2.1. Train track maps

We recall some basic definitions from [Reference Bestvina and HandelBH92]. Identify ${\mathbb {F}_{n}}$ with $\pi _1(\mathrm {R}_n, \ast )$ , where $\mathrm {R}_n$ is a rose with n petals. A marked graph G is a graph of rank n, all of whose vertices have valence at least three, equipped with a homotopy equivalence $m\colon \, \mathrm {R}_n \to G$ called a marking.

A length vector on G is a vector $\lambda \in {\mathbb R}^{|EG|}$ that assigns a positive number, that is, a length, to every edge of G. The volume of G with respect to $\lambda $ is the total length of all the edges of G. This induces a path metric on G where the length of an edge e is $\lambda (e)$ .

A direction d based at a vertex $v \in G$ is an oriented edge of G with initial vertex v. A turn is an unordered pair of distinct directions based at the same vertex. A train track structure on G is an equivalence relation on the set of directions at each vertex $v \in G$ . The classes of this relation are called gates. A turn $(d,d')$ is legal if d and $d'$ do not belong to the same gate, it is called illegal otherwise. A path is legal if it only crosses legal turns.

A map $f \colon \, G \to G'$ between two graphs is called a morphism if it is locally injective on open edges and sends vertices to vertices. If G and $G'$ are metric graphs, then we can homotope f relative to vertices such that it is linear on edges. Similarly, for an ${\mathbb R}$ -tree T, a map $\tilde G\to T$ from the universal cover of G is a morphism if it is injective on open edges. To a morphism $f\colon \, G \to G'$ we associate the transition matrix as follows: Enumerate the (unoriented) edges $e_1,e_2,\cdots ,e_m$ of G and $e_1', e_2', \cdots , e_n'$ of $G'$ . Then the transition matrix M has size $n\times m$ and the $ij$ -entry is the number of times $f(e_j)$ crosses $e_i'$ , that is, it is the cardinality of the set $f^{-1}(x)\cap e_j$ for a point x in the interior of $e_i'$ . If f is in addition a homotopy equivalence, then f is a change-of-marking.

A homotopy equivalence $f\colon \, G \to G$ induces an outer automorphism of $\pi _1(G)$ and hence an element $\phi $ of $\operatorname {\mathrm {Out}}({\mathbb {F}_{n}})$ . If f is a morphism, then we say that f is a topological representative of $\phi $ . A topological representative $f \colon \, G\to G$ induces a train track structure on G as follows: The map f determines a map $Df$ on the directions in G by defining $Df(e)$ to be the first (oriented) edge in the edge path $f(e)$ . We then declare $e_1\sim e_2$ if $(Df)^k(e_1)=(Df)^k(e_2)$ for some $k\geq 1$ .

A topological representative $f\colon \, G \to G$ is called a train track map if every vertex has at least two gates, and f maps legal turns to legal turns and legal paths (equivalently, edges) to legal paths. Equivalently, every positive power $f^k$ is a topological representative. If f is a train track map with transition matrix M, then the transition matrix of $f^k$ is $M^k$ for every $k\geq 1$ . If M is primitive, that is, $M^k$ has positive entries for some $k \ge 1$ , then Perron–Frobenius theory implies that there is an assignment of positive lengths to all the edges of G so that f uniformly expands lengths of legal paths by some factor $\lambda>1$ , called the stretch factor of f.

If $\sigma $ is a path (or a circuit) in G, we denote by $[\sigma ]$ the reduced path homotopic to $\sigma $ (rel endpoints if $\sigma $ is a path). A path or circuit $\sigma $ in G is called a periodic Nielsen path if $[f^k(\sigma )]=\sigma $ for some $k\geq 1$ . If $k=1$ , then $\sigma $ is a Nielsen path. A Nielsen path that cannot be written as a concatenation of nontrivial Nielsen paths is called an indivisible Nielsen path, denoted INP.

The following lemma is an important property of train track maps. For a very rudimentary form, see [Reference Bestvina and HandelBH92, Lemma 3.4] showing that INPs have exactly one illegal turn, and for a more involved version see [Reference Bestvina, Feighn and HandelBFH97] (some details can also be found in [Reference Kapovich and LustigKL14, Proposition 3.27, 3.28]). We will need it for the proof of Lemma 4.8 and include a proof here.

We will use the lemma in the situation that h has no periodic INPs, in which case the conclusion is that whenever $\gamma $ is not legal, then $[h^{R}(\gamma )]$ has fewer illegal turns than $\gamma $ .

2.2. Outer space and its boundary

An ${\mathbb {F}_{n}}$ -tree is an $\mathbb {R}$ -tree with an isometric action of ${\mathbb {F}_{n}}$ . An ${\mathbb {F}_{n}}$ -tree T has dense orbits if some (every) orbit is dense in T. An ${\mathbb {F}_{n}}$ -tree is called very small if the action is minimal, arc stabilizers are either trivial or maximal cyclic and tripod stabilizers are trivial. We review the definition of Outer space first introduced in [Reference Culler and VogtmannCV86].

Unprojectivized Outer space, denoted by ${\text {cv}_{n}} $ , is the set of free, minimal and simplicial ${\mathbb {F}_{n}}$ -trees. By considering the quotient graphs, ${\text {cv}_{n}} $ is also equivalently the set of marked metric graphs, that is, the set of triples $(G,m,\lambda )$ , where G is a graph of rank n with all valences at least $3$ , $m\colon \, \mathrm {R}_n \to G$ is a marking and $\lambda $ is a positive length vector on G. By [Reference Culler and MorganCM87], the map of ${\text {cv}_{n}} \to {\mathbb R}^{{\mathbb {F}_{n}}}$ given by $T \mapsto (\left \lVert g\right \rVert _T)_{g \in {\mathbb {F}_{n}}}$ , where $\left \lVert g\right \rVert _T$ is the translation length of g in T, is an inclusion. This endows ${\text {cv}_{n}} $ with a topology. The closure ${\overline {\text {cv}}_{n}}$ in ${\mathbb R}^{{\mathbb {F}_{n}}}$ is the space of very small ${\mathbb {F}_{n}}$ -trees [Reference Bestvina and FeighnBF94, Reference Cohen and LustigCL95]. The boundary $\partial {\text {cv}_{n}} = {\overline {\text {cv}}_{n}}-{\text {cv}_{n}} $ consists of very small trees that are either not free or not simplicial.

