierarchies

Pretrees are what arises when one wants to discuss ‘treelike’ objects without reference to a metric or topology. In this paper, the pretrees are the ‘primitive’ objects, and metrics/topologies are additional structures on the pretree – think of it the same way a Riemannian metric is a compatible addition to a manifold’s smooth structure.

Fix a set T; an interval function on T is a function $[\cdot , \cdot ]\colon T \times T \to \mathcal P(T)$ , where $\mathcal P(T)$ is the power set of T, that satisfies the following axioms: for all $p,q,r \in T$ ,

1. 1. (symmetric) $[p,q] = [q,p]$ contains $\{p, q\}$ ;

2. 2. (thin) $[p,r] \subset [p,q] \cup [q,r]$ ; and

3. 3. (linear) if $r \in [p,q]$ and $q \in [p,r]$ , then $q = r$ .

A pretree is a pair $(T, [\cdot , \cdot ])$ of a nonempty set T and an interval function $[\cdot , \cdot ]$ on T.

The subsets $[p,q] \subset T$ are called closed intervals, and they should be thought of as the points between p and q (inclusive). We can similarly define open (resp. half-open) intervals by excluding both (resp. exactly one) of $\{p,q\}$ . Generally, ‘interval’ (with no qualifier) refers to any of the three types of intervals we have defined. An interval is degenerate if it is empty or a singleton. We usually omit the interval function and denote a pretree by T. Note that the real line $\mathbb R$ is a pretree.

Any subset $S \subset T$ of a pretree inherits an interval function: for all $u,v \in S$ . A subset $C \subset T$ is convex if $[p,q] \subset C$ for all $p,q \in C$ ; or equivalently, $[\cdot , \cdot ]_C$ is the restriction of $[\cdot , \cdot ]$ to $C \times C \subset T \times T$ . A system of pretrees is a set system $\mathcal T = \bigsqcup _{j \in J} T_j$ and a disjoint union of interval functions on $T_j$ ; we call these systems pretrees for short.

Let $(T, [\cdot , \cdot ])$ and $(T', [\cdot , \cdot ]')$ be pretrees. A pretree-isomorphism is a bijection $f\colon T \to T'$ satisfying $f([p,q]) = [f(p), f(q)]'$ for all $p,q \in T$ . Similarly, a pretree-automorphism of $(T, [\cdot , \cdot ])$ is a pretree-isomorphism $g \colon (T, [\cdot , \cdot ]) \to (T, [\cdot , \cdot ])$ . A pretree is real if its closed intervals are pretree-isomorphic to closed intervals of $\mathbb R$ . By definition, the real line $\mathbb R$ is a real pretree. Note that being real is a property of a pretree, not an added structure like a metric! An arc of a real pretree T is a nonempty union of an ascending chain of nondegenerate intervals. A real pretree is degenerate if it is a singleton; and a system of real pretrees is degenerate if all components are degenerate.

Fix a real pretree T; a convex pseudometric on T is a function $d\colon T \times T \to \mathbb R_{\ge 0}$ satisfying the following axioms: for all $p,q,r \in T$ ,

1. 1. (symmetric) $d(p,q) = d(q,p)$ ;

2. 2. (convex) $d(p,r) = d(p,q) + d(q,r)$ if $q \in [p,r]$ ; and

3. 3. (continuous) $d(p,q) = 2\, d(p,q')$ for some $q' \in [p,q]$ .

For any given convex pseudometric d on T, the preimage $d^{-1}(0) \subset T \times T$ is an equivalence relation on the real pretree T such that each equivalence class is convex and the set $T_d$ of equivalence classes inherits a real pretree structure. A convex metric on T is a convex pseudometric whose equivalence relation $d^{-1}(0)$ is the equality relation on T. A (metric) tree (or $\mathbb R$ -tree) is a real pretree with a convex metric; a forest is a system of trees. For example, the real line $\mathbb R$ is a tree with the standard metric . Note that a convex pseudometric d on a real pretree T induces a convex metric, still denoted d, on the real pretree $T_d$ ; we refer to the tree $(T_d, d)$ as the associated tree.

A λ-homothety of trees $h \colon (T, d) \to (Y, \delta )$ is a pretree-isomorphism $h \colon T \to Y$ that uniformly scales the metric d by $\lambda $ :

$$\begin{align*}\delta(h(p),h(q)) = \lambda \, d(p,q) ~ \text{for all} ~ (p,q) \in \operatorname{dom}(d)= T \times T; \end{align*}$$

equivalently, $h^*\delta = \lambda d$ , where $h^*\delta $ is the pullback of $\delta $ via h. A homothety is a $\lambda $ -homothety for some $\lambda> 0$ ; it is expanding (resp. an isometry) if $\lambda> 1$ (resp. $\lambda = 1$ ). An isometry $\iota \colon (T,d) \to (T, d)$ is elliptic if it fixes a point of T; otherwise, it is loxodromic and acts by a nontrivial translation on its axis, the unique $\iota $ -invariant arc of $(T, d)$ isometric to $(\mathbb R,d_{\mathrm {std}})$ ; the translation distance $\|\iota \|_d \in \mathbb R_{\ge 0}$ is $0$ if $\iota $ is elliptic and equal to the displacement of points in $\iota $ ’s axis if $\iota $ is loxodromic. These definitions extend componentwise to forests.

Let $d_1$ be a nonconstant convex pseudometric on T and $d_{i+1} \colon d_i^{-1}(0) \to \mathbb R_{\ge 0}$ a nonconstant disjoint union of convex pseudometrics for $1 \le i < n$ . The sequence $(d_i)_{i=1}^n$ will be known as an n-level hierarchy of convex pseudometrics on T; we will say just hierarchies for short. A hierarchy $(d_i)_{i=1}^n$ has full support if $d_n$ is a disjoint union of convex metrics. A pseudotree is a pair $(T, (d_i)_{i=1}^n)$ of a real pretree and a hierarchy with full support; a pseudoforest is a system of pseudotrees. A (λ _i ) _i=1 ⁿ -homothety of n-level pseudoforests $h \colon (\mathcal T, (d_i)_{i=1}^n) \to (\mathcal Y, (\delta _i)_{i=1}^n)$ is a system of pretree-isomorphisms $h \colon \mathcal T \to \mathcal Y$ that scales each pseudometric $d_i$ by $\lambda _i$ :

$$\begin{align*}\delta_i(h(p),h(q)) = \lambda_i \, d_i(p,q) ~ \text{for all} ~ i \ge 1 ~ \text{and} ~ (p,q) \in \operatorname{dom}(d_i);\end{align*}$$

a homothety is a $(\lambda _i)_{i=1}^n$ -homothety for some $\lambda _i> 0$ ; it is expanding (resp. isometry) if each $\lambda _i> 1$ (resp. each $\lambda _i = 1$ ). As with trees, an isometry of a pseudotree is either elliptic (fixes a point) or loxodromic (translates a ‘pseudoaxes’). Hierarchies and pseudoforests are the fundamental (perhaps novel) tool in this paper. They are first used in Chapter 3.

