2.1 Marked graphs, circuits and path

A marked graph is a finite graph $G$ which has no valence 1 vertices and is equipped with a homotopy equivalence, called a marking, to the rose $R_n$ given by $\rho : G\to R_n$ (where $n = \text {rank}( {\mathbb {F} } )$). The homotopy inverse of the marking is denoted by the map $\overline {\rho }: R_n \to G$. A circuit in a marked graph is an immersion (i.e. a locally injective continuous map) of $S^1$ into $G$. Here $I$ denotes an interval in $\mathbb {R}$ that is closed as a subset. A path is a locally injective, continuous map $\alpha : I \to G$, such that any lift $\widetilde {\alpha }: I \to \widetilde {G}$ is proper. When $I$ is compact, any continuous map from $I$ can be homotoped, relative to its endpoints, by a process called tightening to a unique path (up to reparametrization) with domain $I$. If $I$ is noncompact, then each lift $\widetilde {\alpha }$ induces an injection from the ends of $I$ to the ends of $\widetilde {G}$. In this case, there is a unique path $[\alpha ]$ which is homotopic to $\alpha$ such that both $[\alpha ]$ and $\alpha$ have lifts to $\widetilde {G}$ with the same finite endpoints and the same infinite ends. If $I$ has two infinite ends, then $\alpha$ is called a line in $G$; otherwise, if $I$ has only one infinite end, then $\alpha$ is called a ray. Given a homotopy equivalence of marked graphs $f: G\to G'$, $f_\#(\alpha )$ denotes the tightened image $[f(\alpha )]$ in $G'$. Similarly, we define $\tilde {f}_\#(\widetilde {\alpha })$ by lifting to the universal cover.

Free factor systems. Given a collection of finitely generated subgroups of $ {\mathbb {F} }$, say $F^1,\ldots, F^k$, the collection of their conjugacy classes denoted by $\{[F^1],\ldots, [F^k]\}$ is called a subgroup system. We say that a circuit $[c]$ is carried by this subgroup system if and only if there exists a representative $H^i$ of some $[F^i]$ such that $c \in H^i$. The subgroup system is called a malnormal subgroup system if $H^i \cap H^j = \emptyset$ for every $i\neq j$, where $H^i$ is any representative of $[F^i]$.

A subgroup system $\{[F^1], [F^2],\ldots, [F^k]\}$ is called a free-factor system if $ {\mathbb {F} } = F^1 \ast F^2 \ast \cdots \ast F^k \ast B$, with $B$ possibly trivial. Note that every free-factor system is always a malnormal subgroup system. Note that given any subgraph $H\subset G$, the fundamental groups of the noncontractible components of $H$ gives rise to a free-factor system, which is denoted by $[H]$.

2.2 Weak topology

Given any finite graph $G$, let $\widehat {\mathcal {B}}(G)$ denote the compact space of equivalence classes of lines, circuits, paths and rays in $G$. We give this space the weak topology. Namely, for each finite path $\gamma$ in $G$, we have one basis element $\widehat {N}(G,\gamma )$ which contains all paths and circuits in $\widehat {\mathcal {B}}(G)$ which have $\gamma$ as its subpath. Let $\mathcal {B}(G)\subset \widehat {\mathcal {B}}(G)$ be the compact subspace of all lines in $G$ with the induced topology. One can give an equivalent description of $\mathcal {B}(G)$ following [Reference Bestvina, Feighn and HandelBFH00]. A line is completely determined, up to reversal of direction, by two distinct points in $\partial \mathbb {F}$, because there is only one line that joins these two points. We can then induce the weak topology on the set of lines coming from the Cantor set $\partial \mathbb {F}$. More explicitly, let $\widetilde {\mathcal {B}}=\{ \partial \mathbb {F} \times \partial \mathbb {F} - \vartriangle \}/(\mathbb {Z}_2)$, where $\vartriangle$ is the diagonal and $\mathbb {Z}_2$ acts by interchanging factors. We can put the weak topology on $\widetilde {\mathcal {B}}$, induced by Cantor topology on $\partial \mathbb {F}$. The group $\mathbb {F}$ acts on $\widetilde {\mathcal {B}}$ with a compact but non-Hausdorff quotient space $\mathcal {B}=\widetilde {\mathcal {B}}/\mathbb {F}$. The quotient topology is also called the weak topology. Elements of $\mathcal {B}$ are called lines. A lift of a line $\gamma \in \mathcal {B}$ is an element $\widetilde {\gamma }\in \widetilde {\mathcal {B}}$ that projects to $\gamma$ under the quotient map and the two elements of $\widetilde {\gamma }$ are called its endpoints.

One can naturally identify the two spaces $\mathcal {B}(G)$ and $\mathcal {B}$ by considering a homeomorphism between the two Cantor sets $\partial \mathbb {F}$ and set of ends of universal cover of $G$, where $G$ is a marked graph. Here $ {\mathrm {Out}(\mathbb {F}) }$ has a natural action on $\mathcal {B}$. The action comes from the action of Aut($\mathbb {F}$) on $\partial \mathbb {F}$. Given any two marked graphs $G,G'$ and a homotopy equivalence $f:G\rightarrow G'$ between them, the induced map $f_\#: \widehat {\mathcal {B}}(G)\rightarrow \widehat {\mathcal {B}}(G')$ is continuous and the restriction $f_\#:\mathcal {B}(G)\rightarrow \mathcal {B}(G')$ is a homeomorphism. With respect to the identification $\mathcal {B}(G)\approx \mathcal {B}\approx \mathcal {B}(G')$, if $f$ preserves the marking, then $f_{\#}:\mathcal {B}(G)\rightarrow \mathcal {B}(G')$ is equal to the identity map on $\mathcal {B}$. When $G=G'$, $f_{\#}$ agree with their homeomorphism $\mathcal {B}\rightarrow \mathcal {B}$ induced by the outer automorphism associated to $f$.

A line (path) $\gamma$ is said to be weakly attracted to a line (path) $\beta$ under the action of $\phi \in {\mathrm {Out}(\mathbb {F}) }$, if the $\phi ^k(\gamma )$ converges to $\beta$ in the weak topology. This is same as saying, for any given finite subpath of $\beta$, $\phi ^k(\gamma )$ contains that subpath for all sufficiently large values of $k$; similarly if we have a homotopy equivalence $f:G\rightarrow G$, a line (path) $\gamma$ is said to be weakly attracted to a line (path) $\beta$ under the action of $f_{\#}$ if the $f_{\#}^k(\gamma )$ converges to $\beta$ in the weak topology.

Topological representative. Given $\phi \in {\mathrm {Out}(\mathbb {F}) }$ a topological representative is a homotopy equivalence $f:G\rightarrow G$ (where $\rho : G \to R_n$ is a marked graph), $f$ takes vertices to vertices and edges to edge paths and $\rho \circ f \circ \overline {\rho }: R_n \rightarrow R_n$ represents $\phi$. A nontrivial path $\gamma$ in $G$ is a periodic Nielsen path if there exists a $k$ such that $f^k_\#(\gamma )=\gamma$; the minimal such $k$ is called the period and if $k=1$, we call such a path Nielsen path. A periodic Nielsen path is indivisible if it cannot be written as a concatenation of two or more nontrivial periodic Nielsen paths.

2.3 EG strata and NEG strata

A filtration of a marked graph $G$ is a strictly increasing sequence of subgraphs $G_0 \subset G_1 \subset \cdots \subset G_k = G$, each with no isolated vertices. The individual terms $G_r$ are called filtration elements, and if $G_r$ is a core graph (i.e. a graph without valence 1 vertices), then it is called a core filtration element. The subgraph $H_r = G_r \setminus G_{r-1}$ together with the vertices which occur as endpoints of edges in $H_r$ is called the stratum of height $r$. The height of subset of $G$ is the minimum $r$ such that the subset is contained in $G_r$. The height of a map to $G$ is the height of the image of the map. A connecting path of a stratum $H_r$ is a nontrivial finite path $\gamma$ of height $< r$ whose endpoints are contained in $H_r$.

Given a topological representative $f : G \to G$ of $\phi \in {\mathrm {Out}(\mathbb {F}) }$, we say that $f$ respects the filtration or that the filtration is $f$-invariant if $f(G_r) \subset G_r$ for all $r$. Given an $f$-invariant filtration, for each stratum $H_r$ with edges $\{E_1,\ldots,E_m\}$, define the transition matrix of $H_r$ to be the square matrix whose $j$th column records the number of times $f(E_j)$ crosses the other edges. If $M_r$ is the zero matrix, then we say that $H_r$ is a zero stratum. If $M_r$ is irreducible, meaning that for each $i,j$ there exists $p$ such that the $i,j$ entry of the $p$th power of the matrix is nonzero, then we say that $H_r$ is irreducible; and if one can furthermore choose $p$ independently of $i,j$, then $H_r$ is aperiodic. Assuming that $H_r$ is irreducible, by the Perron–Frobenius theorem, the matrix $M_r$ a unique eigenvalue $\lambda \ge 1$, called the Perron–Frobenius eigenvalue, for which some associated eigenvector has positive entries: if $\lambda >1$, then we say that $H_r$ is an exponentially growing (EG) stratum; whereas if $\lambda =1$, then $H_r$ is a nonexponentially growing (NEG) stratum. An edge in an NEG stratum will be sometimes referred to as NEG edge. Similarly for edges in EG stratum we shall sometimes use the term EG edge.