Culler Vogtmann’s Outer space, ${\text {CV}_{n}} $ , is the image of ${\text {cv}_{n}} $ in the projective space ${\mathbb P}{\mathbb R}^{{\mathbb {F}_{n}}}$ . Elements in ${\text {CV}_{n}} $ can also be described as free, minimal, simplicial ${\mathbb {F}_{n}}$ -trees with unit covolume. Topologically, ${\text {CV}_{n}} $ is a complex made up of simplices with missing faces, where there is an open simplex for each marked graph $(G,m)$ spanned by positive length vectors on G of unit volume. The closure ${\overline {\text {CV}}_{n}}$ of ${\text {CV}_{n}} $ in ${\mathbb P}{\mathbb R}^{{\mathbb {F}_{n}}}$ is compact and the boundary $\partial {\text {CV}_{n}} = {\overline {\text {CV}}_{n}} -{\text {CV}_{n}} $ is the projectivization of $\partial {\overline {\text {cv}}_{n}}$ .

The spaces ${\text {cv}_{n}} $ and ${\text {CV}_{n}} $ and their closures are equipped with a natural (right) action by $\operatorname {\mathrm {Out}}({\mathbb {F}_{n}})$ . That is, for $\Phi \in \operatorname {\mathrm {Out}}({\mathbb {F}_{n}})$ and $T \in {\overline {\text {cv}}_{n}}$ the translation length function of $T \Phi $ on ${\mathbb {F}_{n}}$ is $\left \lVert g\right \rVert _{T \Phi } = \left \lVert \phi (g)\right \rVert _T$ , where $\phi $ is any lift of $\Phi $ to $\operatorname {\mathrm {Aut}}({\mathbb {F}_{n}})$ .

2.3. Laminations, currents and nonuniquely ergodic trees

In [Reference Bestvina, Feighn and HandelBFH00], Bestvina, Feighn and Handel defined a dynamical invariant called the attracting lamination associated to a train track map. In this article, we will consider the more modern definition of a lamination as given in [Reference Coulbois, Hilion and LustigCHL08a].

Let $\partial {\mathbb {F}_{n}}$ denote the Gromov boundary of ${\mathbb {F}_{n}}$ , and let $\Delta $ be the diagonal in $\partial {\mathbb {F}_{n}} \times \partial {\mathbb {F}_{n}}$ . The double boundary of ${\mathbb {F}_{n}}$ is $\partial ^2 {\mathbb {F}_{n}} = (\partial {\mathbb {F}_{n}} \times \partial {\mathbb {F}_{n}} - \Delta ) / \mathbb {Z}_2$ , which parametrizes the space of unoriented bi-infinite geodesics in a Cayley graph of ${\mathbb {F}_{n}}$ . By an (algebraic) lamination, we mean a nonempty, closed and ${\mathbb {F}_{n}}$ -invariant subset of $\partial ^2 {\mathbb {F}_{n}}$ .

Associated to $T\in {\overline {\text {cv}}_{n}}$ is a dual lamination $L(T)$ , defined as follows in [Reference Coulbois, Hilion and LustigCHL08b]. For $\epsilon> 0$ , let

$$\begin{align*}L_{\epsilon}(T) = \overline{\{(g^{-\infty}, g^{\infty}) \quad | \quad \left\lVert g\right\rVert_T < \epsilon, g \in {\mathbb{F}_{n}} \}}, \end{align*}$$

so $L_{\epsilon }(T)$ is a lamination and set $L(T) = \bigcap _{\epsilon>0}L_{\epsilon }(T)$ . Elements of $L(T)$ are called leaves. For trees in ${\text {cv}_{n}} $ , $L(T)$ is empty.

A current is an additive, nonnegative, ${\mathbb {F}_{n}}$ -invariant function on the set of compact open sets in $\partial ^2 {\mathbb {F}_{n}}$ . Equivalently, it is an ${\mathbb {F}_{n}}$ -invariant Radon measure on the $\sigma $ -algebra of Borel sets of $\partial ^2 {\mathbb {F}_{n}}$ . Let ${\mathrm {Curr}_{n}}$ denote the space of currents, equipped with the weak* topology. The quotient space of ${\mathbb {P}\mathrm {Curr}_{n}}$ of projectivized currents (i.e., homothety classes of nonzero currents) is compact.

For $\mu \in {\mathrm {Curr}_{n}}$ , let $\text {supp}(\mu ) \subset \partial ^2 {\mathbb {F}_{n}}$ denote the support of $\mu $ , which is in fact a lamination. For $T \in {\overline {\text {cv}}_{n}}$ and $\mu \in {\mathrm {Curr}_{n}}$ , if $\text {supp}(\mu ) \subseteq L(T)$ , then we say $\mu $ is dual to T. Denote by $\text {Curr}(T)$ the convex cone of currents dual to T and by ${\mathbb P}\text {Curr}(T)$ the set of projective currents dual to T. ${\mathbb P}\text {Curr}(T)$ is a compact, convex space and its extremal points are called the ergodic currents dual to T. We say T is uniquely ergodic if there is only one projective class of currents dual to T, and nonuniquely ergodic otherwise. In [Reference Coulbois and HilionCH16], the authors show that if $T \in \partial {\text {cv}_{n}} $ has dense orbits, then ${\mathbb P}\text {Curr}(T)$ is the convex hull of at most $3n-5$ projective classes of ergodic currents dual to T.