1.3 Simplicial actions and train tracks

For a pretree T, a direction at $p \in T$ is a maximal subset $D_p \subset T \setminus \{ p \}$ not separated by p (i.e., $p \notin [q,r]$ for all $q,r \in D_p$ ). A branch point is a point with at least three directions, and a branch is a direction at a branch point. An endpoint is a point with at most one direction. A simple pretree is a pretree whose closed intervals are finite subsets. A pretree T is simplicial if it is real, its subset V of branch points and endpoints is a simple pretree, and no convex proper subset contains V; a vertex is a point in V. An (open) edge in a simplicial pretree T is a maximal convex subset $e \subset T$ that contains no vertex. By construction, edges are open intervals; the corresponding closed intervals in T are called closed edges.

An F-pretree is a pretree with an F-action by pretree-automorphisms. An F-pseudotree is a pair of a real F-pretree and an F-invariant hierarchy with full support; equivalently, an F-pseudotree (resp. F-tree) is a pseudotree with an isometric F-action. An F-pseudotree or F-tree is minimal if the underlying F-pretree has no proper nonempty F-invariant convex subset; in this case, the underlying F-pretree has no endpoints. We mostly consider minimal F-pseudotrees with trivial arc (pointwise) stabilizers.

Suppose an F-pseudotree $(T, (d_i)_{i=1}^n)$ has trivial arc stabilizers. For any nontrivial subgroup $G \le F$ , the characteristic convex subset (of T) for G is the unique minimal nonempty G-invariant convex subset $T(G) \subset T$ . In an F-tree $(T, d)$ with trivial arc stabilizers, the restriction of d to $T(G)$ is a G-invariant convex metric, still denoted d; the minimal G-tree $(T(G), d)$ is the characteristic subtree (of $(T, d)$ ) for G.

An F-pretree T is simplicial if T is simplicial and admits an F-invariant convex metric d; equivalently, a simplicial F-pretree is a simplicial pretree with a rigid F-action. Any simplicial F-pretree has an open cone (over a finite dimensional open simplex) worth of F-invariant convex metrics (up to an equivariant isometry isotopic to the identity map). The definitions given so far extend componentwise to systems.

Let $\mathcal T$ and $\mathcal T'$ be simplicial pretrees and $f\colon \mathcal T \to \mathcal T'$ a tight cellular map (i.e., a function that maps vertices to vertices and the restriction to any closed edge is a pretree-embedding – that is, a pretree-isomorphism onto its image). For any choice of convex metrics $d, d'$ on $\mathcal T, \mathcal T'$ , respectively, there is a unique map $(\mathcal T, d) \to (\mathcal T', d')$ that is linear on edges and isotopic to f; whenever a choice of convex metrics is made, we implicitly replace f with this map.

Let $\mathcal T$ be a free splitting of $\mathcal F$ – that is, minimal simplicial $\mathcal F$ -pretrees with trivial edge stabilizers, and suppose $\psi \colon \mathcal F \to \mathcal F$ is an automorphism of a free group system. The $\psi $ -twisted free splitting $\mathcal T\psi $ is the same real pretrees $\mathcal T$ , but the original simplicial $\mathcal F$ -action is precomposed with $\psi $ . A (relative) topological representative for $\psi $ is a $\psi $ -equivariant tight cellular map $f\colon \mathcal T \to \mathcal T$ on a nondegenerate free splitting $\mathcal T$ of $\mathcal F$ : $\psi $ -equivariance means $f(x \cdot p) = \psi (x) \cdot f(p)$ for all $x \in \mathcal F$ and $p \in \mathcal T$ , or equivalently, $f\colon \mathcal T \to \mathcal T \psi $ is equivariant. Given a topological representative $f \colon \mathcal T \to \mathcal T$ for $\psi $ , we let $[f]$ denote the induced map on the quotient $\mathcal F \backslash \mathcal T$ ; we say $[f]$ is a topological representative for the outer class $[\psi ]$ . A (relative) train track for $\psi $ is a topological representative $\tau \colon \mathcal T \to \mathcal T$ for $\psi $ whose iterates $\tau ^m~(m \ge 1)$ are topological representatives for $\phi ^m$ – or equivalently, whose iterates $\tau ^m$ restrict to pretree-embeddings on closed edges.

For any free splitting $\mathcal T$ of $\mathcal F$ , Bass-Serre theory gives a uniform bound on the number of $\mathcal F$ -orbits of edges (linear in $\operatorname {rank}(\mathcal F)$ ) and relates the vertices with nontrivial stabilizers in a (componentwise) connected fundamental domain to a (possibly empty) free factor system $\mathcal F[\mathcal T]$ of $\mathcal F$ – take this as the working definition of free factor systems. The theory also gives a uniform bound on the complexity (e.g., ranks) of free factor systems. A free factor system $\mathcal F[\mathcal T]$ is proper if $\mathcal F[\mathcal T] \neq \mathcal F$ ; equivalently, $\mathcal F[\mathcal T]$ is proper if and only if $\mathcal T$ is not degenerate. Any proper free factor system of $\mathcal F$ has strictly lower complexity than $\mathcal F$ . The trivial free factor system of $\mathcal F$ is the (possibly empty) free factor system consisting of the cyclic $\mathcal F$ -components; it is always proper since we assume $\mathcal F$ has a noncyclic component.

Fix an automorphism $\psi \colon \mathcal F \to \mathcal F$ and a topological representative $f\colon \mathcal T \to \mathcal T$ for $\psi $ . By $\psi $ -equivariance of f, the proper free factor system $\mathcal F[\mathcal T]$ is $[\psi ]$ -invariant – again, we can take this as the definition of $[\psi ]$ -invariance for proper free factor systems. Form a nonnegative integer square matrix $A[f]$ whose rows and columns are indexed by the $\mathcal F$ -orbits of edges in $\mathcal T$ ; and the entry at row- $[e]$ and column- $[e']$ is given by the number of e-translates in the interval $f(e')$ , where $e, e'$ are edges in T. The topological representative f is irreducible if the matrix $A[f]$ is irreducible; or equivalently, if, for any pair of edges $e, e'$ in $\mathcal T$ , a translate of e is contained $f^m(e')$ for some ${m = m(e,e') \ge 1}$ . It is a foundational theorem of Bestvina–Handel that automorphisms have irreducible train tracks.

The proof outline of [Reference Mutanguha22, Theorem I.1] explains how to deduce the theorem as currently stated from the cited theorem.