2.4 Relative train-track maps

Let $f:G\rightarrow G$ be a topological representative for $\phi \in {\mathrm {Out}(\mathbb {F}) }$ and consider a filtration $G_0\subset G_1\subset \cdot \cdot \cdot \subset G_k$ which is preserved by $f$. One can define a map $T_f$ by setting $T_f(E)$ to be the first edge of the edge path $f(E)$. We say $T_f(E)$ is the direction of $f(E)$. If $E_1, E_2$ are two edges in $G$ with the same initial vertex, then the unordered pair $(E_1, E_2)$ is called a turn in $G$. Define $T_f(E_1,E_2) = (T_f(E_1),T_f(E_2))$. Thus, $T_f$ is a map that takes turns to turns. A turn is said to have height $r$ if both the edges defining the turn are of height $r$.

We say that a nondegenerate turn (i.e. $E_1\neq E_2$) is illegal if for some $k>0$ the turn $T^k_f(E_1, E_2)$ becomes degenerate (i.e. $T^k_f(E_1) = T^k_f(E_2)$); otherwise, the turn is legal. A path is said to be a legal path if it contains only legal turns. A path is $r-legal$ if it is of height $r$ and all its height $r$ turns are legal.

Relative train-track map [Reference Feighn and HandelFH11, § 2.6]. Given $\phi \in {\mathrm {Out}(\mathbb {F}) }$ and a topological representative $f:G\rightarrow G$ with a filtration $G_0\subset G_1\subset \cdots \subset G_k$ which is preserved by $f$. The topological representative $f$ is a relative train-track map if every stratum is either a zero stratum or irreducible stratum and, in addition, the following conditions are satisfied for every EG stratum $H_r$:

1. (i) if $E$ is edge in $H_r$, then $T_f(E)$ is also an edge of $H_r$;

2. (ii) $f$ maps $r$-legal paths to legal $r$-paths;

3. (iii) if $\gamma$ is a nontrivial path in $G$ of height less than $r$ with its endpoints in $H_r$, then $f_\#(\gamma )$ has its end points in $H_r$.

2.5 Completely split train-track maps

Given relative train-track map $f:G\rightarrow G$, splitting of a line, path or a circuit $\gamma$ is a decomposition of $\gamma$ into subpaths $\ldots \gamma _0\gamma _1 \cdots \gamma _k\ldots$ such that for all $i\geq 1$ the path $f^i_\#(\gamma ) = \cdots f^i_\#(\gamma _0)f^i_\#(\gamma _1)\cdots f^i_\#(\gamma _k)\ldots.$ The terms $\gamma _i$ are called the terms of the splitting of $\gamma$. Henceforth, the notation $\alpha \cdot \beta$ will be used to denote a splitting and $\alpha \beta$ will denote a concatenation of nontrivial paths $\alpha, \beta$.

Given two NEG edges $E_1, E_2$ and a root-free closed Nielsen path $\rho$ such that $f_\#(E_i) = E_i.\rho ^{p_i}$, then $E_1,E_2$ are said to be in the same linear family and any path of the form $E_1\rho ^m\overline {E}_2$ for some integer $m$ is called an exceptional path.

Complete splittings. A splitting of a path or circuit $\gamma = \gamma _1\cdot \gamma _2\cdot \cdots \cdot \gamma _k$ is called complete splitting if each term $\gamma _i$ falls into one of the following categories:

1. (i) $\gamma _i$ is an edge in some irreducible stratum;

2. (ii) $\gamma _i$ is an indivisible Nielsen path;

3. (iii) $\gamma _i$ is an exceptional path;

4. (iv) $\gamma _i$ is a maximal subpath of $\gamma$ in a zero stratum $H_i$ and $\gamma _i$ is taken (see [Reference Feighn and HandelFH11, Definition 4.4]).

Completely split improved relative train-track maps. A CT or a completely split improved relative train-track map is a topological representative with particularly nice properties. However, CTs do not exist for all outer automorphisms. Only the rotationless (see [Reference Feighn and HandelFH11, Definition 3.13]) outer automorphisms are guaranteed to have a CT representative as has been shown in the following theorem from [Reference Feighn and HandelFH11, Theorem 4.28].

The following results are some properties of CTs defined in the recognition theorem work of Feighn and Handel in [Reference Feighn and HandelFH11]. We only state those we need here.

1. (i) (Rotationless) Each principal vertex is fixed by $f$ and each periodic direction at a principal vertex is fixed by $Tf$.

2. (ii) (Completely split) For each edge $E$ in each irreducible stratum, the path $f(E)$ is completely split.

3. (iii) (Vertices) The endpoints of all indivisible Nielsen paths are vertices. The terminal endpoint of each nonfixed NEG edge is principal.

4. (iv) (Periodic edges) Each periodic edge is fixed.

5. (v) (Zero strata) Each zero strata $H_i$ is contractible and enveloped by an EG strata $H_s,\ s>i$, such that every edge of $H_i$ is a taken in $H_s$. Each vertex of $H_i$ is contained in $H_s$ and link of each vertex in $H_i$ is contained in $H_i \cup H_s$.

6. (vi) (Linear edges) For each linear edge $E_i$ there exists a root-free Nielsen path $w_i$ such that $f_\#(E_i) = E_i w^{d_i}_i$ for some $d_i \neq 0$.

7. (vii) (Nonlinear NEG edges) [Reference Feighn and HandelFH11, Lemma 4.21] Each non-fixed NEG stratum $H_i$ is a single edge with its NEG orientation and has a splitting $f_\#(E_i) = E_i\cdotp u_i$, where $u_i$ is a closed nontrivial completely split circuit and is an Nielsen path if and only if $H_i$ is linear.

We shall call any nonfixed, nonlinear NEG edge a superlinear NEG edge. The advantage of using CT maps rather than using the regular relative train-track maps is the greater control that we get when we iterate the train-track maps, which is very much a necessity here.

2.6 Attracting laminations

A closed subset of $\mathcal {B}=\mathcal {B}( {\mathbb {F} } )$ is called a lamination. An element of a lamination is called a leaf. The action of $ {\mathrm {Out}(\mathbb {F}) }$ on $\mathcal {B}$ induces an action on the set of laminations.

For each marked graph $G$ the homeomorphism $\mathcal {B} \approx \mathcal {B}(G)$ induces a bijection between $ {\mathbb {F} }$-laminations and closed subsets of $\mathcal {B}(G)$. The closed subset of $\mathcal {B}(G)$ corresponding to a lamination $\Lambda \subset \mathcal {B}$ is called the realization of $\Lambda$ in $G$; we generally use the same notation $\Lambda$ for its realizations in marked graphs. In addition, we occasionally use the term lamination to refer to $ {\mathbb {F} }$-invariant, closed subsets of $\widetilde {\mathcal {B}}$; the quotient map $\widetilde {\mathcal {B}} \mapsto \mathcal {B}$ puts these in natural bijection with laminations in $\mathcal {B}$.

Given $\phi \in {\mathrm {Out}(\mathbb {F}) }$ and a lamination $\Lambda \subset \mathcal {B}$, we say that $\Lambda$ is an attracting lamination for $\phi$ if there exists a leaf $\ell \in \Lambda$ satisfying the following:

1. (i) $\Lambda$ is the weak closure of $\ell$;

2. (ii) $\ell$ is a birecurrent, i.e. every finite subpath of $\ell$ occurs infinitely many times in either direction of $\ell$;

3. (iii) $\ell$ is not the axis of the conjugacy class of a generator of a rank-one free factor of $F_n$;

4. (iv) there exists $p \ge 1$ and a weak open set $U \subset \mathcal {B}$ such that $\phi ^p(U) \subset U$ and such that $\{\phi ^{kp}(U) \mid k \ge 1\}$ is a weak neighborhood basis of $\ell$.

Any such leaf $\ell$ is called a generic leaf of $\Lambda$, and any such neighborhood $U$ is called an attracting neighborhood of $\Lambda$ for the action of $\phi ^p$. Let $\mathcal {L}(\phi )$ denote the set of attracting laminations for $\phi$.

Bestvina, Feighn and Handel [Reference Bestvina, Feighn and HandelBFH00] showed that there is bijection between the elements of $\mathcal {L}(\phi )$ and the exponentially growing strata of $G$. In fact, they showed that given a relative train-track map, there is a unique way of associating each attracting lamination to an exponentially growing stratum of $G$. Moreover, they established a bijection between the sets $\mathcal {L}(\phi )$ and $\mathcal {L}(\phi ^{-1})$ by using the notion of free-factor support.

Free factor support. The free-factor support of a line $\ell$ is the conjugacy class of the smallest (with respect to inclusion) free factor $F^k$ such that $\partial \ell \in \partial F^k$. For an attracting lamination associated to some EG stratum $H_r$ of $G$, the free-factor support of $\Lambda ^+_\phi$ is the free-factor support of any generic leaf of $\Lambda ^+_\phi$ and equals to $[G_r]$ (the free-factor system defined by $G_r$ as a subgraph of $G$) (see [Reference Bestvina, Feighn and HandelBFH00, Definition 3.2.3, Lemma 3.2.4]).

In this particular case, one can think of a free-factor support of an attracting lamination, $\Lambda ^+\in \mathcal {L}(\phi )$, to be the conjugacy class of the smallest (in terms of subgroup inclusion) free factor that carries $\Lambda ^+$. Two laminations $\Lambda ^+\in \mathcal {L}(\phi )$ and $\Lambda ^-\in \mathcal {L}(\phi ^{-1})$ are said to be dual if and only if they have the same free-factor support. In [Reference Bestvina, Feighn and HandelBFH00], it is shown that duality induces a bijection between the sets $\mathcal {L}(\phi )$ and $\mathcal {L}(\phi ^{-1})$.

Given a rotationless outer automorphism $\phi \in {\mathrm {Out}(\mathbb {F}) }$ we list a few properties which will be important for us.