In [Reference Kapovich and LustigKL09], Kapovich and Lustig established a length pairing, $\langle \cdot , \cdot \rangle $ , between ${\overline {\text {cv}}_{n}}$ and the space of measured currents ${\mathrm {Curr}_{n}}$ . They also showed in [Reference Kapovich and LustigKL10, Theorem 1.1] that for $T \in {\overline {\text {cv}}_{n}}$ and $\mu \in {\mathrm {Curr}_{n}}$ , $\langle T, \mu \rangle = 0$ if and only if $\mu $ is dual to T.

Given two trees T and $T'$ , we say a map $h\colon \, T \to T'$ is alignment-preserving if whenever $b \in T$ is contained in an arc $[a,c] \subset T$ , then $h(b)$ is contained in the arc $[h(a),h(c)]$ .

2.4. Length measures and nonuniquely ergometric trees

Since ${\mathbb R}$ -trees need not be locally compact, classical measure theory is not well suited for them. In [Reference PaulinPau95], a length measure was introduced for ${\mathbb R}$ -trees. See [Reference GuirardelGui00] for details.

A length measure on an ${\mathbb {F}_{n}}$ -tree T is a collection of finite Borel measures $\lambda _I$ for every compact interval I in T such that if $J \subset I$ , then $\lambda _J = (\lambda _I)|_J$ . We require the length measure to be invariant under the ${\mathbb {F}_{n}}$ action. The collection of the Lebesgue measures of the intervals of T is ${\mathbb {F}_{n}}$ -invariant, and this will be called the Lebesgue measure of T. A length measure $\lambda $ is nonatomic or positive if every $\lambda _I$ is nonatomic or positive. If every orbit is dense in some segment of T, then T cannot have an invariant measure with atoms. Further, if T is indecomposable, that is, if for any pair of nondegenerate arcs I and J in T, there exist $g_1,\ldots ,g_m \in {\mathbb {F}_{n}}$ such that $I \subset \bigcup g_iJ$ and $g_i J \cap g_{i+1} J$ is nondegenerate, then every nonzero length measure is positive (in fact, the condition of mixing [Reference GuirardelGui00] suffices).

Let ${\mathcal D}(T)$ be the cone of ${\mathbb {F}_{n}}$ -invariant length measures on T, with projectivization ${\mathbb P}{\mathcal D}(T)$ , that is, the homothety classes of ${\mathbb {F}_{n}}$ -invariant length measures on T. ${\mathbb P}{\mathcal D}(T)$ is a compact convex set and we will call its extremal points the ergodic length measures on T. When T has dense orbits there are at most $3n-4$ such measures for any T (see [Reference GuirardelGui00, Corollary 5.2, Lemma 5.3]) and ${\mathcal D}(T)$ is naturally a subset of $\partial cv_n$ . In fact,

We say T is uniquely ergometric if there is only one projective class of length measures on T, which necessarily is the homothety class of the Lebesgue measure on T. It is called nonuniquely ergometric otherwise.

2.5. Arational trees and the free factor complex

For a tree $T \in {\overline {\text {cv}}_{n}}$ and a free factor H of ${\mathbb {F}_{n}}$ , let $T_H$ denote the minimal H-invariant subtree of T (this tree is unique unless H fixes an arc). A tree $T \in \partial {\text {cv}_{n}} $ is arational if every proper free factor H of ${\mathbb {F}_{n}}$ has a free and simplicial action on $T_H$ . By [Reference ReynoldsRey12], every arational tree is free and indecomposable or it is the dual tree to an arational measured lamination on a surface with one puncture. The arational trees of the first kind are either Levitt type or nongeometric.

Let $\mathcal {AT} \subset \partial {\text {CV}_{n}} $ denote the set of arational trees with the subspace topology. Using Lemma 2.3, define an equivalence relation $\sim $ on $\mathcal {AT}$ by ‘forgetting the metric’, that is, $T \sim T'$ if $T' \in {\mathbb P} {\mathcal D}(T)$ , and endow $\mathcal {AT}/\sim $ with the quotient topology. The following lemma is implicit in [Reference GuirardelGui00] and we include a proof for completeness.

The free factor complex ${\mathcal {FF}_{n}}$ is a simplicial complex whose vertices are given by conjugacy classes of proper free factors of ${\mathbb {F}_{n}}$ and a k-simplex is given by a nested chain $[A_0] \subset [A_1] \subset \cdots \subset [A_k]$ . When the rank $n=2$ the definition is modified and an edge connects two conjugacy classes of rank 1 factors if they have complementary representatives. The free factor complex can be given a metric as follows: Identify each simplex with a standard simplex and endow the resulting space with path metric. By result of [Reference Bestvina and FeighnBF14a], the metric space ${\mathcal {FF}_{n}}$ is Gromov hyperbolic. The Gromov boundary of ${\mathcal {FF}_{n}}$ was identified with $\mathcal {AT}/\sim $ in [Reference Bestvina and ReynoldsBR15] and [Reference HamenstädtHam16].

There is a projection map $\pi \colon \, {\text {CV}_{n}} \to {\mathcal {FF}_{n}}$ defined as follows [Reference Bestvina and FeighnBF14a, Section 3]: for $G \in {\text {CV}_{n}} $ , $\pi (G)$ is the collection of free factors given by the fundamental group of proper subgraphs of G which are not forests. This map is coarsely well defined, that is, $\operatorname {\mathrm {diam}}_{{\mathcal {FF}_{n}}}(\pi (G)) \le K$ for some universal K. Note that if $G, G'$ belong to the same open simplex of ${\text {CV}_{n}} $ , then $\pi (G)=\pi (G')$ , so the projection of a simplex of ${\text {CV}_{n}} $ has uniformly bounded diameter.

3.1. Isomorphism between length measures

In this section, we identify the space of length measures on a folding sequence with that of the limiting tree when it is an arational tree.