Suppose $\psi \colon \mathcal F \to \mathcal F$ is an automorphism with an irreducible topological representative $f\colon \mathcal T \to \mathcal T$ . Perron–Frobenius theory implies the matrix $A[f]$ has a unique real eigenvalue $\lambda = \lambda [f] \ge 1$ with a unique positive left eigenvector $\nu [f]$ whose entries sum to $1$ . From the eigenvector $\nu [f]$ , we get an $\mathcal F$ -invariant convex metric $d_f$ on $\mathcal T$ (well-defined up to an equivariant isometry isotopic to the identity map). The restriction of f to any edge is a $\lambda $ -homothetic embedding with respect to $d_f$ ; the metric $d_f$ is the eigenmetric (on $\mathcal T$ ) for $[f]$ . If $\lambda = 1$ , then f is a $\psi $ -equivariant simplicial automorphism of $\mathcal T$ .

1.4 Growth types and limit trees

Since the introduction of train tracks, it has been standard to construct limit forests by iterating an expanding irreducible train track (Section 2.1). Unfortunately, such a construction is not canonical as it can depend on the initial train track. The main idea of the paper: patch together a ‘descending’ sequence of limit trees to get a limit pseudoforest and inductively ‘normalize’ its hierarchy into a canonical limit pseudoforest.

Fix a free group system ${\mathcal {G}}$ of finite type (unlike $\mathcal F$ , all components of ${\mathcal {G}}$ can be cyclic), an automorphism $\psi \colon {\mathcal {G}} \to {\mathcal {G}}$ , and a metric free splitting $(\mathcal T,d)$ of ${\mathcal {G}}$ whose free factor system is $[\psi ]$ -invariant. An element $x \in {\mathcal {G}}$ [ψ]-grows exponentially rel. d with rate λ _x if it is $\mathcal T$ -loxodromic and the limit inferior of the sequence $\left (m^{-1} \log \| \psi ^m(x) \|_d\right )_{m \ge 0}$ is $\log \lambda _x> 0$ . If an element $[\psi ]$ -grows exponentially rel. d, then it $[\psi ]$ -grows exponentially rel. $d'$ with the same rate for any metric free splitting $(\mathcal T',d')$ of ${\mathcal {G}}$ with $\mathcal F[\mathcal T'] = {\mathcal {Z}}$ ; say the element [ψ]-grows exponentially rel. ${\mathcal {Z}}$ . An element $x\in {\mathcal {G}}$ [ψ]-grows polynomially rel. ${\mathcal {Z}}$ with degree < n if the sequence $\left (m^{-n} \| \psi ^m(x)\|_d\right )_{n \ge 0}$ converges to $0$ . Any element of ${\mathcal {G}} [\psi ]$ -grows either exponentially or polynomially rel. ${\mathcal {Z}}$ [Reference Mutanguha22, Corollary III.4]. The growth type of an element is preserved when passing to invariant subgroup systems of finite type.

The automorphism $\psi $ is exponentially growing rel. ${\mathcal {Z}}$ if some element $[\psi ]$ -grows exponentially rel. ${\mathcal {Z}}$ ; otherwise, $\psi $ is polynomially growing rel. ${\mathcal {Z}}$ . The growth type of an outer class $[\psi ]$ is also well-defined. The ‘rel. ${\mathcal {Z}}$ ’ in our terminology may be omitted when ${\mathcal {Z}}$ is trivial. The next proposition deals with the first obstacle:

The constructed $\mathcal F$ -forest $(\mathcal Y, \delta )$ is the limit forest for $[\psi ]$ rel. ${\mathcal {Z}}'$ , for some $[\psi ]$ -invariant proper free factor system ${\mathcal {Z}}'$ that supports ${\mathcal {Z}}$ (see Sections 2.1 and 2.4). Unfortunately, these limit forests depend on the choice of ${\mathcal {Z}}'$ ; our goal is to give a canonical contruction.

Given the central tool (hierarchies) and objective (universal limit trees), we outline again how these two fit together. Gaboriau–Levitt’s index theory [Reference Gaboriau and Levitt11] gives a uniform bound on the complexity of the point stabilizers system ${\mathcal {G}}[\mathcal Y]$ for a minimal $\mathcal F$ -forest $(\mathcal Y,\delta )$ with trivial arc stabilizers – this is a partial generalization of Bass–Serre theory. When $\mathcal Y$ is not degenerate, the subgroup system ${\mathcal {G}}[\mathcal Y]$ has strictly lower complexity than $\mathcal F$ . This allows us to induct on complexity (see Chapter 3).

Suppose the automorphism $\psi \colon \mathcal F \to \mathcal F$ has a nondegenerate limit forest $(\mathcal Y_1,\delta _1)$ with nontrivial point stabilizers; the system of stabilizers has strictly smaller complexity than $\mathcal F$ . By $\psi $ -equivariance of $\lambda _1$ -homothety $h_1\colon (\mathcal Y_1, \delta _1) \to (\mathcal Y_1, \delta _1)$ , the $\mathcal F$ -orbits of points with nontrivial stabilizers are permuted by $[h_1]$ , the subgroup system ${\mathcal {G}}$ is $[\psi ]$ -invariant, and the restriction of $\psi $ to ${\mathcal {G}}$ determines a unique outer automorphism $[\varphi ]$ of ${\mathcal {G}}$ .

Suppose $\varphi \colon {\mathcal {G}} \to {\mathcal {G}}$ has a nondegenerate limit forest $(\mathcal Y_2, \delta _2)$ with stretch factor $\lambda _2$ . Using the blow-up construction from [Reference Mutanguha22], we equivariantly blow up $\mathcal Y_1$ with respect to $h_i \colon \mathcal Y_i \to \mathcal Y_i ~ (i = 1,2)$ to get real pretrees $\mathcal T$ with a minimal rigid $\mathcal F$ -action and a $\psi $ -equivariant ‘ $\mathcal F$ -expanding’ pretree-isomorphism $f\colon \mathcal T \to \mathcal T$ induced by $h_1$ and $h_2$ . In fact, the blow-up construction implies the $\mathcal F$ -pretrees $\mathcal T$ inherit an $\mathcal F$ -invariant $2$ -level hierarchy $(\delta _1, \delta _2)$ with full support and f is an expanding homothety with respect to this hierarchy. So we have a limit pseudoforest $(\mathcal T, (\delta _1,\delta _2))$ for $[\psi ]$ (see Section 3.1). Under what conditions can we construct an $\mathcal F$ -invariant convex metric on $\mathcal T$ from $(\delta _1, \delta _2)$ ? The heart of the paper is the following observation: the two limit forests $(\mathcal Y_i, \delta _i)$ are paired with attracting laminations $\mathcal L_i^+[\psi ]$ partially ordered by inclusion; an $\mathcal F$ -invariant convex metric on $\mathcal T$ can be naturally constructed from $(\delta _1, \delta _2)$ if $\mathcal L_2^+[\psi ]$ is not in $\mathcal L_1^+[\psi ]$ (see Section 3.4) or $\lambda _1 < \lambda _2$ (see Section 3.5)!