2.7 Relatively hyperbolic groups

Given a group $\Gamma$ and a collection of subgroups $H_\alpha < \Gamma$, the coned-off Cayley graph of $\Gamma$ or the electric space of $\Gamma$ relative to the collection $\{H_\alpha \}$ is a metric space which consists of the Cayley graph of $\Gamma$ and a collection of vertices $v_\alpha$ (one for each left coset of $H_\alpha$) such that each point of $H_\alpha$ is joined to (or coned-off at) $v_\alpha$ by an edge of length $1/2$. The resulting metric space is denoted by $(\widehat {\Gamma }, {|\cdot |}_{\rm el})$.

A group $\Gamma$ is said to be (weakly) hyperbolic relative to a finite collection of finitely generated subgroups $\{H_\alpha \}$ if $\widehat {\Gamma }$ is a $\delta$-hyperbolic metric space, in the sense of Gromov. Here $\Gamma$ is said to be strongly hyperbolic relative to the collection $\{H_\alpha \}$ if the coned-off space $\widehat {\Gamma }$ is hyperbolic and it satisfies the bounded coset penetration property (see [Reference FarbFar98]), but this bounded coset penetration property is a very hard condition to check for random groups $\Gamma$. However, it is well known that if the group $\Gamma$ is weakly relatively hyperbolic with respect to the collection of subgroups $\{H_\alpha \}$ and this collection is mutually malnormal and quasiconvex, then $\Gamma$ is strongly relatively hyperbolic. We shall be using this result for our constructions here.

In our main result, Proposition 3.13, the bounded coset penetration property is verified by the cone-bounded hallways strictly flare condition due to [Reference Mj and ReevesMR08]. It was shown in [Reference Mj and ReevesMR08] that this flaring property establishes a condition (namely, mutual coboundedness) due to Bowditch [Reference BowditchBow12] which implies the strong relative hyperbolicity.

3.1 Nonattracting subgroup system

The nonattracting subgroup system of an attracting lamination contains information about lines and circuits which are not attracted to the lamination. This is a crucial construction from the train-track theory that lies in the heart of our proof here. First introduced by Bestvina, Feighn and Handel in their Tit's alternative work [Reference Bestvina, Feighn and HandelBFH00], it was later studied in more details by Handel and Mosher in [Reference Handel and MosherHM20]. We urge the reader unfamiliar with this construction to look into [Reference Handel and MosherHM20] where it has been explored in great detail.

The construction of the nonattracting subgraph is as follows.

Suppose $\phi \in {\mathrm {Out}(\mathbb {F}) }$ is rotationless and $f:G\rightarrow G$ is a CT representing $\phi$ such that $\Lambda ^+_\phi$ is an invariant attracting lamination which corresponds to the EG stratum $H_s\subset G$. The nonattracting subgraph Z of $G$ is defined as a union of irreducible strata $H_i$ of $G$ such that no edge in $H_i$ is weakly attracted to $\Lambda ^+_\phi$. This is equivalent to saying that a strata $H_i$ is contained in $G\setminus Z$ if and only if there exists some $k\geq 0$ such that for some edge $E_i$ of $H_i$, a term in the complete splitting of $f^k_\#(E_i)$ is an edge of $H_s$.

Define the path $\widehat {\rho }_s$ to be trivial path at an arbitrarily chosen vertex if there does not exist any indivisible Nielsen path of height $s$, otherwise $\widehat {\rho }_s$ is the unique indivisible path of height $s$ (see [Reference Feighn and HandelFH11, Corollary 4.19]).

The groupoid $\langle \mathcal {Z}, \widehat {\rho }_s \rangle$. Let $\langle \mathcal {Z}, \widehat {\rho }_s \rangle$ be the set of lines, rays, circuits and finite paths in $G$ which can be written as a concatenation of subpaths, each of which is an edge in $Z$, the path $\widehat {\rho }_s$ or its inverse. Under the operation of tightened concatenation of paths in $G$, this set forms a groupoid [Reference Handel and MosherHM20, Lemma 5.6]. We say that a path, circuit, ray or line is carried by $\langle \mathcal {Z}, \widehat {\rho _s}\rangle$ if it can be written as a concatenation of paths in $\langle \mathcal {Z}, \widehat {\rho _s}\rangle$.

Define the graph $K$ by setting $K=Z$ if $\widehat {\rho }_s$ is trivial and let $h:K \rightarrow G$ be the inclusion map. Otherwise, define an edge $E_\rho$ representing the domain of the Nielsen path $\rho _s:E_\rho \rightarrow G_s$, and let $K$ be the disjoint union of $Z$ and $E_\rho$ with the following identification. Given an endpoint $x\in E_\rho$, if $\rho _s(x)\in Z$, then identify $x\sim \rho _s(x)$. Given distinct endpoints $x,y\in E_\rho$, if $\rho _s(x)=\rho _s(y)\notin Z$, then identify $x \sim y$. In this case, define $h:K\rightarrow G$ to be the inclusion map on $K$ and the map $\rho _s$ on $E_\rho$. It is not difficult to see that the map $h$ is an immersion. Hence, restricting $h$ to each component of $K$, we get an injection at the level of fundamental groups. The nonattracting subgroup system $\mathcal {A}_{\rm na}(\Lambda ^+_\phi )$ is defined to be the subgroup system defined by this immersion.

Handel and Mosher proved that the nonattracting subgroup system has a maximality property with respect to subgroup inclusion (see [Reference Handel and MosherHM20, Corollary 1.8]).

We leave it to the reader to look it up in [Reference Handel and MosherHM20] where it is explored in detail. We do, however, list some key properties which we will use and these results exhibit the importance of this subgroup system. For ease of notation, we simply write $\hat {\sigma } = \widehat {\rho }_s$ when it is clear what height or Nielsen path we are working with.

In light of this lemma, we sometimes use the phrase ‘$\alpha$ is an element of $\langle \mathcal {Z}, \hat {\sigma }\rangle$’ and ‘$\alpha$ is carried by $\langle \mathcal {Z}, \hat {\sigma }\rangle$’ interchangeably when $\alpha$ is a path, line or circuit in $G$. Similarly, when $\alpha$ is a line or a circuit, the phrase ‘$\alpha$ is carried by $\mathcal {A}_{\rm na}(\Lambda ^+_\phi )$’ and ‘$\alpha$ is carried by $\langle \mathcal {Z}, \hat {\sigma }\rangle$’ mean the same thing.

3.2 Weak attraction theorem

3.3 Free-by-cyclic extensions for exponentially growing $\phi$

The method of proof followed in this work was developed by the author in [Reference GhoshGho18], where examples of free-by-free (strongly) relatively hyperbolic extensions are constructed, which, in turn, was inspired by the work of Bestvina, Feighn and Handel in [Reference Bestvina, Handel and FeighnBHF97], where they constructed free-by-free hyperbolic extensions.

For the rest of this section, we assume that $\phi \in {\mathrm {Out}(\mathbb {F}) }$ is an exponentially growing and rotationless outer automorphism. Let $\Lambda ^\pm _\phi$ be a dual lamination pair associated to this automorphism. In addition, let $f:G\to G$ be a CT map representing $\phi$ and $H_r$ be the unique exponentially growing strata associated to $\Lambda ^+_\phi$ and $\mathcal {A}_{\rm na}(\Lambda ^+_\phi )$ be the nonattracting subgroup system of $\Lambda ^+_\phi$. Recall that, by construction, any conjugacy class is not weakly attracted to $\Lambda ^+_\phi$ under iterates of $\phi$ if and only if it is carried by $\mathcal {A}_{\rm na}(\Lambda ^+_\phi$). Similarly, let $f': G' \to G'$ be a CT representing $\phi ^{-1}$ and $H'_s$ be the unique exponentially growing strata in $G'$ that is associated to $\Lambda ^-_\phi$.

We use the term generic leaf segment of $\alpha$ repeatedly, by which we mean we are considering a subpath of $\alpha$ of height $r$ (or height $s$) which is also a subpath of some generic leaf of $\Lambda ^+_\phi$ (or $\Lambda ^-_\phi$) and the first and last edges of this subpath are in $H_r$ (or $H'_s$, depending on the context). In general, when we talk about subpaths of generic leaves we are always considering leaf segments with their initial and terminal edges in $H_r$ (or $H'_s$).

Convention. We use the notation ${|\alpha |}_{H_r}$ to denote the $r$-length of a path $\alpha$ in $G$, i.e. we only count the edges of $\alpha$ contained in $H_r$ for computing this length.

Bounded cancelation constant. Let $f: G \to G$ be a CT representing some $\phi \in {\mathrm {Out}(\mathbb {F}) }$. The bounded cancelation constant for $f$, denoted by $BCC(f)$ is a constant such that given any concatenation of paths $\alpha \beta$ (with endpoints at vertices) in $G$, $f_\#(\alpha \beta )$ is obtained from the concatenation $f_\#(\alpha ) f_\#(\beta )$ by canceling at most $BCC(f)$ edges from terminal end of $f_\#(\alpha )$ with at most $BCC(f)$ edges from initial end of $f_\#(\beta )$ (see [Reference Bestvina, Feighn and HandelBFH00, Lemma 2.3.1]).

We recall the notion of critical constant from [Reference Bestvina, Handel and FeighnBHF97, p. 219].

Critical constant. Let $f:G\to G$ be a CT for some exponentially growing $\phi \in {\mathrm {Out}(\mathbb {F}) }$ with $H_r$ being an exponentially growing strata with associated Perron–Frobenius eigenvalue $\lambda$ and $\Lambda ^+_\phi$ be the attracting lamination associated to $H_r$. The number ${2BCC(f)}/{\lambda -1}$ is called the critical constant for $f$ corresponding to $H_r$.