Consider a folding sequence of marked graphs of rank n

Let $\tilde G_i$ be the universal cover of $G_i$ , and let $\tilde f_i$ be a lift of $f_i$ . For any positive length measure $(\lambda _i)_i \in {\mathcal D}((G_i)_i)$ , we can realize $(\tilde G_i, \tilde \lambda _i)_i$ as a sequence in ${\text {cv}_{n}} $ , which can be ‘filled in’ by a folding path in ${\text {cv}_{n}} $ (see [Reference Bestvina and FeighnBF14a] for details on folding paths). In particular, $(\tilde G_i,\tilde \lambda _i)_i$ always converges to a point $T \in \partial {\text {cv}_{n}} $ . Furthermore, we have morphisms $h_i \colon \, \tilde G_i \to T$ such that $h_i = h_{i+1} \tilde f_{i+1}$ . With respect to the length measure $\tilde \lambda _i$ , $\tilde f_i$ and $h_i$ restrict to isometries on edges [Reference Bestvina and ReynoldsBR15, Lemma 7.6].

Let $(U_i)_i$ be the sequence of open simplices ${\text {CV}_{n}} $ associated to the sequence $(G_i)_i$ . Recall the projection map $\pi \colon \, {\text {CV}_{n}} \to {\mathcal {FF}_{n}}$ is coarsely well defined on simplices of ${\text {CV}_{n}} $ . We will say the folding sequence $(G_i)_i$ converges to an arational tree T if $\pi (U_i)$ converges to $[T] \in \partial {\mathcal {FF}_{n}}$ .

3.2. Isomorphism between currents

In this section, we state an analogous result identifying the space of currents on an unfolding sequence with the space of currents of a legal lamination associated to a unfolding sequence. We record some definitions from [Reference Namazi, Pettet and ReynoldsNPR14] first.

Consider an unfolding sequence of marked graphs of rank n

Denote the composition $F_i = f_1 \circ \cdots \circ f_i$ . Let $\Omega ^L_{\infty }(G_i)$ denote the set of bi-infinite legal paths in $G_i$ . Define the legal lamination of the unfolding sequence $(G_i)_i$ to be

$$\begin{align*}\Lambda = \bigcap_{i} F_i(\Omega^L_{\infty}(G_i)) \subseteq \Omega_{\infty}^L(G_0).\end{align*}$$

Use the marking on $G_0$ to identify $\partial ^2 \pi _1(G_0)$ with $\partial ^2 {\mathbb {F}_{n}}$ . The preimage, in $\partial ^2 {\mathbb {F}_{n}}$ , of the lift of $\Lambda $ to $\partial ^2 \pi _1(G_0)$ is a lamination $\tilde {\Lambda }$ . We denote by $\text {Curr}(\Lambda )$ the convex cone of currents supported on $\tilde {\Lambda }$ , with projectivization ${\mathbb P}\text {Curr}(\Lambda )$ .

An invariant sequence of subgraphs is a sequence of nondegenerate (i.e., not forests) proper subgraphs $H_i \subset G_i$ such that $f_i$ restricts to a morphism $H_i \to H_{i-1}$ . We will need the following theorem from [Reference Namazi, Pettet and ReynoldsNPR14], which we will include a sketch of the proof for completeness.

Sketch of proof.

The lamination $\Lambda $ consists of biinfinite lines in $G_0$ that lift to every $G_i$ . All such lines are legal, and we view $\Lambda $ as a subset of $(\partial \mathbb F)^2$ invariant under the involution that flips the factors. An element in $\text {Curr}((G_i)_i)$ is a compatible sequence $(\mu _i)_i$ , where $\mu _i$ assigns a nonnegative weight to each edge of $G_i$ . The compatibility condition is that the transition matrix of $G_{i+1}\to G_i$ takes the vector $\mu _{i+1}$ to the vector $\mu _i$ . An alternative way to describe compatibility is this. Let $\tilde G_i$ be the universal cover of $G_i$ , and let $F_{i+1}:\tilde G_{i+1}\to \tilde G_i$ be a lift of the folding map. The weights $\mu _{i+1},\mu _i$ lift to the edges of $\tilde G_{i+1},\tilde G_i$ . If e is an edge of $\tilde G_i$ , then $F_{i+1}^{-1}(e)$ is a finite collection of partial edges in $\tilde G_{i+1}$ , and we complete them to edges, say $e_1,e_2,\cdots ,e_k$ . The compatibility condition is

$$ \begin{align*}\mu_i(e)=\mu_{i+1}(e_1)+\mu_{i+1}(e_2)+\cdots+\mu_{i+1}(e_k)\end{align*} $$

Since $F_{i+1}$ is injective on the leaves of $\Lambda $ , no such leaf passes through more than one of the $e_j$ ’s. Let $\text {Cyl}_{\Lambda }(e)$ be the set of leaves of $\Lambda $ that pass through e and similarly for $\text {Cyl}_\Lambda (e_j)$ . Thus, we have

$$ \begin{align*}\text{Cyl}_\Lambda(e)=\bigsqcup_j \text{Cyl}_\Lambda(e_j)\end{align*} $$