1.5 Bounded cancellation and laminations

Suppose a minimal $\mathcal F$ -forest $(\mathcal Y, \delta )$ is very small – that is, nontrivial arc stabilizers are maximal cyclic subgroups and the fixed point subset for a nontrivial elliptic element is an arc. Let $(\mathcal T, d)$ be a metric free splitting of $\mathcal F$ and $[\cdot ,\cdot ]_T$ (resp. $[\cdot , \cdot ]_Y$ ) denote the interval function for $\mathcal T$ (resp. $\mathcal Y$ ). A map $f \colon (\mathcal T,d) \to (\mathcal Y, \delta )$ is piecewise-linear (PL) if the restriction to any closed edge is a linear map; an equivariant PL-map exists if and only if $\mathcal T$ -elliptic elements in $\mathcal F$ are $\mathcal Y$ -elliptic. Equivariant PL-maps $(\mathcal T,d) \to (\mathcal Y, \delta )$ are surjective and Lipschitz since the isometric $\mathcal F$ -action on $(\mathcal Y, \delta )$ is minimal and there are only finitely many $\mathcal F$ -orbits of edges in $\mathcal T$ ; $1$ -Lipschitz maps are also known as metric maps. Generally, if $\mathcal T, \mathcal Y$ are free splittings of $\mathcal F$ , then an equivariant function $f \colon \mathcal T \to \mathcal Y$ is a (simplicial) PL-map if its restrictions to any closed edge is isotopic to a linear map with respect to some/any $\mathcal F$ -invariant convex metrics $d, \delta $ on $\mathcal T, \mathcal Y$ , respectively.

Such a $C[f]$ is a cancellation constant for f. This proof is due to Bestvina–Feighn–Handel.

A line in a forest is an arc that is isometric to $(\mathbb R, d_{\mathrm {std}})$ ; a ray in a forest is an arc that is isometric to $(\mathbb R_{\ge 0}, d_{\mathrm {std}})$ , and its origin is the point corresponding to $0$ under the isometry. Two rays are end-equivalent if their intersection is a ray; an end of a forest is an end-equivalence class of rays in the forest. Note that there is a natural bijection between the set of lines in a forest and set of unordered pairs of distinct ends in the same component of the forest. For simplicial $\mathcal F$ -pretrees $\mathcal T$ , the notions of line/ray/end are well-defined for the cone of $\mathcal F$ -invariant convex metrics on $\mathcal T$ .

Sketch of proof

(1): Let $\rho $ be a ray in $(\mathcal T,d)$ , $p_{0} \in \rho $ its origin, $f(\rho )$ unbounded, and $s_0 = f(p_0)$ . Use Figure 1 for reference. By bounded cancellation and the Lipschitz property, $f(\rho )$ has at most one end of $(\mathcal Y, \delta )$ . For some $n \ge 0$ , assume $s_{n} \in [s_0, f(p)]_Y$ for all $p \in \rho \setminus [p_0, p_{n}]_T$ . Set . Since $f(\rho )$ is unbounded,

$$\begin{align*}\delta(s_0, f(p_{n+1}))> 2\, \delta(s_0, s_{n}) + C\end{align*}$$

for some $p_{n+1} \in \rho \setminus [p_{0},p_{n}]_T$ . Pick $s_{n+1} \in [s_0, f(p_{n+1})]_Y$ satisfying $\delta (s_0, s_{n+1})> 2\,\delta (s_0, s_{n})$ and $\delta (s_{n+1},f(p_{n+1}))> C$ ; so $s_{n} \in [s_0, s_{n+1}]_Y$ . By bounded cancellation, the interval $[s_0,s_{n+1}]_Y \text{ in } f([p_0, p_{n+1}]_T)$ is disjoint from $f(\rho \setminus [p_0,p_{n+1}]_T)$ . So the union $\bigcup _{n \ge 0} [s_0, s_n]_Y$ is a ray $f_*(\rho )$ in $f(\rho )$ with origin $s_0$ . By construction, $f(\rho )$ is in the C-neighborhood of $f_*(\rho )$ . Any bounded neighborhood of a ray contains a unique ray, up to end-equivalence.

Figure 1 The ray $f_*(\rho )$ with origin $s_0 = f(p_0)$ is built inductively in the image $f(\rho )$ .

(2): Represent both ends of l with rays $\rho ^\pm \subset l$ sharing the same origin. By Part 1 and bounded cancellation, we have rays $f_*(\rho ^\pm )$ representing unique distinct ends $\epsilon ^\pm $ of $(\mathcal Y, \delta )$ ; moreover, $f(l)=f(\rho ^-)\cup f(\rho ^+)$ is in the C-neighborhood of $f_*(\rho ^-) \cup f_*(\rho ^+) \subset f(l)$ . Let $f_*(l) \subset f_*(\rho ^-) \cup f_*(\rho ^+)$ be the line determined by the ends $\epsilon ^\pm $ . Then $f(l)$ is in the C-neighborhood of $f_*(l)$ . Any bounded neighborhood of a line contains a unique line.

(3): The argument is almost the same. Let $\rho '$ be a ray representing $\epsilon $ and $s_0 = q_0$ its origin. Pick points $q_{n+1}, s_{n+1} \in \rho '$ with $\delta (s_0, s_{n+1})> 2\,\delta (s_0, s_{n})$ , $\delta (s_0, q_{n+1})> 2\, \delta (s_0, s_n) + C$ and $\delta (s_{n+1},q_{n+1})> C$ . Since $f\colon \mathcal T \to \mathcal Y$ is surjective, we can lift $q_n$ to $p_n \in T$ . By bounded cancellation and K-Lipschitz property, the distance $d(p_0, [p_n, p_{n+m}]_T)> \frac {1}{K} \delta (s_0,s_n)$ . Thus, $(p_n)_{n\ge 0}$ determines an end e of $(\mathcal T,d)$ with unbounded f-image. Let $\rho $ be a ray representing e with origin $p_0$ . As $\rho ' \subset f(\rho )$ by construction, we get $f_*(\rho ) = \rho '$ by Part 1. By Part 2, the end e is the unique end with $f_*(e) = \epsilon $ , and we denote it by  $f^*(\epsilon )$ .

The corollary defines the equivariant lifting (resp. projecting) function $f^*$ (resp. $f_*$ ), where the domain $\operatorname {dom}(f^*)$ of $f^*$ is the set of lines in $(\mathcal Y, \delta )$ and the domain $\operatorname {dom}(f_*)$ of $f_*$ is the set of lines in $(\mathcal T, d)$ whose ends both have unbounded f-images. Note that the image $\operatorname {im}(f^*)$ is $\operatorname {dom}(f_*)$ ; moreover, $f^*$ and $f_*$ are inverses of each other. Both lifting and projecting functions are canonical: $f^* = g^*$ and $f_* = g_*$ for any equivariant maps $f,g \colon (\mathcal T, d) \to (\mathcal Y, \delta )$ since $f,g$ will be a bounded $\delta $ -distance from some equivariant PL-map; for lack of better notation, we still denote the functions by $f^*, f_*$ despite this independence.