Let $C$ be some number greater than critical constant for $f$ corresponding to $H_r$. Suppose $\alpha \beta \gamma$ is a concatenation of height $r$ paths where $\beta$ is some height $r$ legal segment with $r$-length $\geq C$ and $\alpha, \gamma$ are height $r$ legal paths. Note that $f_\#(\beta )$ is a legal path of height $r$. Then bounded cancelation implies that $f^k_\#(\alpha \beta \gamma )$ contains a legal segment of height $r$. It can be easily seen that because $C$ exceeds the critical constant, there is some $1\geq \mu >0$ (depending on $C$) such that after tightening operation, the subpath of $f^k_\#(\beta )$ which survives as a height $r$ legal segment in $f^k_\#(\alpha \beta \gamma )$ has $r$-length $\geq \mu \lambda ^k{|\beta |}_{H_r} \geq \mu \lambda ^k C$.

It is perhaps worth noting here that the $r$-length of $\beta$ grows exponentially fast under iteration by $f$, whereas the cancelations due to tightening at both ends of iterates of $\beta$ happen at most by a fixed amount determined by the bounded cancelation constant of $f$. Thus, we may remove the requirement that $\alpha, \gamma$ are $r$-legal paths and instead assume that they are height $r$ paths such that for some $M$ we have $|f^M_\#(\alpha )|, |f^M_\#(\gamma )| > BCC(f^M)$. Then for that same $M$, the subpath of $f^M_\#(\beta )$ which survives as height $r$ legal segment in $f^M_\#(\alpha \beta \gamma )$ has $r$-length at least $\mu \lambda ^M C$.

To summarize: if we have a path in $G$ which has some $r$-legal ‘central’ subsegment (such as $\beta$ above) of length greater than the critical constant, with sufficient ‘protection’ on both sides (such as $\alpha, \gamma$ absorbing the cancelations above), then the bounded cancelation lemma protects this ‘central’ segment from completely canceling out under the tightening operation while iterating under $f$.

Following the work of Bestvina, Feighn and Handel [Reference Bestvina, Handel and FeighnBHF97] we define the following notion of legality for any number $C>0$ which exceeds the critical constant for $f$.

Notation. For any path $\alpha$ in $G$, we decompose $\alpha = \epsilon _1\delta _1\cdots \epsilon _k\delta _k$. where each $\epsilon _i$ is a path which does not contain any subsegment that is an element of $\langle \mathcal {Z}, \hat {\sigma }\rangle$ and each $\delta _i$ is a path that is entirely carried by $\langle \mathcal {Z}, \hat {\sigma }\rangle$.

Let ${|\alpha |}_{\langle \mathcal {Z}, \hat {\sigma }\rangle } = \sum {|\epsilon _i|}_G$, where ${|\alpha _i|}_G$ denotes the length of $\alpha _i$ in $G$. Recall that no edge of $H_r$ is individually an element by $\langle \mathcal {Z}, \hat {\sigma }\rangle$ because it grows exponentially under iteration and limits to a generic leaf of the attracting lamination associated to $H_r$. However, a concatenation of edges of $H_r$ can give us an indivisible Nielsen path (in that case, it would be one of the $\delta _i$) which is represented by an element of $\langle \mathcal {Z}, \hat {\sigma }\rangle$.

Therefore, ${|\alpha |}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }$ denotes the edge length of $\alpha$ in $G$ relative to ${\langle \mathcal {Z}, \hat {\sigma }\rangle }$, i.e. length of $\alpha$ in $G$, not counting the copies of subpaths of $\alpha$ which are carried by ${\langle \mathcal {Z}, \hat {\sigma }\rangle }$, where $\sigma$ is the unique indivisible Nielsen path of height $r$ (if it exists). In what follows, $\rho$ will denote the unique indivisible Nielsen path of height $s$ in the CT map $f':G'\to G'$ for $\phi ^{-1}$. From train-track theory we know that if $\rho$ is a closed Nielsen path, then so is $\sigma$ and the conjugacy classes of $\rho$ and $\sigma$ are same up to reversal, because the laminations associated to the strata $H_r$ and $H'_s$ are dual to each other. In addition, recall from Bestvina–Feighn–Handel train-track theory that $\sigma$ has exactly one illegal turn in $H_r$ and, hence, does not occur as a subpath of any generic leaf of $\Lambda ^+_\phi$.

Standing assumptions for this section. For the rest of this section, fix some $C$ much larger than the critical constants of $f, f'$ corresponding to the EG strata $H_r, H'_s$, respectively. By increasing $C$ if necessary, also assume that $C > \max \{2BCC(f), 2BCC(f') \}$. In addition, fix the $\mu$ corresponding to this choice of $C$ arising from the notion of critical constant.

Definition 3.4 For any circuit $\alpha$ in $G$, decompose $\alpha$ into a concatenation of paths each of which is either contained in the complement of $G_{r}$ or a path in $G_{r-1}$ or a path of height $r$. By $\alpha _i$ we denote a component (if it exists) in this decomposition of $\alpha$ which is a generic leaf segment of height $r$ and ${|\alpha _i|}_{H_r} \geq C$. We look at the following ratio for this decomposition:

\[ \frac{\sum{|\alpha_i|}_{H_r}}{{|\alpha|}_{\langle\mathcal{Z}, \hat{\sigma}\rangle}}. \]

The $H_{r}$-legality of $\alpha$ is defined as the maximum of the above ratio over all such decompositions of $\alpha$ and denoted by ${LEG}_{H_r^\sigma }(\alpha )$. Define ${LEG}_{H_r^\sigma }(\alpha )=0$ if ${|\alpha |}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }= 0$.

The following lemma shows that there exists some uniform exponent $M$, such that iterating $\alpha$ by $\phi ^{\pm M}$ gives us enough legality to eventually be proportional to the relative length of $\phi ^{\pm M}$ in their respective marked graphs.

Proof. We choose a long generic leaf segment $\gamma ^+$ of some generic leaf of $\Lambda ^+_\phi$ and let ${|\gamma ^+|}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }\gg 3C$. Use $\gamma ^+$ to define a weak attracting neighborhood $V^+_\phi$ for $\Lambda ^+_\phi$. Similarly, choose a long generic leaf segment $\gamma ^-$ such that ${|\gamma ^-|}_{\langle \mathcal {Z}', \hat {\rho }\rangle }\gg 3C$ and define an attracting neighborhood $V^-_\phi$. The weak attraction theorem tells us that there exists some $M^+>0$ such that for any conjugacy class $\alpha$ that is not carried by $\mathcal {A}_{\rm na}(\Lambda ^\pm _\phi )$ either $\alpha \in V^-_\phi$ or $\phi ^m_\#(\alpha )\in V^+_\phi$ for every $m\geq M^+$. By a symmetric argument applied on $\phi ^{-1}$, we get $M^- > 0$. Let $M = \mathrm {max} \{M^+, M^-\}$. By increasing $M$ if necessary, we also assume that $\mu \lambda ^M > 2$, where $\lambda$ is the Perron–Frobenius eigenvalue for $H_r$, $\mu$ is the constant arising out from the discussion on critical constant. We now show that there exists some $\epsilon _1>0$ such that for every conjugacy class $\alpha$ not carried by the nonattracting subgroup system either ${LEG}_{H_r^\sigma }(\phi ^M_\#(\alpha )) \geq \epsilon _1$ or ${LEG}_{H_s^{'\rho }}(\phi ^{-M}_\#(\alpha ))\geq \epsilon _1$. We use the change of markings map $\tau$ to push paths in $G$ to paths in $G'$ as follows: if $\alpha$ is a path in $G$, then we use the notation $\alpha '$ to denote the path $\tau _\#(\alpha )$ in $G'$.

Suppose there does not exist any such $\epsilon _1 >0$ which satisfies the property stated previously, i.e. for each positive integer $i$, there exists a conjugacy class which is represented by a circuit $\delta _i$ in $G$ such that ${|\delta _i|}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }$, ${|\delta '_i|}_{\langle \mathcal {Z}', \hat {\rho }\rangle } > 0$ (because they are not carried by the nonattracting subgroup system) and legality ratios of $f^M_\#(\delta _i)$ in $G$ and ${f'}^M_\#(\delta '_i))$ in $G'$ are both less than ${1}/{i}$. We argue to a contradiction by constructing a line which violates the weak attraction theorem.

First we handle the case when ${|\delta _i|}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }$ (respectively, ${|\delta '_i|}_{\langle \mathcal {Z}', \hat {\rho }\rangle }$) is unbounded. Pass to a subsequence, if necessary, and assume that both ${|\delta _i|}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }$ and ${|\delta '_i|}_{\langle \mathcal {Z}', \hat {\rho }\rangle }$ diverge to $\infty$ as $i\to \infty$. As $\delta _i$ are not carried by the nonattracting subgroup system, we have either $f^M_\#(\delta _i)\in V^+_\phi$ or $\delta '_i \in V^-_\phi$ (which implies ${{f'}_\#^{M}}(\delta '_i)\in V^-_\phi$). However, given our assumption that ${|\delta _i|}_{\langle \mathcal {Z}, \hat {\sigma }\rangle } \to \infty$ and that the legality ratio of $f^M_\#(\delta _i)$ is small for all sufficiently large $i$, the height $r$ leaf segments of $f^M_\#(\delta _i)$ contributing to its legality ratio is small compared with ${|f^M_\#(\delta _i)|}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }$ (which diverges to $\infty$). We claim that we can find some sequence of subpaths $\alpha _i\subset \delta _i$ which has the following properties for all sufficiently large $i$:

1. (i) $\alpha _i$ not carried by $\langle \mathcal {Z}, \hat {\sigma }\rangle$;

2. (ii) ${|\alpha _i|}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }$ diverges to $\infty$;

3. (iii) for all sufficiently large $i$, $f^M_\#(\alpha _i)$ does not contain any height $r$ generic leaf segment of $\Lambda ^+_\phi$ of $r$-length $\geq C$;

4. (iv) for all sufficiently large $i$, $\alpha _i$ cannot be written as a concatenation $\alpha _i = \alpha _{x_i} \alpha _{y_i} \alpha _{z_i}$, where $\alpha _{y_i}$ is a generic leaf segment of $\Lambda ^+_\phi$ of height $r$ with ${|\alpha _{y_i}|}_{H_r} \geq C$ and $\alpha _{x_i}, \alpha _{z_i}$ are paths of height $r$ with $|f^M_\#({\alpha }_{x_i})|, |f^M_\#({\alpha }_{z_i})| > BCC(f^M)$.