Define measure $\mu $ on the cylinder sets corresponding to edges:

$$ \begin{align*}\mu(\text{Cyl}_\Lambda(e))=\mu_i(e).\end{align*} $$

The compatibility condition states that this measure is additive. The assumption that the sequence has no invariant subgraphs implies that cylinder sets corresponding to edges form a basis for the topology on $\Lambda $ . This allows us to extend $\mu $ to general cylinder sets $\text {Cyl}_{\Lambda }(\gamma )$ , where $\gamma $ is a finite segment of $\Lambda $ in $\tilde {G}_i$ . The key is that folding cannot identify vertices in the same orbit. Thus, there is a uniform upper bound on the number of vertices that map to the same vertex for any $\tilde G_j\to \tilde G_i$ . When $\gamma $ is a segment, the preimages of $\gamma $ in $\tilde G_j$ , for j sufficiently large, will be contained in either single edges or concatenations of two edges (see Lemma 8.2). By the above remark, the number of the length-2 paths is bounded by the combinatorial length of $\gamma $ times the number of vertices in $G_j$ . While we don’t have enough information from $\mu _j$ alone to assign measure to these cylinder sets, we know their contribution goes to 0 as $j\to \infty $ . So for each j, we take the sum of the measures of cylinder sets of the single edges in the preimage of $\gamma $ . This is an increasing and bounded sequence as $j\to \infty $ , so we define $\mu (\text {Cyl}_\Lambda (\gamma ))$ to be the limit, and this is the only possible definition. It is now an exercise to check that $\mu $ induces a premeasure on the semiring of cylinder sets $\text {Cyl}_{\Lambda }(\gamma )$ . Carathéodory’s theorem then implies that $\mu $ extends to a unique (Radon) measure on $\Lambda $ , which finishes the proof.

4.1. The automorphisms

Let ${\mathbb {F}_{7}}=\langle a,b,c,d,e,f,g\rangle $ . Denote by $\bar {x}$ the inverse of $x \in {\mathbb {F}_{7}}$ . First, consider the map induced on the three-petaled rose by the automorphism

$$\begin{align*}\theta \colon\, a \mapsto b, b \mapsto c, c \mapsto ca \in \operatorname{\mathrm{Aut}}(\mathbb{F}_3)\end{align*}$$

and the map induced by the inverse automorphism

$$\begin{align*}\vartheta \colon\, a \mapsto \bar{b}c, b \mapsto a, c \mapsto b.\end{align*}$$

Using $\theta $ and $\vartheta $ to also denote the corresponding graph maps and using the convention that a also denotes the initial direction of the oriented edge a, while $\bar {a}$ denotes the terminal direction, we have the maps $D\theta ^3$ and $D\vartheta ^3$ given, respectively, as:

Now, let $\phi \in \operatorname {\mathrm {Aut}}({\mathbb {F}_{7}})$ be the automorphism:

$$ \begin{align*}a\mapsto b,b\mapsto c,c\mapsto ca,d\mapsto d,e\mapsto e,f\mapsto f,g\mapsto g\end{align*} $$

and $\rho \in \operatorname {\mathrm {Aut}}(\mathbb {F}_7)$ be the rotation by four clicks:

$$ \begin{align*}a\mapsto e, b\mapsto f, c\mapsto g, d\mapsto a, e\mapsto b, f\mapsto c, g\mapsto d.\end{align*} $$

Thus, $\phi $ is the extension of $\theta $ by identity, and $\rho $ rotates the support of $\phi $ off itself.

Lemma 4.3. For any $r \ge 3$ , the map on the seven-petaled rose induced by $\phi _r = \rho \phi ^r$ is a train track map with respect to the train track structure with gates

$$ \begin{align*}\{a,b,c\},\{d,e,f\}\end{align*} $$

and eight more gates consisting of single half edges. The transition matrix $M_r$ has block form

$$\begin{align*}\begin{pmatrix} 0&I\\ B^r&0 \end{pmatrix}, \end{align*}$$

where I is the $4\times 4$ identity matrix, and B is the transition matrix of $\theta $ :

$$\begin{align*}B= \begin{pmatrix} 0&0&1\\ 1&0&0\\ 0&1&1 \end{pmatrix}. \end{align*}$$

Lemma 4.4. For any $r \ge 3$ and $r \equiv 0$ mod 3, the map on the seven-petaled rose induced by $\psi _r = (\rho \phi ^r)^{-1}$ is a train track map with respect to the train track structure with gates

$$ \begin{align*}\{a,e,\bar{g}\},\{b, \bar{d}\},\{c, \bar{b}\},\{d, \bar{c}\},\{f,\bar{e}\},\{g,\bar{f}\},\{ \bar{a}\}\end{align*} $$

The transition matrix $N_r$ has block form

$$\begin{align*}\begin{pmatrix} 0&C^r\\ I&0 \end{pmatrix}, \end{align*}$$

where I is the $4\times 4$ identity matrix, and C is the transition matrix of $\vartheta $ :

$$\begin{align*}C= \begin{pmatrix} 0&1&0\\ 1&0&1\\ 1&0&0 \end{pmatrix}. \end{align*}$$

4.2. Asymptotics of transition matrices

Let $\theta $ , $\vartheta $ , $\phi _r$ , $\psi _r$ be the maps defined in the last section. We now analyze the behavior of the transition matrices $M_r$ and $N_r$ for $\phi _r$ and $\psi _r$ , respectively.

Lemma 4.5. Let B be the transition matrix for $\theta $ , with Perron–Frobenius eigenvalue $\lambda _B$ . There exists a constant $\kappa _B>0$ such that if $r,s-r\to \infty $ , then

$$ \begin{align*}\frac 1{\kappa_B \lambda_B^s} M_r M_s\to Y,\end{align*} $$

where Y is an idempotent matrix of the form

$$\begin{align*}Y = \begin{pmatrix} u & p u & q u & 0 & 0 & 0 & 0 \end{pmatrix} \text{ with } u = \begin{pmatrix} 0, u_1, u_2, u_3, 0, 0, 0 \end{pmatrix}^T \text{ and } p, q> 0,\end{align*}$$

and $\begin {pmatrix} u_1,u_2,u_3\end {pmatrix}^T$ is a Perron–Frobenius eigenvector of B.

