The dynamics and geometry of free group endomorphisms

Word-hyperbolic groups are groups that act geometrically (properly and cocompactly) on proper $\delta$-hyperbolic spaces and these spaces are the coarse equivalents of negatively curved geometric spaces. Word-hyperbolic groups are an important class of groups introduced by Misha Gromov [14] and one of the fundamental problems in geometric group theory is determining necessary and sufficient algebraic conditions for a group to be word-hyperbolic.

Finite type can be thought of as a strengthening of finite presentation for torsion-free groups. Baumslag-Solitar groups are two-generator one-relator groups with the presentation $BS(m,n)=\langle a,t~{}|~{}t^{-1}a^{m}t=a^{n}\rangle$ for $m,n\neq 0$. It is now a classical fact that a word-hyperbolic group cannot have subgroups isomorphic to Baumslag-Solitar groups. The question, historically attributed to Gromov, asks if Baumslag-Solitar subgroups are the only essential obstruction to word-hyperbolicity for finite type groups. It has been answered in the affirmative for certain classes of groups: the fundamental groups of closed $3$-manifolds (Thurston [30, 31], Perelman) and free-by-cyclic groups (Brinkmann [8]). The main result of this paper extends the latter to a larger class of HNN extensions.

The HNN extension has the presentation $F*_{A}=\langle F,t~{}|~{}t^{-1}at=\phi(a),~{}\forall a\in A\rangle$. If $A=F=\phi(A)$, then $F*_{A}=F\rtimes_{\phi}\mathbb{Z}$ is a free-by-cyclic group and this case was proven by Peter Brinkmann. Our proof of the main theorem above uses Brinkmann’s result. If $A=F\neq\phi(A)$, then $F*_{A}=F*_{\phi}$ is a strictly ascending HNN extension and coarse hyperbolization for this class of groups was our original motivation. Finally, when $A\neq F$, we more or less reduce this to the case $A=F$.

The case $A=F$ of the main theorem (Theorem 6.7) proved to be difficult since it required an understanding of the dynamics of injective nonsurjective endomorphisms of free groups and no such complete study had been carried out yet. Patrick Reynolds studied the dynamics of irreducible nonsurjective endomorphisms of free groups [27] and we previously used this work to prove an instance of the main theorem where $\phi:F\to F$ is irreducible but not surjective [23, 24]. In this case, we showed that the strictly ascending HNN extension $F*_{\phi}$ is always word-hyperbolic. Unfortunately, it remained unclear how Reynolds’ work could be generalized to all injective nonsurjective endomorphisms and no further progress on the problem had been made.

The first part of this paper (Sections 1–4) is a self-contained systematic study of all injective nonsurjective endomorphisms of free groups and is the main novel contribution of the paper. We hereby present a summary of the important results from these sections. Note that the statements given here are not the complete statements of the cited results.

This means any injective nonsurjective endomorphism $\phi:F\to F$ is induced by a graph map $f:\Gamma\to\Gamma$ such that: 1) some possibly empty or disconnected proper subgraph $\Gamma^{\prime}\subset\Gamma$ is $f$-invariant and the restriction of $f$ to $\Gamma^{\prime}$ is a homotopy equivalence; 2) some $f$-invariant proper subgraph $\Gamma^{\prime\prime}\supset\Gamma^{\prime}$ has an $f^{n}$-image contained in $\Gamma^{\prime}$ for some $n\geq 1$; and 3) collapsing the lifts of the subgraph $\Gamma^{\prime\prime}$ in the universal cover $\tilde{\Gamma}$ induces an expanding immersion $\bar{f}:T\to T$ on a simplicial $F$-tree. In particular, if $\Gamma^{\prime}$ is empty, then $f$ is an expanding graph immersion.

In our previous work [23], we extended a result of Ilya Kapovich [17] and proved the special case of the main theorem with $A=F$ and $\phi:F\to F$ induced by an expanding graph immersion. The expanding immersion was crucial to the proof and so we developed expanding relative immersions with the intent of, in a sense, relativizing our previous proof and extending it to all injective nonsurjective endomorphisms. This application of the structure theorem is carried out in the second part of the paper (Sections 5–7).