It is clear that items (i) and (ii) are easily satisfied. To see why we can choose the $\alpha _i$ so that item (iii) is also true, we argue by contradiction. If property (iii) fails, then there exists some $L > 0$ such that for each sufficiently large $i$, and for any subpath $\beta _i$ of $\delta _i$ with $|\beta _i|_{\langle \mathcal {Z}, \hat {\sigma }\rangle }\geq L$, $|f^M_\#(\beta _i)|_{\langle \mathcal {Z}, \hat {\sigma }\rangle }$ contains a height $r$ leaf segment of $r$-length $\geq C$. For every sufficiently large $i$, write $\delta _i$ as a concatenation of subpaths $\beta ^j_i$ such that $|\beta ^j_i|_{\langle \mathcal {Z}, \hat {\sigma }\rangle } = L$ for each $j$, except perhaps for at most one value of $j$. We then have that for each $j$, $f^M_\#(\beta ^j_i)$ contains a $r$-legal generic leaf segment, say $\gamma ^j_i$, having $r$-length $\geq C$. Fixing $i$ and summing over the $j$, we obtain

\[ {LEG}_{H_r^\sigma}(f^M_\#(\delta_i)) \geq \frac{\sum_j {|\gamma^j_i|}_{H_r}}{|f^M_\#(\delta_i)|_{\langle\mathcal{Z}, \hat{\sigma}\rangle}} \geq \frac{C}{|f^M_\#(\delta_i)|_{\langle\mathcal{Z}, \hat{\sigma}\rangle}} \frac{|\delta_i|_{\langle\mathcal{Z}, \hat{\sigma}\rangle}}{L+1} \geq \frac{C}{\lambda^M (L+1)} > 0, \]

where the first inequality is due to the maximality clause in the definition of legality ratio. The third inequality is obtained by using Lemma 3.3. However, our assumption that the legality ratio of $f^M_\#(\delta _i)$ is less than ${1}/{i}$ for all sufficiently large $i$ now gives us the desired contradiction.

If $\alpha _i$ satisfy items (i), (ii), (iii), then item (iv) must also be satisfied. Suppose property (iv) fails and let ${\alpha }_{y_i}$ be a subsegment of $\alpha _i$ such that ${\alpha }_{y_i}$ is a height $r$ generic leaf segment with $r$-length $\geq C$. We can then write $\alpha _i = {\alpha }_{x_i} {\alpha }_{y_i} {\alpha }_{z_i}$ where ${\alpha }_{x_i}, {\alpha }_{z_i}$ are paths of height $r$ with $|f^M_\#({\alpha }_{x_i})|, |f^M_\#({\alpha }_{z_i})| > BCC(f^M)$. This implies that $f^M_\#(\alpha _i)$ contains a generic leaf segment of $r$-length at least $\mu \lambda ^M {{|\alpha }_{y_i}|}_{H_r}$ (follows from definition of critical constant). Since ${|{\alpha }_{y_i}|}_{H_r} \geq C$ and $\mu \lambda ^M > 2$, we see that $f^M_\#(\alpha _i)$ contains a generic leaf segment of $r$-length greater than $C$, contradicting property (iii).

Suppose that for every such sequence of $\alpha _i$ that satisfies items (i)–(iv), $\alpha '_i$ fails to satisfy item (ii) for the map $f': G'\to G'$ for all sufficiently large $i$. This would imply that for any sufficiently large subpath of $\delta _i$ whose $f^M_\#$ image does not contain a central generic leaf segment of $r$-length at least $C$, the ${f'}^M_\# \tau _\#$ image contains a central long generic leaf segment of $s$-length $\geq C$. However, this would violate that the legality ratios of both $f^M_\#(\delta _i)$ and ${f'}^M_\#(\delta '_i)$ converge to $0$. Therefore, we assume that we have a sequence $\alpha _i\subset \delta _i$ which satisfies items (i)–(iv) for $f:G\to G$ and $\alpha '_i$ satisfies similar properties in $f': G' \to G'$.

As the lengths of $\alpha _i$ in $G$ diverge to $\infty$, we may assume (after perhaps adding a few edges and changing the length of $\alpha _i$ by at most $|G|$, where $|G|$ denotes the total number of edges in $G$) that for all sufficiently large $i$, $\alpha _i$ is a circuit not carried by $\langle \mathcal {Z}, \hat {\sigma }\rangle$, because ${|\alpha _i|}_{\langle \mathcal {Z}, \hat {\sigma }\rangle } \to \infty$. In addition, because the circuits $\alpha _i$ are not carried by $\langle \mathcal {Z}, \hat {\sigma }\rangle$, [Reference Handel and MosherHM20, Lemma 1.11] tells us that there exists a line $\ell$ which is a weak limit of some subsequence of $\alpha _i$ and $\ell$ is not carried by $\langle \mathcal {Z}, \hat {\sigma }\rangle$. This, by definition, implies that $\ell$ is not carried by $\mathcal {A}_{\rm na}(\Lambda ^\pm _\phi )$. Therefore, neither is $\ell$ carried by the nonattracting subgroup system nor does the realization of $\ell$ in $G$ or $G'$ contain $\gamma ^+$ or $\gamma ^-$ as a subpath (i.e. $\ell$ does not belong to the chosen attracting neighborhoods) by using properties (ii) and (iv) of $\alpha _i$. In addition, we have that $f^M_\#(\ell )\notin V^+_\phi$ and ${f'}^{M}_\#(\ell )\notin V^-_\phi$ (by property (iii) of $\alpha _i$), but this violates the weak attraction theorem.

Next we deal with the case when ${|\delta _i|}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }$ (respectively, ${|\delta '_i|}_{\langle \mathcal {Z}', \hat {\rho }\rangle }$) are bounded above. It directly follows that ${|f^M_\#(\delta _i)|}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }$ and ${|f'^M_\#(\delta '_i)|}_{\langle \mathcal {Z}', \hat {\rho }\rangle }$ are both bounded above. Applying the weak attraction theorem, we also have that $f^M_\#(\delta _i)$ has a central leaf segment of length at least $C$, hence the numerator in the legality ratio of $f^M_\#(\delta _i)$ is at least $1$. Similarly for legality ratio of $f'^M_\#(\delta '_i)$. Hence, this contradicts our assumption that both of these legality ratios converge to zero as $i \to \infty$. Thus, the case when the ${|\delta _i|}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }$ is bounded is ruled out.

Hence, we conclude that an $\epsilon _1> 0$ does indeed exist so that either ${LEG}_{H_r^\sigma }(\phi ^M_\#(\alpha )) \geq \epsilon _1$ or ${LEG}_{H_s^{'\rho }}(\phi ^{-M}_\#(\alpha ))\geq \epsilon _1$.

For the final step, take any conjugacy class $\alpha$ not carried by the nonattracting subgroup system and assume without loss that ${LEG}_{H_r^\sigma }(\phi ^M_\#(\alpha )) \geq \epsilon _1$. As we are working with a topmost lamination it follows that ${|f^k_\#(\phi ^M_\#(\alpha ))|}_{\langle \mathcal {Z}, \hat {\sigma }\rangle } \leq \lambda ^k{|\phi ^M_\#(\alpha )|}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }$ for every $k>0$ by using Lemma 3.3. If $m>M$, it now follows from the definitions that ${LEG}_{H_r^\sigma }(\phi ^m_\#(\alpha )) \geq \epsilon$, where $\epsilon = \epsilon _1\mu$. Recall that $\mu \leq 1$ and, therefore, we set $\epsilon = \epsilon _1\mu$ to complete the proof.

Given a conjugacy class $\alpha$ not carried by $\mathcal {A}_{\rm na}(\Lambda ^\pm _\phi )$, the following lemma compares the growth of legal segments of $\alpha$ relative to the size of $\alpha$, when both are being measured in terms of paths not carried by $\langle \mathcal {Z}, \hat {\sigma }\rangle$, i.e. ${|\cdot |}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }$.

Before we state the lemma, we would like to point out a small subtlety regarding the topmost lamination hypothesis. As $\Lambda ^+_\phi$ is a topmost lamination, any exponentially growing strata of height greater than $r$ is carried by $\langle \mathcal {Z}, \hat {\sigma }\rangle$ (see proof of Lemma 3.3). Thus, the only strata of height $s > r$ which can get attracted to $\Lambda ^+_\phi$ are superlinear NEG edges of the form $E_s\mapsto E_su_s$, where $u_s$ is a circuit contained in $G_{s-1}$ and gets attracted to $\Lambda ^+_\phi$. Hence, neither $E_s$ nor $u_s$ are carried by $\langle \mathcal {Z}, \hat {\sigma }\rangle$. In this situation, for any circuit $\alpha$ not carried by the nonattracting subgroup system, the part of $\alpha$ that lies above $H_r$ only grows polynomially and the part that intersects with $H_r$ grows exponentially. This enables us to control the exponent for flaring in the following lemma. However, if we work with a circuit that does not get attracted to any lamination which properly contains $\Lambda ^+_\phi$, the lemma will still be true for all such circuits without the topmost assumption.