Proof. There exists a Perron–Frobenius eigenvector $x=(x_1,x_2,x_3)^T$ for B and constants $p, q> 0$ such that

$$\begin{align*}P = \lim_{s \to \infty} \frac{B^s}{\lambda_B^s} = \begin{pmatrix} x & px & qx \end{pmatrix}. \end{align*}$$

We have

The square of the limiting matrix above has a nonzero block where P is of the form

$$ \begin{align*}(px_1+qx_2)P,\end{align*} $$

and zero elsewhere, so we set

$$\begin{align*}\kappa_B=px_1+qx_2 \qquad \text{and} \qquad (u_1,u_2,u_3)^T = \frac{1}{\kappa_B} (x_1,x_2,x_3)^T.\\[-41pt] \end{align*}$$

We have a similar statement for the matrices $N_r$ .

Lemma 4.6. Let C be the transition matrix for $\vartheta = \theta ^{-1}$ , with Perron–Frobenius eigenvalue $\lambda _C$ . There exists a constant $\kappa _C>0$ such that if $r, s-r\to \infty $ , then

$$ \begin{align*}\frac 1{\kappa_C \lambda_C^s} N_s N_r \to Z,\end{align*} $$

where Z is an idempotent matrix of the form

$$\begin{align*}Z = \begin{pmatrix} 0 & v & p v & q v & 0 & 0 & 0 \end{pmatrix} \text{ with } v = \begin{pmatrix} v_1 , v_2 , v_3 , 0 , 0 , 0 , 0 \end{pmatrix}^T \text{ and } p, q> 0, \end{align*}$$

and $(v_1,v_2,v_3)^T$ is a Perron–Frobenius eigenvector of C.

Proof. We observe that the matrix $N_s N_r$ has shape that is the transpose of the matrix in Lemma 4.5, with powers of the PF matrix C forming the nonzero blocks:

For future reference, we also record the following. Let $P = \lim _{r \to \infty } B^r/\lambda _B^r$ and ${Q = \lim _{r \to \infty } C^r/\lambda _C^r}$ . Set

$$\begin{align*}M_\infty = \lim_{r \to \infty} \frac{M_r}{\lambda^r_B} =\begin{pmatrix} 0 & 0\\ P & 0 \end{pmatrix} \qquad \text{and} \qquad N_\infty = \lim_{r \to \infty} \frac{N_r}{\lambda^r_C} =\begin{pmatrix} 0 & Q \\ 0 & 0 \end{pmatrix}. \end{align*}$$

Lemma 4.7. There are $p,q,r,s> 0$ such that

$$\begin{align*}M_\infty Y = \begin{pmatrix} y & py & qy & 0 & 0 & 0 & 0 \end{pmatrix} \text{ with } y = \begin{pmatrix} 0, 0, 0, 0, y_1, y_2, y_3 \end{pmatrix}^T, \end{align*}$$

$(y_1,y_2,y_3)^T$ is a Perron–Frobenius eigenvector of B, and

$$\begin{align*}ZN_\infty = \begin{pmatrix} 0 & 0 & 0 & 0 & z & rz & sz \end{pmatrix} \text{ with } z = \begin{pmatrix} z_1, z_2, z_3, 0 , 0 , 0 , 0 \end{pmatrix}^T, \end{align*}$$

and $(z_1,z_2,z_3)^T$ is a Perron–Frobenius eigenvector of C.

4.3. Folding and unfolding sequence

Consider a sequence of positive integers $(r_i)_{i \ge 1}$ and the sequence of automorphisms $\phi _{r_i}$ , with transition matrix $M_{r_i}$ and $\phi _{r_i}^{-1} = \psi _{r_i}$ with transition matrix $N_{r_i}$ . Let $\tau _i \to \tau _{i-1}$ (resp. $\tau _{i-1}' \to \tau _i'$ ) be the train track map induced on the rose by $\phi _{r_i}$ (resp. $\psi _{r_i}$ ) as given by Lemma 4.3 (resp. Lemma 4.4). Thus, we have an unfolding sequence

and a folding sequence

Let $\Phi _i = \phi _{r_1} \circ \ldots \circ \phi _{r_i}$ and $\Phi _i^{-1} = \Psi _i = \psi _{r_i} \circ \ldots \circ \psi _{r_1}$ . Here, $\tau _0$ is a rose with petals labeled by elements in $\{a,b,c,d,e,f,g\}$ and hence for $i \ge 1$ , $\tau _i$ is a rose labeled by $\{\Phi _i(a), \ldots , \Phi _i(g)\}$ . Also, $\tau _0'$ is a rose labeled by $\{a,b,c,d,e,f,g\}$ , so $\tau _i'$ is also a rose labeled by $\{\Phi _i(a), \ldots , \Phi _i(g)\}$ . Thus, for every $i\geq 0$ , $\tau _i$ and $\tau _i'$ have the same marking but different train track structures. In other words, they belong to the same simplex in ${\text {CV}_{7}}$ .

The next lemma studies the behavior of illegal turns in a path along the folding sequence. This will be used in the proof of Proposition 5.10 to show that the limit tree of the folding sequence is nongeometric.

Proof. By Lemma 4.4, the illegal turns in $\tau ^{\prime }_{j}$ are

$$ \begin{align*}\{a,e\}, \{a,\bar{g}\},\{e,\bar{g}\}, \{b,\bar{d}\},\{c,\bar{b}\},\{d,\bar{c}\},\{f,\bar{e}\},\{g,\bar{f}\}\end{align*} $$

and we have

$$\begin{align*}\{a,e\} & \xrightarrow{\psi_{r_{j+1}}} \{d,\bar{c}\} \xrightarrow{\psi_{r_{j+2}}} \{g,\bar{f}\} \\[2pt] \{a,\bar{g}\} & \xrightarrow{\psi_{r_{j+1}}} \{d,\bar{c}\} \xrightarrow{\psi_{r_{j+2}}} \{g,\bar{f}\} \\[2pt] \{b,\bar{d}\} & \xrightarrow{\psi_{r_{j+1}}} \{e,\bar{g}\} \\[2pt] \{c,\bar{b}\} & \xrightarrow{\psi_{r_{j+1}}} \{f,\bar{e}\} \\[2pt] \{d,\bar{c}\} & \xrightarrow{\psi_{r_{j+1}}} \{g,\bar{f}\}. \end{align*}$$

Thus, for any illegal edge path $\beta \subset \tau _{j}'$ , one of $\beta , \psi _{r_{j+1}}(\beta ), \psi _{r_{j+2}}\psi _{r_{j+1}}(\beta )$ has an illegal turn $\{x, y\}$ , where $x, y \in \{e, f, g, \bar {e}, \bar {f}, \bar {g}\}$ .