Expanding relative immersions are interesting in their own right as they can potentially be applied to many currently open problems about nonsurjective endomorphisms. To name a few: relative immersions may be needed to extend Hagen-Wise’s cubulation of word-hyperbolic free-by-cyclic groups [16, 15] to the groups $F*_{A}$; we suspect that they could be used to extend the main theorem to a characterization of the possible Dehn/isoperimetric functions of the groups $F*_{A}$ following work by Bridson-Groves [7] (See also Problem 2); they provide a framework for generalizing Bestvina-Feighn-Handel’s construction of improved relative train tracks for automorphisms [5], which were used in Brinkmann’s, Hagen-Wise’s, and Bridson-Groves’ results; for a different direction of generalization, the main theorem may be extended to HNN extensions $G*_{A}$ of torsion-free word-hyperbolic groups $G$ over free factors $A\leq G$; and finally, the virtual fibering question is a particularly interesting problem that was first posed to us by Dawid Kielak and our structure theorem along with the cubulation problem could be significant steps towards its resolution:

Besides Section 7, the results in this paper were the author’s doctoral thesis. With future applications of the structure theorem in mind, we have chosen to emphasize in this paper the independence of the “theory” (first part) and “application” (second part). The two parts can be read independently. If you choose to skip the first part, then we encourage you to read the prologue and interlude; these optional sections contain several examples that capture the key points. The main results of the first part are summarized again in the interlude with a bit more detail since we are assuming you will, by then, be familiar with the terms defined in the definitions and conventions section. The summary is all you will need to use expanding relative immersions in your own work. The epilogue is a brief discussion of some questions about endomorphisms in the context of expanding relative immersions.

The main tools used in the first part are Bass-Serre theory, Stallings graphs, bounded cancellation, and train track theory. The second part uses some Bass-Serre theory and the Bestvina-Feighn combination theorem.

We will start with a few motivating examples to illustrate the main construction of the first part of this paper. Let $F=F(a,b)$ be the free group on two generators and $\phi,\varphi,\psi:F\to F$ be injective nonsurjective endomorphisms given by

The standard rose is a rose $R$ with two petals and an identification $F=\pi_{1}(R)$ such that the basis $\{a,b\}$ corresponds to the petals; it will be graphically represented by a rose with two oriented petals labelled by $a$ and $b$ respectively.

For each integer $k\geq 1$, let $\hat{R}_{k}$ be the cover of $R$ corresponding to the subgroup $\phi^{k}(F)$, i.e., it is the quotient of the universal cover $\tilde{R}$ by the action of the subgroup $\phi^{k}(F)$. The Stallings graph $S(\phi^{k}(F))$ is the core of $\hat{R}_{k}$, i.e., the smallest connected subgraph of $\hat{R}_{k}$ with rank 2. Alternatively, we can define $\tilde{R}(\phi^{k}(F))\subset\tilde{R}$ to be the smallest subtree invariant under the $\phi^{k}(F)$-action and $S(\phi^{k}(F))$ to be the quotient $\phi^{k}(F)\backslash\tilde{R}(\phi^{k}(F))$. Evidently, there is a natural isomorphism $\phi^{k}(F)\cong\pi_{1}(S(\phi^{k}(F)))$. A Stallings graph $S$ is graphically represented as an $R$-digraph, i.e., $S$ will be an oriented graph whose oriented edges are labelled by $a$ or $b$ as shown in Figure 2.

The Stallings graphs $S(\phi^{k}(F))$ are roses for all $k\geq 1$ and each petal doubles in size with each iteration. Since $\phi$ is injective, it is an isomorphism onto its image. In particular, we may use $\phi$ to get an isomorphism $F\cong\phi^{k}(F)\cong\pi_{1}(S(\phi^{k}(F)))$ for $k\geq 1$. Under this isomorphism, the basis $\{a,b\}$ corresponds to the petals of $S(\phi^{k}(F))$ and we recover the standard rose for all $k\geq 1$. This is equivalent to the existence of an immersion (locally injective map) on the rose $f:R\to R$ such that $\pi_{1}(f)=\phi$ [24, Lemma 3.2]. Indeed, the obvious map $f:R\to R$ that maps the $a$-petal to the path $ab$ and the $b$-petal to the path $ba$ is an immersion. The petals of $S(\phi^{k}(F))$ doubling in size with each iteration implies $f$ is in fact an expanding immersion, i.e., all edges expand under $f$-iteration.

The Stallings graphs $S(\varphi^{k}(F))$ are barbells for all $k\geq 1$, the middle bars roughly double in size with each iteration, and the plates are labelled by $a$ (See Figure 3). Under the isomorphism $F\cong\varphi^{k}(F)\cong\pi_{1}(S(\varphi^{k}(F)))$, the conjugacy classes $\{[a],[b]\}$ correspond to the two plates of the barbell $S(\varphi^{k}(F))$ and we recover the so called standard barbell for all $k\geq 1$. Just as in the first example, this is equivalent to the existence of an immersion on the barbell $g:B\to B$ such that $\pi_{1}(g)=\varphi$ under some identification $F=\pi_{1}(B)$. However, the immersion is not expanding since the plates are not growing in size with each iteration.