If we remove the topmost assumption on $\Lambda ^+_\phi$, then we could have a circuit $\alpha$ that has a long overlap with both $\Lambda ^-$ (which properly contains $\Lambda ^-_\phi$) and $\Lambda ^+_\phi$. It would be very tricky to control the exponents of the following lemma in that case. However, if we consider a conjugacy classes which are attracted to $\Lambda ^+_\phi$, but not weakly attracted to any lamination properly containing $\Lambda ^+_\phi$, both Lemmas 3.5 and 3.6 will still be true for such circuits. This observation will be useful in the proof of Proposition 5.5.

Lemma 3.6 Suppose $\phi \in {\mathrm {Out}(\mathbb {F}) }$ is rotationless and exponentially growing with a lamination pair $\Lambda ^\pm _\phi$ which are topmost for $\phi, \phi ^{-1}$. Then for every $\epsilon >0$ and $A>0$, there is $M_1$ depending only on $\epsilon, A$ such that if ${LEG}_{H_r^\sigma }(\alpha )\geq \epsilon$ for some circuit $\alpha$, then

\[ {|f^m_\#(\alpha)|}_{\langle\mathcal{Z}, \hat{\sigma}\rangle}\geq A {|\alpha|}_{\langle\mathcal{Z}, \hat{\sigma}\rangle} \]

for every $m\geq M_1$.

Proof. Using the description of critical constant we can see that

\begin{align*} {|f^m_\#(\alpha)|}_{\langle\mathcal{Z}, \hat{\sigma}\rangle} &\geq \mu\lambda^m\biggl\{\sum {|\text{generic leaf segments of $\alpha$ with $r$-length} \geq C|}_{H_r}\biggr\} \\ &\geq \mu\lambda^m\epsilon{|\alpha|}_{\langle\mathcal{Z}, \hat{\sigma}\rangle}. \end{align*}

Choose $m$ to be large enough so that $\mu \lambda ^m\epsilon \geq A$.

Recall the nonattracting subgroup system $\mathcal {A}_{\rm na}(\Lambda ^\pm _\phi ) = \{[F^1], [F^2],\ldots,[F^k]\}$ is a mutually malnormal subgroup system of finitely generated subgroups of $ {\mathbb {F} }$ (hence, quasiconvex). Hence, one can start with the Cayley graph of $ {\mathbb {F} }$ and construct the coned-off space $\widehat { {\mathbb {F} } }$ with respect to the collection $\{F^i\}$ and form a electrocuted metric space denoted by ($\widehat { {\mathbb {F} } }, {|\cdot |}_{\rm el}$). The group $ {\mathbb {F} }$ is then strongly hyperbolic relative to the collection $\{F^i\}$. Note, that this property of $\widehat { {\mathbb {F} } }$ is still true if we replace any of the $F^i$ with a conjugate of itself. We stress the fact here that $\mathcal {A}_{\rm na}(\Lambda ^\pm _\phi )$ is not necessarily a free-factor system. For a conjugacy class not carried by $\mathcal {A}_{\rm na}(\Lambda ^\pm _\phi )$, by ${|\alpha |}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }$ we denote the length (relative to $\langle \mathcal {Z}, \hat {\sigma }\rangle$) of the circuit in $G$ that realizes this conjugacy class. By ${\|\alpha \|}_{\rm el}$ we denote the length of the shortest representative of the conjugacy class $\alpha$ in the electrocuted metric space ($\widehat { {\mathbb {F} } }, {|\cdot |}_{\rm el}$). Note that ${\|\alpha \|}_{\rm el}$ is independent of the graph $G$ on which we have the relative train-track map. The following lemma establishes that these two measurements are comparable and, hence, our results are independent of the choice of train-track map. Before we begin, we would like to state an example which will perhaps be useful for the reader to understand the way we are computing the bounds.

The first half of the proof gives an upper bound to the electrocuted length of lifts (to the electrocuted universal cover of marked graph $G$) of circuits $\alpha$ in the graph $G$ in terms of ${|\alpha |}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }$ because lifting a circuit can cause its length to blow up in the electrocuted metric of the universal cover, we need to show that we can control this. Consider the automorphism which is realized by a train-track map on the rose with two edges $f(a) = ab, f(b) = bab$. Here the indivisible Nielsen path $\sigma = abAB$, where $A, B$ are inverses of $a, b$, respectively. Consider the conjugacy class of the circuit $\alpha = a b \sigma A b \sigma$. Then ${|\alpha |}_{\langle \mathcal {Z}, \hat {\sigma }\rangle } = 4$ and ${\|\alpha \|}_{\rm el} = 6$. All the embedded paths in the rose here can be listed as $ab, aB, Ab, AB$ and their inverses. We can get a bound on ${\|\alpha \|}_{\rm el}$ in terms of a multiple of ${|\alpha |}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }$ by counting the number of such embedded paths and copies of the path $\sigma$ each of which can appear at most ${|\alpha |}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }$ times.

The second half of the proof deals with bounding a similar problem which can happen when we project paths from the electrocuted universal cover to the marked graph $G$. An example to keep in mind would be the geodesic $a\overline {\sigma }a$ which gives ${\|a\overline {\sigma }a\|}_{\rm el} = 3$, but its projection has length 4 in the rose. By rewriting $a\overline {\sigma }a$ as $abaBAa$ we see that when we tighten this path in the projection, there are no copies of $\sigma$ in it. Thus, each segment of such type will cause a discrepancy of length at most $4$ in this example. Then we can bound ${|\alpha |}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }$ by a multiple of ${\|\alpha \|}_{\rm el}$ by counting the maximum number of times such discrepancies can occur.

Suppose $\Lambda ^+$ is an attracting lamination for $\phi \in {\mathrm {Out}(\mathbb {F}) }$ and $H_p$ is the EG strata associated to $\Lambda ^+$. If there is a closed indivisible Nielsen path of height $p$ (equivalently, if $H_p$ is a geometric strata [Reference Handel and MosherHM20, Proposition 2.18]), then $\Lambda ^+$ is minimal (i.e. does not contain any other attracting lamination (see [Reference Handel and MosherHM09, Corollary 2.32] and [Reference Handel and MosherHM20, Proposition 2.15])). It is due to this property that we do not need the ‘topmost lamination’ assumption for the following lemma.

Proof. As we are working with a conjugacy class which is not carried by $\mathcal {A}_{\rm na}(\Lambda ^\pm _\phi )$, the circuit $\alpha$ representing such a conjugacy class is not carried by $\langle \mathcal {Z}, \hat {\sigma }\rangle$, hence ${|\alpha |}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }\geq 1$. Let $H_r$ denote the exponentially growing stratum corresponding to the lamination $\Lambda ^+_\phi$.

Let $L'_2 = \max \{{|\tilde {\epsilon }_i|}^{\tilde {G}}_{\rm el}\}$ where $\epsilon _i$ varies over finitely many embedded paths (i.e. no repetition of an edge or its inverse) in $G$ which consist entirely of edges of $G$ which are not carried by $\langle \mathcal {Z}, \hat {\sigma }\rangle$. Each such $\tilde {\epsilon }_i$ is a geodesic (in the electrocuted universal cover) that does not pass through any conepoint. The electrocuted geodesic representing $\tilde {\alpha }$ can be written as a concatenation of paths of type $\tilde {\epsilon }_i$ and geodesics of length one connecting two points inside the copy of a coset, via the conepoint. The number of such $\tilde {\epsilon }_i$ appearing in a decomposition of $\tilde {\alpha }$ is at most ${|\alpha |}_{\langle \mathcal {Z}, \hat {\sigma }\rangle } + 1$ and the number of components of the other type (via the conepoints) is at most ${|\alpha |}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }$. Hence, if $\alpha$ is any circuit in $G$ which is not carried by the nonattracting subgroup system, then

\[ {|\tilde{\alpha}|}^{\tilde{G}}_{\rm el}\leq L'_2 ({|\alpha|}_{\langle\mathcal{Z}, \hat{\sigma}\rangle} + 1) + {|\alpha|}_{\langle\mathcal{Z}, \hat{\sigma}\rangle} \implies {|\tilde{\alpha}|}^{\tilde{G}}_{\rm el}/ {|\alpha|}_{\langle\mathcal{Z}, \hat{\sigma}\rangle}\leq L_2, \]

for some $L_2(=3L'_2)$ independent of $\alpha$.

Suppose next that $w\in {\mathbb {F} }$ is some cyclically reduced word not in the union of $F^i$. Let $\tilde {\alpha }$ be a path in electrocuted universal cover $\tilde {G}$ that represents the geodesic connecting the identity element to $w$, under the lift of the marking on $G$. Suppose $\tilde {\alpha }= \tilde {u}_1 X_1 \tilde {u}_2 X_2\cdots \tilde {u}_n X_n$, where $X_i$ are geodesic paths in $\tilde {G}$ connecting two points in a copy of some coset via the attached conepoint and $u_i$ are geodesic paths in $\tilde {G}$ which connect copies of two electrocuted cosets and does not pass through any conepoint. $\tilde {u}_1, X_n$ could possibly be trivial.