Consider the automorphism $\vartheta $ and corresponding train track map $h \colon \, \mathrm {R}_3 \to \mathrm {R}_3$ as in Lemma 4.2. Then h does not have any periodic INPs. Since R is the constant from Lemma 2.1, we get that one of $[\psi _{r_{j+1}}(\beta )], [\psi _{r_{j+2}}\psi _{r_{j+1}}(\beta )], [\psi _{r_{j+3}}\psi _{r_{j+2}}\psi _{r_{j+1}}(\beta )]$ has fewer illegal turns than $\beta $ .

5.1. Sequence of free factors

Given a sequence $(r_i)_{i \ge 1}$ , recall that $\Phi _i = \phi _{r_1}\phi _{r_1}\cdots \phi _{r_i}$ , where $\phi _r = \rho \phi ^r$ . For convenience, also set $\Phi _0 = \operatorname {\mathrm {id}}$ . We have the folding sequence

where $\tau _i'$ is a rose labeled by $\{ \Phi _i(a),\cdots ,\Phi _i(g)\}$ , and $\psi _r = \phi _r^{-1}$ . From the markings, we can associate $\tau _i'$ to an open simplex $U_i$ in ${\text {CV}_{7}}$ . Consider a sequence of free factors $A_i \in \pi (U_i)$ , where $\pi \colon \, {\text {CV}_{7}} \to {\mathcal {FF}_{7}}$ . For an appropriate sequence of $(r_i)_i$ , we will see that $(A_i)_i$ is a quasi-geodesic (with infinite diameter). The key will be Lemma 5.3 which is the main goal of this section.

We now consider the following explicit sequence of free factors. Let $A_0 = \langle d,e,f\rangle $ be the free factor in ${\mathbb {F}_{7}}$ , and define

$$ \begin{align*}A_i:= \Phi_i (A_0) = \langle \Phi_i(d), \Phi_i(e), \Phi_i(f)\rangle.\end{align*} $$

Note that for any $r,s,t>0$ , the following holds:

(1) $$ \begin{align} \begin{aligned} A_0 &= \langle d, e, f \rangle \\ A_1 &= \phi_r(A_0) = \langle a, b, c \rangle \\ A_2 &= \phi_s\phi_r(A_0) = \langle e, f, g \rangle \\ A_3 &= \phi_t\phi_s\phi_r(A_0) = \langle b,c,d \rangle. \end{aligned} \end{align} $$

Thus, for any sequence $(r_i)_i$ ,

(2) $$ \begin{align} A_i = \Phi_i(A_0) = \Phi_{i-1}(A_1) = \Phi_{i-2}(A_2) = \Phi_{i-3}(A_3). \end{align} $$

We say two free factors A and $A'$ are disjoint if (possibly after conjugating) ${\mathbb {F}_{n}}=A*A'*B$ for a (possibly trivial) free factor B, and $A'$ is compatible with A if it either contains A (up to conjugation) or is disjoint from A.

Recall the transition matrix $M_r$ for $\phi _r$ , and the $3\times 3$ matrix B whose power $B^r$ forms a block of $M_r$ . For each $i \ge 1$ , let $\overline {M}_i = M_i$ mod 2. By a simple computation, we see that $B^7 = I$ mod 2. Thus, when $i = j$ mod 7, $\overline {M}_i = \overline {M}_j$ . We have the following lemma.

Lemma 5.2. Let $V_0$ be the three-dimensional vector space of $(\mathbb {Z}/2\mathbb {Z})^7$ spanned by the vectors $(0,0,0,1,0,0,0)^T,(0,0,0,0,1,0,0)^T,(0,0,0,0,0,1,0)^T$ . Then for all $i \ge 0$ ,

$$\begin{align*}V_0 \bigcup \left( \bigcup_{j=0}^{107} \overline{M}_i \overline{M}_{i+1}\ldots \overline{M}_{i+j} V_0 \right) = (\mathbb{Z}/2\mathbb{Z})^7. \end{align*}$$

Proof. Since $\overline {M}_i=\overline {M}_j$ whenever $i = j$ mod 7, it is enough to verify the statement for $i \in \{0,\ldots ,6\}$ . In these cases, we can check the validity of the statement using Sage with the following code:

Proof. For any $i \ge 1$ and $k \ge 0$ , let

$$\begin{align*}B_{i+k} = \phi_i \phi_{i+1} \cdots \phi_{i+k} A_0. \end{align*}$$

Abelianizing and reducing mod 2, we have $A_0 \equiv V_0$ , and $B_{i+k} \equiv \overline {M}_i \cdots \overline {M}_{i+k} V_0$ . Thus, by Lemma 5.2, the sequence $\{A_0, B_i, \ldots , B_{i+107}\}$ cannot be contained in the same free factor or be disjoint from a common factor.