Alternatively, we view $f^*$ and $f_*$ as functions on the sets of $\mathcal F$ -orbits of lines. We can equip these sets with a natural topology. The set $\mathbb R(\mathcal Y, \delta )$ of $\mathcal F$ -orbits of lines in $(\mathcal Y, \delta )$ has the following topology: for any $p,q \in \mathcal Y$ , let $U[p,q]$ be the $\mathcal F$ -orbit of lines that contain a translate of $[p,q]$ ; the collection $\{ U[p,q]:p,q \in \mathcal Y \}$ is a basis for the space of ( $\underline{\mathcal{F}}$ -orbits of) lines. This space is well-defined for the equivariant homothetic class of $(\mathcal Y, \delta )$ . The space of lines is also well-defined for the free splitting $\mathcal T$ and denoted $\mathbb R(\mathcal T)$ .

Henceforth, we identify $\mathbb R(\mathcal Y, \delta )$ with a subspace of $\mathbb R(\mathcal T)$ using the canonical embedding $f^*$ .

Now assume $\mathcal T'$ is a free splitting of $\mathcal F$ with $\mathcal F[\mathcal T] = \mathcal F[\mathcal T']$ and let $f\colon \mathcal T \to \mathcal T'$ be an equivariant PL-map. The folds in the factorization of f never identify points in the same $\mathcal F$ -orbit. For $[l] \in \mathbb R(\mathcal T)$ , each end of l has unbounded f-image (i.e., $\operatorname {dom}(f_*) = \mathbb R(\mathcal T)$ ); so $f_* \colon \mathbb R(\mathcal T) \to \mathbb R(\mathcal T')$ is a canonical homeomorphism (with inverse $f^*$ ). Similarly, if $g \colon \mathcal T \to \mathcal T$ is a $\psi $ -equivariant PL-map for some automorphism $\psi \colon \mathcal F \to \mathcal F$ , then $g_*, g^* \colon \mathbb R(\mathcal T) \to \mathbb R(\mathcal T)$ are canonical homeomorphisms for $[\psi ]$ .

A lamination in $(\mathcal Y, \delta )$ (resp. $\mathcal T$ ) is a nonempty closed subset of $\mathbb R(\mathcal Y, \delta )$ (resp. $\mathbb R(\mathcal T)$ ); when the $\mathcal F$ -forest in question is clear, we say lamination with no qualifier. An element of a lamination is called a leaf; a leaf segment of a lamination $\Lambda $ is a nondegenerate closed interval in a line representing a leaf of $\Lambda $ . A lamination is minimal if each leaf is dense in the lamination; a lamination is perfect if it has no isolated leaves.

Let $[l]$ be a line and $\Lambda $ a lamination in $\mathbb R(\mathcal Y, \delta )$ (or $\mathbb R(\mathcal T)$ ). A sequence $[l_m]_{m \ge 0}$ in the space of lines weakly limits to $[l]$ if some subsequence converges to $[l]$ ; we say $[l]$ is a weak limit of the sequence. The sequence $[l_m]_{m \ge 0}$ weakly limits to $\Lambda $ if it weakly limits to every leaf of $\Lambda $ . The ‘weak’ terminology is used to highlight that the space of lines is not Hausdorff – a convergent sequence may have multiple limits!

More generally, a sequence $[p_m, q_m]_{m \ge 0}$ of intervals converges to $[l]$ if, for any closed interval $[a,b] \subset l$ , $[p_m, q_m]$ contains a translate of $[a,b]$ for $m \gg 1$ (i.e., for large enough m) – precisely, there is an $M[a,b] \ge 1$ such that $U[a, b]$ contains $U[p_m,q_m]$ for $m \ge M[a,b]$ . Again, a sequence of intervals weakly limits to $[l]$ if some subsequence converges to $[l]$ , and it weakly limits to $\Lambda $ if it weakly limits to every leaf of $\Lambda $ .

2.1 Constructing limit forests (1)

This is a summary of [Reference Gaboriau, Levitt and Lustig12, Appendix]; the reader is invited to read that appendix.

Fix an automorphism $\psi \colon \mathcal F \to \mathcal F$ with an expanding irreducible train track $\tau \colon \mathcal T \to \mathcal T$ . Set and let $d_\tau $ be the eigenmetric on $\mathcal T$ for $[\tau ]$ . For $m \ge 0$ , let $d_m$ be the pullback of $\lambda ^{-m}d_\tau $ via $\tau ^m$ :

By definition, the pullback $d_m$ is an $\mathcal F$ -invariant (not necessarily convex) pseudometric on $\mathcal T$ whose quotient metric space is equivariantly isometric to $(\mathcal T\psi ^m, \lambda ^{-m}d_\tau )$ . The $\lambda $ -Lipschitz property for $\tau $ with respect to $d_\tau $ implies the sequence of pseudometrics $d_m$ is (pointwise) monotone decreasing. The limit $d_\infty $ is an $\mathcal F$ -invariant pseudometric on $\mathcal T$ , the quotient metric space $(\mathcal T_\infty , d_\infty )$ is an $\mathcal F$ -forest, and the $\psi $ -equivariant $\lambda $ -Lipschitz train track $\tau $ induces a $\psi $ -equivariant $\lambda $ -homothety $h \colon (\mathcal T_\infty , d_\infty ) \to (\mathcal T_\infty , d_\infty )$ ; in particular, the equivariant metric surjection $\pi \colon (\mathcal T, d_\tau ) \to (\mathcal T_\infty , d_\infty )$ semiconjugates $\tau $ to h: $\pi \circ \tau = h \circ \pi $ .

As $\tau $ is a train track, the restriction of $\pi $ to any edge of $\mathcal T$ is an isometric embedding. So $\mathcal T_\infty $ is not degenerate. In fact, the $\pi $ -image of any edge of $\mathcal T$ can be extended to an axis for a $\mathcal T_\infty $ -loxodromic element in $\mathcal F$ . Thus, the $\mathcal F$ -forest $(\mathcal T_\infty , d_\infty )$ is minimal, and the uniqueness of h follows from [Reference Culler and Morgan6, Theorem 3.7]. Finally, the minimal $\mathcal F$ -forest $(\mathcal T_\infty , d_\infty )$ has trivial arc stabilizers. This sketches the first case of Proposition 1.2. The $\mathcal F$ -forest is the (forward) limit forest for $[\tau ]$ .

2.2 Stable laminations (1)

The first part of this section is mostly adapted from Section 1 of [Reference Bestvina, Feighn and Handel2]. The following definition of stable laminations is from [Reference Bestvina, Feighn and Handel2, Definition 1.3].