Under this setup we have

\[ {|\tilde{\alpha}|}^{\tilde{G}}_{\rm el}={|\tilde{u}_1|}^{\tilde{G}}_{\rm el} + {|\tilde{u}_2|}^{\tilde{G}}_{\rm el} +\cdots+ {|\tilde{u}_n|}^{\tilde{G}}_{\rm el} + n. \]

Modify $\tilde {\alpha }$ by replacing each $X_i$ with a path $\tilde {v}_i$ inside the corresponding coset, such that $\tilde {v}_i$ is a geodesic in the standard metric on $\tilde {G}$ (i.e. before electrocution). Consider projection of this modified path obtained from $\tilde {\alpha }$ to $G$, and tighten the projection of $\tilde {u}_i$ to get the path $u_i$. This gives the inequality ${|u_i|}_{\langle \mathcal {Z}, \hat {\sigma }\rangle } \leq {|\tilde {u}_i|}^{\tilde {G}}_{\rm el}$. The projection of each $\tilde {v}_i$ is a path carried by $\langle \mathcal {Z}, \hat {\sigma }\rangle$, so it does not contribute to the relative length of the tightened projection of $\tilde {\alpha }$ unless the projection of $\tilde {v}_i$, call it $v_i^*$, is a closed indivisible Nielsen path of height $r$ and there is cancelation between $v^*_i$ and $u_i$ and/or $u_{i+1}$ so that after tightening, the path $u_iv^*_iu_{i+1}$ has no copies of the closed indivisible Nielsen path of height $r$ (this is where we use the property that an EG strata of height $p > r$, which is not contained in $\mathcal {Z}$, has no corresponding closed indivisible Nielsen path of height $p$). This could bump up the length of the projected path. However, each such $v^*_i$ can contribute a maximum of length $2|G| +1$ to the tightened (modified) projection of $\tilde {\alpha }$ (call it $\alpha$).

Thus, $\alpha$ is obtained by tightening of $u_1v^*_1u_2v^*_2\cdots u_nv^*_n$, to give us a circuit which represents the conjugacy class of the word determined by $\tilde {\alpha }$. Let $j$ vary over all such indices of $v^*_i$ where $v^*_i$ is the indivisible Nielsen path of height $r$. We then obtain

\[ {|\alpha|}_{\langle\mathcal{Z}, \hat{\sigma}\rangle} \leq \sum {|u_i|}_{\langle\mathcal{Z}, \hat{\sigma}\rangle} + \sum {|v^*_{j}|}_{G} \leq {|\tilde{\alpha}|}^{\tilde{G}}_{\rm el} + {|\tilde{\alpha}|}^{\tilde{G}}_{\rm el}\cdot (2|G|+1) \leq 2{|\tilde{\alpha}|}^{\tilde{G}}_{\rm el} (1 + |G|). \]

Therefore, ${|\alpha |}_{\langle \mathcal {Z}, \hat {\sigma }\rangle } / {|\tilde {\alpha }|}^{\tilde {G}}_{\rm el} \leq L_1 (=2(1+|G|))$. Combining the two inequalities, we obtain

(1)\begin{equation} \frac{1}{L_1} \leq {|\tilde{\alpha}|}^{\tilde{G}}_{\rm el} / {|\alpha|}_{\langle\mathcal{Z}, \hat{\sigma}\rangle} \leq L_2. \end{equation}

For the last step use the lift of marking map on $G$ to $\tilde {G}$ and observe that there is an $ {\mathbb {F} }$-equivariant quasi-isometry between $\tilde {G}$, with electrocuted metric, and the universal cover of the standard rose for $ {\mathbb {F} }$ with the electrocuted metric obtained by electrocuting cosets of the collection of subgroups $\{F^i\}$. Hence, there exists some $K'>0$ such that for every cyclically reduced word $w\in {\mathbb {F} } \setminus \cup F^i$ we have

(2)\begin{equation} \frac{1}{K'}\leq {|w|}_{\rm el}/{|\tilde{\alpha}_w|}^{\tilde{G}}_{\rm el} \leq K', \end{equation}

where $\tilde {\alpha }_w$ is the electrocuted geodesic in $\tilde {G}$ connecting the image of identity element to image of $w$ under the marking map on $\tilde {G}$ and ${|w|}_{\rm el}$ is the length of the electrocuted geodesic in $\hat { {\mathbb {F} } }$ connecting identity element and $w$. Hence, combining the inequalities (1) and (2) above we can conclude that there exists some $K>0$ such that

\[ K \geq {|\alpha|}_{\langle\mathcal{Z}, \hat{\sigma}\rangle}/{\|\alpha\|}_{\rm el} \geq \frac{1}{K}, \]

where ${\|\alpha \|}_{\rm el}$ is the electrocuted length, in $\hat { {\mathbb {F} } }$, of the cyclically reduced word whose conjugacy class is represented by $\alpha$ in $G$.

Observe that the proof actually tells us that ${|x_j\Phi (x_j)\Phi ^2(x_j)\cdots \Phi ^n(x_j)|}_{\rm el}$ grows exponentially fast as $n\to \infty$. This observation will be useful when we try to prove the cone-bounded hallway flaring condition in Lemma 3.12.

In addition, note that this corollary is trivially true if there are no peripheral subgroups, i.e. if the nonattracting subgroup system is trivial. This is because the triviality of the nonattracting subgroup system implies that every conjugacy class is attracted to the corresponding attracting lamination and, thus, will be exponentially growing.

Proposition 3.9 (Conjugacy flaring)

Suppose $\phi \in {\mathrm {Out}(\mathbb {F}) }$ is rotationless and exponentially growing with a lamination pair $\Lambda ^\pm _\phi$ which are topmost. Then there exists some $M_0>0$ such that for every conjugacy class $\alpha$ not carried by $\mathcal {A}_{\rm na}(\Lambda ^\pm _\phi )$, we have

\[ 3{\|\alpha\|}_{\rm el} \leq \max \{{\|\phi^m_\#(\alpha)\|}_{\rm el}, {\|\phi^{-m}_\#(\alpha)\|}_{\rm el}\} \]

for every $m\geq M_0$.

Proof. Let $M, \epsilon$ be as in Lemma 3.5 and let $\lambda _r, \lambda _s$ be stretch factors associated to the EG strata for $\Lambda ^+_\phi$ and $\Lambda ^-_\phi$, respectively. Every conjugacy class not carried by $\mathcal {A}_{\rm na}(\Lambda ^\pm _\phi )$ is stretched at most by a factor of $\lambda _r$ under $\phi$ and by factor of $\lambda _s$ under $\phi ^{-1}$. Thus, for every conjugacy class $\alpha$ not carried by $\mathcal {A}_{\rm na}(\Lambda ^\pm _\phi )$, we get ${|\phi ^M_\#(\alpha )|}_{\langle \mathcal {Z}, \hat {\sigma }\rangle } \leq \lambda ^M_r {|\alpha |}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }$, which implies ${|\alpha |}_{\langle \mathcal {Z}, \hat {\sigma }\rangle } \leq \lambda _r^M {|\phi ^{-M}_\#(\alpha )|}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }$ by replacing $\alpha$ with $\phi ^{-M}(\alpha )$. By a symmetric argument we obtain an inequality involving $\lambda _s$. Use Lemma 3.7 to choose some number $D>0$ such that we have ${\|\phi ^{M}_\#(\alpha )\|}_{\rm el} \geq {\|\alpha \|}_{\rm el}/D$ and ${\|\phi ^{-M}_\#(\alpha )\|}_{\rm el} \geq {\|\alpha \|}_{\rm el}/D$ for every conjugacy class $\alpha$ not carried by $\mathcal {A}_{\rm na}(\Lambda ^\pm _\phi )$. Note that $D$ is chosen arbitrarily here, only to track the ratio of the numbers above and it depends only on $M, \lambda _r, \lambda _s$. We also have ${LEG}_{H_r^\sigma }(\phi ^M_\#(\alpha )) \geq \epsilon$ or ${LEG}_{H^{'\rho }_s}(\phi ^{-M}_\#(\alpha )) \geq \epsilon$.

By applying Lemma 3.7, we may choose some constant $K$ such that for every conjugacy class $\alpha$ as above, either $K \geq {|\phi ^M_\#(\alpha )|}_{\langle \mathcal {Z}, \hat {\sigma }\rangle }/{\|\phi ^M_\#(\alpha )\|}_{\rm el}\geq {1}/{K}$ or $K\geq {|\phi ^{-M}_\#(\alpha )|}_{\langle \mathcal {Z}', \hat {\rho }\rangle }/{\|\phi ^{-M}_\#(\alpha )\|}_{\rm el} \geq {1}/{K}$.

For concreteness assume that ${LEG}_{H_r^\sigma }(\phi ^m_\#(\alpha )) \geq \epsilon$. Then by applying Lemma 3.6 with $\epsilon$ and $A= 3DK^2$, we get that there exists some $M_1$ such that for all $m\geq M_0 > M + M_1$, such that

(3)\begin{align} {\|\phi^m_\#(\alpha)\|}_{\rm el} &\geq \frac{1}{K}{|\phi^m_\#(\alpha)|}_{\langle\mathcal{Z}, \hat{\sigma}\rangle} \nonumber\\ &\geq \frac{1}{K}3DK^2{|\phi^{M}_\#(\alpha)|}_{\langle\mathcal{Z}, \hat{\sigma}\rangle} \nonumber\\ &\geq 3DK \frac{1}{K} {\|\phi^{M}_\#(\alpha)\|}_{\rm el}= 3D{\|\phi^{M}_\#(\alpha)\|}_{\rm el} \nonumber\\ &\geq 3D\frac{1}{D}{\|\alpha\|}_{\rm el} = 3{\|\alpha\|}_{\rm el}. \end{align}

The following lemma proves that conjugacy flaring implies the Mj–Reeves cone-bounded hallways strictly flaring condition. The technique used in the proof is similar to that used by Bestvina, Feighn and Handel in [Reference Bestvina, Handel and FeighnBHF97, Theorem 5.1]. Recall that we are working with an exponentially growing outer automorphism with a dual lamination pair $\Lambda ^\pm _\phi$ such that $\mathcal {A}_{\rm na}(\Lambda ^\pm _\phi ) = \{[F^1], [F^2],\ldots,[F^k]\}$.