Now, consider any sequence $(r_i)_i$ with $r_i \equiv i$ mod $7$ so that $\overline {M}_{r_i} = \overline {M}_i$ for all i. Let $A_i = \Phi _i A_0 = \phi _{r_1} \cdots \phi _{r_i}A_0$ . Set $\Phi _0 = \operatorname {\mathrm {id}}$ . For any $i \ge 1$ , by applying the automorphism $\Phi _{i-1}^{-1}$ , the sequence of free factors $\{A_{i-1},\ldots ,A_{i+107}\}$ is isomorphic to the sequence

$$\begin{align*}\{A_0, \phi_{r_i}A_0, \ldots, \phi_{r_i}\phi_{r_{i+1}} \cdots \phi_{r_{i+107}}A_0 \}. \end{align*}$$

The latter sequence after abelianization and reducing mod 2 is equivalent to the sequence $\{A_0,B_i,\ldots ,B_{i+107}\}$ . Thus, $\{A_{i-1},\ldots ,A_{i+107}\}$ cannot be contained in the same factor or be disjoint from a common factor.here

5.2. Subfactor projection

We will now use subfactor projection theory originally introduced in [Reference Bestvina and FeighnBF14b] and further developed in [Reference TaylorTay14] to show that $(A_i)_i$ is a quasi-geodesic for appropriate choices of sequence $(r_i)_i$ .

We first define subfactor projection and recall the main results about them. For $G \in {\text {CV}_{n}} $ and a rank $\geq 2$ free factor A, let $A|G$ denote the core subgraph of the cover of G corresponding to the conjugacy class of A. Pulling back the metric on G, we obtain $A|G \in \mathrm {CV}(A)$ . Denote by $\pi _A(G):= \pi (A|G) \subset {\mathcal F}(A)$ the projection of $A|G$ to ${\mathcal F}(A)$ . Here, $\mathrm {CV}(A)$ is the Outer space of the free group A and ${\mathcal F}(A)$ is the corresponding free factor complex.

Recall two free factors A and B are disjoint if they are distinct vertex stabilizers of a free splitting of ${\mathbb {F}_{n}}$ . If B is not compatible with A, then we say B meets A, that is, B and A are not disjoint and A is not contained in B, up to conjugation. In this case, define the projection of B to ${\mathcal F}(A)$ as follows:

$$\begin{align*}\pi_A(B):=\{\pi_A(G) | G \in {\text{CV}_{n}} \text{ and } B|G \subset G \text{ is embedded }\}\end{align*}$$

If B is compatible with A, then define $\pi _A(B)$ to be empty. If A meets B and B meets A, then we say A and B overlap.

We now prove the following lemma.

We are now ready to prove the main results of this section.

Recall that ${\mathcal {FF}_{n}}$ is Gromov hyperbolic and that its Gromov boundary is the space of equivalence class of arational trees. Also, recall we say a folding sequence $(G_i)_i$ converges to an arational tree T, if $\pi (U_i)$ converges to $[T] \in \partial {\mathcal {FF}_{n}}$ , where $U_i$ is the open simplex in in ${\text {CV}_{n}} $ associated to $G_i$ . We have the following corollary.

5.3. Nongeometric tree

We will now show that the arational tree obtained in the previous section as the limit of the free factors $(A_i)_i$ is nongeometric. This section will use the terminology of band complexes and resolutions; for details see [Reference Bestvina and FeighnBF95].

Definition 5.9 (Geometric tree)

[Reference Bestvina and FeighnBF94, Reference Levitt and PaulinLP97] Let X be a band complex and T a $G=\pi _1(X)$ -tree. A resolution $f :\widetilde {X} \to T$ is exact if for every G-tree $T'$ and equivariant factorization

$$\begin{align*}\widetilde{X} \stackrel{f'}{\longrightarrow} T' \stackrel{h}{\longrightarrow} T \end{align*}$$

of f with $f'$ a surjective resolution it follows that h is an isometry onto its image. We say T is geometric if every resolution is exact.

The proof of the following proposition is based on [Reference Bestvina and FeighnBF94, Proposition 3.6].

Proof. Let $\tilde \psi _i \colon \, \tilde \tau _{i-1}' \to \tilde \tau _i'$ be a lift of the train track map to the universal covers fixing a base vertex. Pick a length measure on $(\tau _i')_i$ , so we get a folding sequence $\tilde \tau _0' \stackrel {\tilde \psi _1}{\longrightarrow } \tilde \tau _1' \stackrel {\tilde \psi _2}{\longrightarrow } \cdots $ in ${\text {cv}_{7}}$ that converges to T. Recall that there are morphisms $h_i \colon \, \tilde {\tau }_i' \to T$ such that $h_i = h_{i+1} \tilde \psi _{i+1}$ . Since T is arational, $h_i$ ’s are not isometries though they restrict to isometries on edges. Let X be a finite band complex with resolution $f \colon \, \tilde X \to T$ . We will show that the resolution factors through $\tilde \tau _i'$ for sufficiently large i. This will imply T is not geometric.

Let $\Gamma $ be the underlying real graph of X (disjoint union of metric arcs) with preimage $\tilde \Gamma $ in $\tilde X$ . We may assume f embeds the components of $\tilde \Gamma $ . A vertex v of $\tilde X$ is either a vertex of $\tilde \Gamma $ or a corner of a band or a 0-cell of $\tilde X$ . For every such vertex v, choose a point $f_0(v) \in \tilde \tau _0$ so that $f_0$ is equivariant and $f=h_0 f_0$ on the vertices of $\tilde X$ .

An edge in $\tilde X$ is either a subarc of $\tilde \Gamma $ or a vertical boundary component of a band or a one-cell in $\tilde X$ . Up to the action of ${\mathbb {F}_{7}}$ , there are only finitely many edges. Using Lemma 4.8, we can find $i>0$ such that for every edge e in $\tilde X$ , the edge path in $\tilde \tau _i'$ joining the two vertices of $\tilde \psi _i \cdots \tilde \psi _1 f_0(\partial e)$ is legal. Now, extend $\tilde \psi _i \cdots \tilde \psi _1 f_0$ to an equivariant map $f_i \colon \, \tilde X \to \tilde \tau _i'$ that sends edges to legal paths (or points) and is constant on the leaves. Thus, $f_i$ is a resolution of $\tilde \tau _i'$ .

This yields a factorization

but $h_i$ is not an isometry. This shows T is nongeometric.