Fix an automorphism $\psi \colon \mathcal F \to \mathcal F$ with an expanding irreducible train track $\tau \colon \mathcal T \to \mathcal T$ . Set , let $d_\tau $ be the eigenmetric on $\mathcal T$ for $[\tau ]$ , and pick an edge $e \subset \mathcal T$ . Expanding irreducibility implies the interval $\tau ^k(e)$ contains at least three translates of e for some $k \ge 1$ . By the intermediate value theorem, $\tau ^k(p) = x \cdot p$ for some $x \in \mathcal F$ and $p \in e$ . Recall that edges are open intervals; since the restriction of $x^{-1} \cdot \tau ^k$ to the edge e is an expanding $\lambda ^k$ -homothetic embedding $e \to \mathcal T$ (with respect to $d_\tau $ ) that fixes p and has e in its image, we can extend e to a line $l_p \subset \mathcal T$ by iterating $x^{-1} \cdot \tau ^k$ . By construction, the restriction of $x^{-1} \cdot \tau ^k$ to $l_p$ is a $\lambda ^k$ -homothety $l_p \to l_p$ with respect to the eigenmetric $d_\tau $ for $[\tau ]$ ; the $\mathcal F$ -orbit $[l_p]$ is an eigenline of $[\tau ^k]$ based at $[p]$ (in $\mathcal F \backslash \mathcal T$ ). A $\underline{\mathrm{stable}\ \mathcal{T}\mathrm{-lamination}}$ $\Lambda ^+$ for $[\tau ]$ is the closure of an eigenline of $[\tau ^k]$ for some $k \ge 1$ . By $\phi $ -equivariance of $\tau $ , the restriction of $\tau $ to l representing a leaf of a stable lamination $\Lambda ^+$ is a $\lambda $ -homothetic embedding. In fact, $[\tau ]$ maps eigenlines to eigenlines, and the image is also a stable lamination for $[\tau ]$ .

As the transition matrix $A[\tau ]$ is irreducible, it is a block transitive permutation matrix, and the ‘first return’ matrix for each block is primitive (i.e., has a positive power). There is a bijective correspondence between the stable laminations for $[\tau ]$ and the blocks of $A[\tau ]$ . In particular, there are finitely many stable laminations for $[\tau ]$ , these laminations are pairwise disjoint, and $\tau _*$ transitively permutes them [Reference Bestvina, Feighn and Handel2, Lemma 1.2]. By finiteness, their union $\mathcal L^+[\tau ]$ is a lamination and is called the system of stable laminations for $[\tau ]$ .

2.2.3 Iterated turns

We have already shown how iterating an edge in $\mathcal T$ by the train track $\tau $ produces the system of stable laminations $\mathcal L^+[\tau ]$ . Later, we will consider how $\mathcal L^+[\tau ]$ determines laminations in (a free splitting of) the subgroup system ${\mathcal {G}}[\mathcal Y_\tau ]$ .

Let $\mathcal T'$ be a free splitting of $\mathcal F$ whose free factor system $\mathcal F[\mathcal T']$ is trivial. Then there is an equivariant PL-map $f \colon (\mathcal T', d') \to (\mathcal T, d_\tau )$ . Let $\gamma $ be a line in $(\mathcal Y_\tau , d_\infty )$ , $\pi ^*(\gamma )$ its lift to $(\mathcal T, d_\tau )$ , and $f^*(\pi ^*(\gamma ))$ its lift to $(\mathcal T', d')$ . Denote the ends of $\gamma $ by $\varepsilon _i~(i = 1,2)$ . Let $T \subset \mathcal T$ be the component containing $\pi ^*(\gamma )$ , and $T' \subset \mathcal T'$ , $Y_\tau \subset \mathcal Y_\tau $ , and $F \subset \mathcal F$ be the matching components. Denote the first return maps of $\tau $ , h and $\psi $ on T, $Y_\tau $ and F by $\tilde \tau, \tilde h$ and $\varphi $ respectively. For the rest of the section, redefine $\lambda $ to be the stretch factor of the expanding homothety $\tilde h$ .

Suppose $\circ $ is a point on the line $\gamma $ with a nontrivial stabilizers . Let $d_i~(i = 1,2)$ be the direction at $\circ $ containing $\varepsilon _i$ . By Gaboriau–Levitt index theory, $\tilde h^j(\circ ) = y \cdot \circ $ and $\tilde h^j(d_i) = ys_i \cdot d_i$ for some $y \in F$ , $s_i \in G_\circ $ , and minimal $j \ge 1$ . Since F acts on $Y_\tau $ with trivial arc stabilizers, the elements $ys_1, ys_2$ are unique and $s_1^{-1}s_2 \in G_\circ $ is independent of the chosen $y \in F$ .

Set to be the trivial element and for $m \ge 0$ . Let $T'(G)$ be the characteristic convex subset of $T'$ for a nontrivial subgroup $G \le F$ . Since $T'$ is simplicial, the characteristic convex subset $T'(G)$ is closed, and we have the closest point retraction $T' \to T'(G)$ ; it extends uniquely to the ends-completions. Let $q_{i,m}'$ be the closest point projection of $f^*(\pi ^*(\tilde h_*^{mj}(\varepsilon _i)))$ to $T'(\varphi ^{mj}(G_\circ ))$ . Set and to be $\psi _\circ $ -equivariant maps for an automorphism $\psi _\circ \colon F \to F$ in the outer class $[\varphi ^j]$ . As $h_\circ $ fixes $\circ $ , we get $\psi _\circ (G_\circ ) = G_\circ $ and $y_{m}^{-1} \cdot T'(\varphi ^{mj}(G_\circ ))$ is the characteristic convex subset for $\psi _\circ ^{m}(G_\circ ) = G_\circ $ . Thus, is in $T'(G_\circ )$ for $i = 1,2$ and $m \ge 0$ . The interval $[q_{1,m}, q_{2,m}]$ in $T'(G_\circ )$ (i.e., the closest point projection of $f^*(\pi ^*(h_\circ ^m(\gamma )))$ ) is the turn in $f^*(\pi ^*(h_\circ ^m(\gamma )))$ about $T'(G_\circ )$ .