Proposition 3.10 (Hallway flaring)

Suppose $\phi \in {\mathrm {Out}(\mathbb {F}) }$ is rotationless and exponentially growing with a lamination pair $\Lambda ^\pm _\phi$ which are topmost. Choose a lift $\Phi \in {\mathrm {Aut}(\mathbb {F}) }$ of $\phi$. There exists numbers $N_\Phi >0$ and $L_\Phi >0$ such that for every word $w\in {\mathbb {F} } \setminus \bigcup _i F^i$ with ${|w|}_{\rm el} \geq L_\Phi$ we have

\[ 2{|w|}_{\rm el} \leq \max\{{|\Phi^n_\#(w)|}_{\rm el}, {|\Phi^{-n}_\#(w)|}_{\rm el}\} \]

for every $n \geq N_\Phi$. Moreover, if $w$ is cyclically reduced, then the result holds for all $w$ such that ${|w|}_{\rm el} > 1$.

Proof. Recall that any subgraph $H\subset G$ determines a free-factor system $\mathcal {F}_r$, the components of $\mathcal {F}_r$ being determined by the connected components of $H$. Consider the filtration element $G_r$ corresponding to $H_r$ and let $\mathcal {F}_r$ be the free-factor system determined by $G_r$. The free-factor support of $\Lambda ^\pm _\phi$, denoted by $[B_r]$ say, has the property $[B_r] \sqsubset \mathcal {F}_r$ (see [Reference Bestvina, Feighn and HandelBFH00, § 2.6]). As $\Phi$ leaves some conjugate of $B_r$ invariant, let $B$ denote such an invariant free factor. We note here that $B$ is not necessarily a proper free factor. However, we can choose a basis of $B$ so that none of the conjugacy classes of the basis elements are carried by $\mathcal {A}_{\rm na}(\Lambda ^\pm _\phi )$. We work with such a basis.

Let $L' = \max \{{|\Phi ^i_\#(b_j)|}_{\rm el}\mid i=0,\pm 1,\pm 2,\ldots,\pm M_0\}$ where $b_j\in B$ varies over all the basis elements of $B$ and $M_0$ is the constant from Proposition 3.9. By the description of $L'$, we have that ${|\Phi ^{i}_\#(b_j)|}_{\rm el}\geq 1/L'$ for all $i, j$ as described previously.

Assume $w\in {\mathbb {F} } \setminus \bigcup _i F^i$ and ${|w|}_{\rm el} \geq L'-3$.

The proof is by induction. For the base case let $n = M_0$.

If $w$ is a cyclically reduced word, then conjugacy class of $w$ is not carried by $\mathcal {A}_{\rm na}(\Lambda ^\pm _\phi )$ and so by using Proposition 3.9 we have

\[ \max\{{|\Phi^n_\#(w)|}_{\rm el}, {|\Phi^{-n}(w)|}_{\rm el}\} \geq 3{|w|}_{\rm el} \geq 2{|w|}_{\rm el}. \]

If $w$ is not cyclically reduced, then we can choose a basis element $k\in B$, such that $kw\in {\mathbb {F} } \setminus \cup F^i$ is a cyclically reduced word. Hence, we get the same inequality as previously, but with $w$ being substituted by $kw$.

For concreteness, suppose that ${|\Phi ^n_\#(kw)|}_{\rm el} \geq 3{|kw|}_{\rm el}$. Then we have $3{|kw|}_{\rm el}\leq {|\Phi ^n_\#(kw)|}_{\rm el}\leq {|\Phi ^n_\#(w)|}_{\rm el}+{|\Phi ^n_\#(k)|}_{\rm el}\leq {|\Phi ^n_\#(w)|}_{\rm el} +L'$. This implies that $3 + 3{|w|}_{\rm el} - L' \leq 3{|kw|}_{\rm el} - L' \leq {|\Phi ^n_\#(w)|}_{\rm el}$ since ${|k|}_{\rm el} = 1$ (because $k$ is a basis element) and there is no cancelation between $k$ and $w$. As we have ${|w|}_{\rm el} \geq L'-3$, the above inequality then implies $2{|w|}_{\rm el}\leq {|\Phi ^n_\#(w)|}_{\rm el}$ and we are done with the base case for our inductive argument. In addition, note that $L'-3 < {|\Phi ^n_\#(w)|}_{\rm el}$ and so $\Phi ^n_\#(w)\notin \cup F^i$. By repeatedly applying the above argument, one concludes that $\Phi ^{sn}_\#(w) \notin \cup F^i$ for all $s>0$.

Now assume that $M_0< n$ for the inductive step. First observe that from what we have proven so far, given any integer $s>0$ we have either ${|\Phi ^{sM_0}_\#(w)|}_{\rm el}\geq 2^{s}{|w|}_{\rm el}$ or ${|\Phi ^{-sM_0}_\#(w)|}_{\rm el} \geq 2^{s}{|w|}_{\rm el}$. Fix some positive integer $s_0$ such that $2^{s_0}> 2L'$. Any integer $n>s_0M_0$ can be written as $n= sM_0 + t$ where $0\leq t < M_0$ and $s_0\leq s$. If ${|\Phi ^{sM_0}_\#(w)|}_{\rm el}\geq 2^{s}{|w|}_{\rm el}$, then we can deduce

\[ {|\Phi^n_\#(w)|}_{\rm el} = {|\Phi^{sM_0+t}_\#(w)|}_{\rm el}\geq 2^s{|w|}_{\rm el}/L' \geq 2{|w|}_{\rm el}. \]

Similarly, when ${|\Phi ^{-sM_0}_\#(w)|}_{\rm el}\geq 2^{s}{|w|}_{\rm el}$, we can prove by a symmetric argument that ${|\Phi ^{-n}_\#(w)|}_{\rm el} \geq 2{|w|}_{\rm el}$. Set $L_\Phi \geq L'-3$ and $N_\Phi > s_0M_0$ to conclude the proof.

For the rest of the paper, fix some lift $\Phi \in {\mathrm {Aut}(\mathbb {F}) }$ of $\phi$. Observe that the construction of the nonattracting subgroup system (in particular, the fact that it is a vertex group system) implies that $\Phi$ can leave at most one of $F^i$ invariant (see Lemma 3.1(iv)). This implies that if $\Phi (F^i)= w_iF^i w_i^{-1}$ and $\Phi (F^j) = w_j F^j w_j^{-1}$, then $w_i\neq w_j$ if $i\neq j$. For any $w\in {\mathbb {F} }$, use the notation $w^{-1}\Phi$ to denote the composition of the inner automorphism $x\mapsto w^{-1} x w$ with $\Phi$. Let $\Phi _i = w_i^{-1}\Phi$ and $\Phi _j= w_j^{-1} \Phi$ in $\Gamma$. Applying Corollary 3.8, we see that $w_j^{-1}w_i\notin F^j$ and $w_i^{-1}w_j\notin F^i$. Using the action of $\Phi$ on $ {\mathbb {F} }$ we get $\Phi ^n_i = w_i^{-1}\Phi (w_i^{-1})\cdots \Phi ^{(n-1)}(w_i^{-1})\Phi ^n$.

To be in sync with the framework for the cone-bounded hallway flaring condition, we work with a lift $\Phi _i$ (constructed from $\Phi$) such that $\Phi _i(F^i)=F^i$ and $\Phi _i(F^j) = x_j F^j x_j^{-1}$. Then $\Phi _j = x_j^{-1}\Phi _i$ is a lift which leaves $F^j$ invariant and $\Phi _j(F^i)= x_j^{-1} F^i x_j$. A simple computation using the action of the automorphism $\Phi _i$ on $\Gamma$, we get that $\Phi ^{-n}_i\Phi ^n_j = \Phi _i^{-n}(x_j^{-1})\Phi _i^{-(n-1)}(x_j^{-1})\cdots \Phi ^{-1}_i(x_j^{-1})$. This means that $\Phi ^{-n}_i\Phi ^n_j$ is an inner automorphism which is defined by the conjugating word $\Phi _i^{-n}(x_j^{-1})\Phi _i^{-(n-1)}(x_j^{-1})\cdots \Phi ^{-1}_i(x_j^{-1}) \in {\mathbb {F} }$. Recalling Corollary 3.8 (and the remark just after it) we have that this conjugator word grows exponentially as $n\to \infty$.

We have to bound the exponent to get flaring, which is done in the following lemma, and it gives the cone-bounded hallways strictly flaring condition as we soon show.

Lemma 3.12 (Cone-bounded hallway flaring)

Suppose $\phi \in {\mathrm {Out}(\mathbb {F}) }$ is rotationless and exponentially growing with a lamination pair $\Lambda ^\pm _\phi$ which are topmost. Let $\Phi _i$ be a lift such that $\Phi _i(F^i)=F^i$ and $\Phi _i(F^j) = x_j F^j x_j^{-1}$ for some $x_j\in {\mathbb {F} }$ when $j\neq i$. Then there exists some $N_c > 0$ such that

\[ 2{|x_j|}_{\rm el} \leq \max\{{|{\Phi_i}^n_\#(x_j)|}_{\rm el}, {|{\Phi_i}^{-n}_\#(x_j)|}_{\rm el}\} \]

for every $n\geq N_c$ (independent of $i,j$) and for every such $j\neq i$.