Since $h_\circ (d_1) = d_1$ , the ends $h_{\circ *}^m(\varepsilon _1)~(m \ge 0)$ are in fact ends of $d_1$ . If $h_{\circ *}(\varepsilon _1) = \varepsilon _1$ , then the sequence $(q_{1,m})_{m \ge 0}$ is constant. Otherwise, the ends $h_{\circ *}^m(\varepsilon _1)$ are distinct for $m \ge 0$ . Let $\gamma _{1,m}$ be the line in $d_1$ determined by $h_{\circ *}^{m+1}(\varepsilon _1)$ and $h_{\circ *}^m(\varepsilon _1)$ . As $h_\circ $ is an expanding homothety, the distance $d_\infty (\circ , \gamma _{1,m})> 0$ from $\circ $ to $\gamma _{1,m}$ grows exponentially in m. So $d_\infty (\circ , \gamma _{1,M_1})> 2C[\pi \circ f]$ for some minimal $M_1 \ge 0$ , and the line $f^*(\pi ^*(\gamma _{1,m}))$ is disjoint from $T'(G_\circ )$ for $m \ge M_1$ by bounded cancellation (see Figure 2). In particular, the ends $f^*(\pi ^*(h_{\circ *}^{m+1}(\varepsilon _1)))$ and $f^*(\pi ^*(h_{\circ *}^m(\varepsilon _1)))$ have the same closest point projection to $T'(G_\circ )$ , and the sequence $(q_{1,m})_{m \ge M_1}$ is constant.

Figure 2 For $m \ge M_1$ , the line $f^*(\pi ^*(\gamma _{1,m}))$ cannot intersect $T'(G_\circ )$ .

Since $h_\circ (d_2) = s_1^{-1}s_2 \cdot d_2$ , the ends $f^*(\pi ^*(h_{\circ *}^{m+1}(\varepsilon _2)))$ and $\psi _\circ ^{m}(s_1^{-1}s_2) \cdot f^*(\pi ^*(h_{\circ *}^{m}(\varepsilon _2)))$ have the same closest point projection to $T'(G_\circ )$ for $m \gg 1$ by similar bounded cancellation reasoning (i.e., $q_{2,m+1} = \psi _\circ ^m(s_1^{-1}s_2)\cdot q_{2,m}$ for some minimal $M_2 \ge 0$ and all $m \ge M_2$ ).

Set $M = \max (M_1, M_2)$ . The sequence $[q_{1,M+m}, q_{2,M+m}]_{m \ge 0}$ of intervals is well-defined for the line $\gamma $ and point $\circ \in \gamma $ as $M_1$ and $M_2$ were chosen minimally. An iterated turn over $T'(G_\circ )$ rel. $\left .\psi _\circ \right |{}_{G_\circ }$ is any such sequence of intervals. More generally, we define an iterated turn over T rel. $\varphi $ : pick arbitrary points $p_i \in T~(i=1,2)$ and elements $x_i \in F$ ; set and for $m \ge 0$ ; the sequence $[p_{1,m}, p_{2,m}]_{m \ge 0}$ is the iterated turn denoted by $(p_1, p_2: x_1, x_2; \varphi )_{T}$ . Any iterated turn $(p_1, p_2: x_1, x_2; \varphi )_{T}$ translates to a unique normal form $(p_1, p_2: \epsilon , x_1^{-1}x_2; \tilde \varphi )_{T}$ with $\tilde \varphi \colon y \mapsto x_1^{-1}\varphi (y)x_1$ .

We now characterize the growth of an iterated turn over T rel. $\varphi $ :

Proposition 2.4. Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism, $\tau \colon \mathcal T \to \mathcal T$ an expanding irreducible train track for $\psi $ with eigenmetric $d_\tau $ , and $(\mathcal Y_\tau , d_\infty )$ the limit forest for $[\tau ]$ . Choose a nondegenerate component $T \subset \mathcal T$ , corresponding components $F \subset \mathcal F$ , $Y_\tau \subset \mathcal Y_\tau $ , and a positive iterate $\psi ^k$ that preserves F. Let $\tilde h \colon (Y_\tau , d_\infty ) \to (Y_\tau , d_\infty )$ be the $\varphi $ -equivariant $\lambda $ -homothety, where $\varphi $ is in the outer automorphism $[\left .\psi ^k\right |{}_F]$ and . Finally, for $i = 1,2$ , pick $p_i \in T$ and $x_i \in F$ .

The point in $(T\varphi ^m, \lambda ^{-m}d_\tau )$ converges to $\star _i$ in $(\overline {Y}_\tau , d_\infty )$ as $m \to \infty $ , where $\star _i$ is the unique fixed point of $x_i^{-1} \cdot \tilde h$ in the metric completion $(\overline Y_\tau , d_\infty )$ ; concretely:

$$\begin{align*}\lim_{m \to \infty} \lambda^{-m} d_\tau(p_{1,m}, p_{2,m}) = d_\infty(\star_1, \star_2).\end{align*}$$

If $x_1^{-1}x_2$ fixes $\star _1$ , then $\star _1 = \star _2$ and the $m^{th}$ term $[p_{1,m}, p_{2,m}]$ of the iterated turn $(p_1, p_2: x_1, x_2; \varphi )_T$ has $d_\tau $ -length $ \le (m+1)A$ for some constant $A \ge 1$ . Otherwise, $\star _1 \neq \star _2$ , and the iterated turn has arbitrarily long leaf segments of $\mathcal L^+[\tau ]$ .

The limit $[\star _1, \star _2] \subset \overline {Y}_\tau $ of an iterated turn is independent of the points $p_1, p_2 \in T$ . Thus, we introduce the notion of an algebraic iterated turn over F rel. $\varphi $ , denoted $(x_1, x_2; \varphi )_F$ .

Proof. Let $p_1, p_2 \in T$ , $x_1, x_2 \in F$ and $\pi \colon (\mathcal T, d_\tau ) \to (\mathcal Y_\tau , d_\infty )$ be the constructed equivariant metric PL-map. For $i = 1,2$ , set and for $m \ge 0$ . Recall that T and $T\varphi ^m$ are the same pretrees but with different actions; thus, in $T\varphi ^m$ , we have $p_{i,m} = \varphi ^{-1}(x_i)\cdots \varphi ^{-m}(x_i) \cdot p_i$ for $m \ge 0$ . As $\pi \colon (T, d_\tau ) \to (Y_\tau , d_\infty )$ is an equivariant metric PL-map, so is the composition

$$\begin{align*}\pi_m \colon (T\varphi^m, \lambda^{-m}d_\tau) \overset{\pi}\longrightarrow (Y_\tau\varphi^m, \lambda^{-m}d_\infty) \overset{\tilde h^{-m}}\longrightarrow (Y_\tau, d_\infty). \end{align*}$$

The point $p_i$ in $(T, d_\tau )$ projects (via $\pi $ ) to $\pi (p_i)$ in $(Y_\tau , d_\infty )$ ; the point $p_{i,m}$ in $(T \varphi ^m, \lambda ^{-m}d_\tau )$ projects (via $\pi _m$ ) to

in $(Y_\tau , d_\infty )$ for $m \ge 1$ – in the last line, $x_i^{-1} \cdot \tilde h$ is a $\lambda $ -homothety $(Y_\tau , d_\infty ) \to (Y_\tau , d_\infty )$ . Since $(x_i^{-1} \cdot \tilde h)^{-1}$ is contracting, the point $\pi _m(p_{i,m})$ converges (as $m \to \infty $ ) to the unique fixed point $\star _i$ of $(x_i^{-1}\cdot \tilde h)^{-1}$ (and $x_i^{-1} \cdot \tilde h$ ) in the metric completion $(\overline {Y}_\tau , d_\infty )$ by the contraction mapping theorem; note that $x_1^{-1}x_2 \cdot \star _1 = \star _1$ if and only if $\star _1 = \star _2$ . Thus, the $\pi _m$ -projection of the point $p_{i,m}$ in $(T\varphi ^{m}, \lambda ^{-m} d_\tau )$ converges (as $m \to \infty $ ) to $\star _i$ in $(\overline {Y}_\tau , d_\infty )$ ; in particular,

$$\begin{align*}\lim_{m \to \infty} \lambda^{-m} d_\infty(\pi(p_{1,m}), \pi(p_{2,m})) = \lim_{m \to \infty} d_\infty(\pi_m(p_{1,m}), \pi_m(p_{2,m})) = d_\infty(\star_1, \star_2). \end{align*}$$

Let $\tilde \tau \colon T \to T$ be the $\varphi $ -equivariant translate of a component of $\tau ^k$ . The interval $[p_{1,m}, p_{2,m}] \subset T$ , the $m^{th}$ term in $(p_1, p_2:x_1,x_2;\varphi )_T$ , is covered by these $2m+1$ intervals:

$$\begin{align*}\begin{aligned} &\varphi^{m-1}(x_1) \cdots \varphi(x_1) \cdot [x_1 \cdot p_1, \tilde \tau(p_1)], \ldots,~\varphi^{m-1}(x_1) \cdot [\tilde \tau^{m-2}(x_1 \cdot p_1), \tilde \tau^{m-1}(p_1)],\\ &[\tilde \tau^{m-1}(x_1\cdot p_1), \tilde \tau^{m}(p_1)],~[\tilde \tau^{m}(p_1), \tilde \tau^{m}(p_2)],~[\tilde \tau^{m}(p_2), \tilde \tau^{m-1}(x_2 \cdot p_2)], \\ &\varphi^{m-1}(x_2) \cdot [\tilde \tau^{m-1}(p_2), \tilde \tau^{m-2}(x_2 \cdot p_2)],\ldots,~\varphi^{m-1}(x_2) \cdots \varphi(x_2) \cdot [\tilde \tau(p_2), x_2\cdot p_2]. \end{aligned} \end{align*}$$

Set and .

Recall that $\underset {m' \to \infty }\lim \lambda ^{-m'} d_\tau (\tilde \tau ^{m'}(p_{1,m}), \tilde \tau ^{m'}(p_{2,m})) = d_\infty (\pi (p_{1,m}), \pi (p_{2,m}))$ . For $m' \ge 0$ , we get a similar covering of $[p_{1,m+m'}, p_{2,m+m'}]$ by $2m'+1$ intervals with the ‘middle’ $[\tilde \tau ^{m'}(p_{1,m}), \tilde \tau ^{m'}(p_{2,m})]$ . Since $\tilde \tau $ is $\lambda $ -Lipschitz with respect to $d_\tau $ , the sum of the $d_\tau $ -lengths of all intervals but the middle in this covering is $\le \lambda ^{m'} 2D'$ . By the triangle inequality,

$$\begin{align*}\lambda^{-(m+m')} \left| d_\tau(p_{1,m+m'}, p_{2,m+m'}) - d_\tau(\tilde \tau^{m'}(p_{1,m}), \tilde \tau^{m'}(p_{2,m})) \right| \le \lambda^{-m} 2D'. \end{align*}$$

Fix $\epsilon> 0$ ; then $\lambda ^{-m} 2D' < \epsilon $ and $\left | \lambda ^{-m} d_\infty (\pi (p_{1,m}), \pi (p_{2,m})) - d_\infty (\star _1, \star _2) \right | < \epsilon $ for some $m \gg 1$ .

Similarly,

$$\begin{align*}\begin{aligned} \lambda^{-m}\left| \lambda^{-m'} d_\tau(\tilde \tau^{m'}(p_{1,m}), \tilde \tau^{m'}(p_{2,m})) - d_\infty(\pi(p_{1,m}), \pi(p_{2,m})) \right| &< \epsilon \\ \text{and } \left|\lambda^{-(m+m')} d_\tau(p_{1,m+m'}, p_{2,m+m'}) - d_\infty(\star_1, \star_2)\right| &< 3\epsilon \text{ for } m' \gg 1; \\ \text{that is,}~ \lim_{m \to \infty} \lambda^{-m} d_\tau(p_{1,m}, p_{2,m}) = d_\infty(\star_1, \star_2).& \end{aligned} \end{align*}$$

Let $N(u,v)$ be the number of vertices in an interval $(u,v) \subset T$ ; set N to be the maximum of $N(p_1,p_2)$ , $N(x_1 \cdot p_1, \tilde \tau (p_1))$ , and $N(\tilde \tau (p_2), x_2 \cdot p_2)$ . As $\tilde \tau $ is a train track, the interval $[p_{1,m}, p_{2,m}]$ is covered by $(2m+1)(N+1)$ leaf segments.

Suppose $\star _1 = \star _2$ . We claim that any leaf segment (of $\mathcal L^+[\tau ]$ ) in $[p_{1,m}, p_{2,m}]$ has uniformly (in $m\ge 0$ ) bounded $d_\tau $ -length – this implies $[p_{1,m}, p_{2,m}]$ has $d_\tau $ -length $\le (2m+1)(N+1)B$ for some bounding constant $B \ge 1$ . We mimic Case 2 from the proof of Proposition 2.2. For the contrapositive, suppose some term $[p_{1,m}, p_{2,m}]$ has a leaf segment with $d_\tau $ -length $L> 2(C[\pi ] + D')$ . By the train track property, bounded cancellation and interval covering, $[p_{1, m+m'}, p_{2, m+m'}]$ has a leaf segment with $d_\tau $ -length $\ge \lambda ^{m'} (L - 2C[\pi ] - 2D')$ for $m' \ge 0$ ; in $(T\varphi ^{m+m'}, \lambda ^{-(m+m')}d_\tau )$ , $[p_{1, m+m'}, p_{2, m+m'}]$ has length $ \ge \lambda ^{-m}(L-2C[\pi ] - 2D')$ . In the limit (as $m' \to \infty $ ), $d_\infty (\star _1, \star _2) \ge \lambda ^{-m}(L-2C[\pi ] - 2D')> 0$ .

Suppose $\star _1 \neq \star _2$ . Set ; then $\lambda ^{-m}d_\tau (p_{1,m}, p_{2,m})> L$ for some $m \gg 1$ . By the pigeonhole principle, the interval $[p_{1,m}, p_{2,m}]$ has a leaf segment with $d_\tau $ -length $\frac {\lambda ^{m}L}{(2m+1)(N+1)}$ , which can be arbitrarily large (in m).