What we are now about to do will be the crux of the first part of the paper. Since the (nongrowing) plates of $S(\varphi^{k}(F))$ are all labelled $a$, we deduce that $\langle a\rangle$ is a free factor fixed by $\varphi$ and $\langle b\rangle$ is mapped to a conjugate of $\langle a\rangle$. Thus, $\{\langle a\rangle,\langle b\rangle\}$ forms an invariant free factor system that contains a fixed free factor. Now consider $\tilde{g}:\tilde{B}\to\tilde{B}$ to be the lift of $g$ to the universal cover $\tilde{B}$. Collapsing all translates of axes of $a$ and $b$ in $\tilde{B}$, i.e., collapsing all edges of $\tilde{B}$ labelled by the plates of $B$, will produce a so-called $(F,\{\langle a\rangle,\langle b\rangle\})$-tree $T$ with a nonfree $F$-action and $\tilde{g}$ induces an expanding immersion $\bar{g}:T\to T$ where each edge doubles in size under $\bar{g}$-iteration. The map $\bar{g}$ will be referred to as a relative immersion.

For the final example, the Stallings graphs $S(\psi^{k}(F))$ are roses again where one petal roughly doubles in size with each iteration and another is labelled by $a$ (See Figure 4). Unlike the first example, the roses with the isomorphisms $F\cong\psi^{k}(F)\cong\pi_{1}(S(\psi^{k}(F)))$ are not standard roses since $b$ does not correspond to a petal for $k\geq 1$. In fact, applying $\psi$ to the labels of $S(\psi^{k}(F))$ forces a full fold with one petal to get $S(\psi^{k+1}(F))$ which means the Stallings graphs are all distinct marked roses. In particular, $\psi$ cannot be induced by a graph immersion! But the observation that the nongrowing petals of $S(\psi^{k}(F))$ are all labelled by $a$ implies $\langle a\rangle$ is a fixed free factor. Despite $\psi$ not being induced by an immersion, the obvious map on the standard rose $h:R\to R$ will induce an expanding immersion $\bar{h}:Y\to Y$ on an $(F,\{\langle a\rangle\})$-tree $Y$ where every edge doubles in size.

The main result of the first part of this paper is that this construction always produces expanding relative immersions, Theorem 4.5: any injective nonsurjective endomorphism of a free group is induced by a graph map $f:\Gamma\to\Gamma$ such that collapsing the translates of axes in $\tilde{\Gamma}$ of some canonical invariant free factor system containing a fixed free factor system will induce an expanding relative immersion $\bar{f}:T\to T$. This can be summarized into two steps: first establishing the existence of a unique (possibly trivial) maximal fixed free factor system (Section 2); then showing that collapsing this free factor system and its preimages in the universal cover of an appropriately chosen graph $\Gamma$ will produce a tree $T$ on which we can define an expanding relative immersion (Section 4). The two guiding principles will be bounded cancellation (Lemmas 1.4 & 3.1) and the fact that Stallings graphs for iterated injective nonsurjective endomorphisms have unbounded size (Lemma 1.1 & Proposition 3.9).

$F$ will always be a free group with finite rank at least $2$. A nontrivial subgroup system of $F$ is a nonempty finite collection of nontrivial subgroups $\mathcal{A}=\{A_{1},\ldots,A_{l}\}$ of $F$. A nontrivial free factor system of $F$ is a nonempty collection of nontrivial free factors $\mathcal{A}=\{A_{1},\ldots,A_{l}\}$ of $F$ such that $A_{i}\cap A_{j}$ is trivial if $i\neq j$ and $\langle A_{1},\ldots,A_{l}\rangle$ is free factor of $F$. We define the trivial system to be the collection consisting of the trivial subgroup; this will allow us to treat absolute and relative cases simultaneously in our proofs. For any subgroup system $\mathcal{A}$, the subgroups $A_{i}\in\mathcal{A}$ are called the components of $\boldsymbol{\mathcal{A}}$ or $\boldsymbol{\mathcal{A}}$-components. A subgroup system $\mathcal{A}$ is finitely generated if its components are finitely generated. A free factor system $\mathcal{A}$ is proper if some component is proper, i.e., $\mathcal{A}\neq\{F\}$. Let $\mathcal{A}$ and $\mathcal{B}$ be subgroup systems of $F$. We shall say $\boldsymbol{\mathcal{A}}$ carries $\boldsymbol{\mathcal{B}}$ if for every component $B\in\mathcal{B}$, there is a component $A\in\mathcal{A}$ and element $x\in F$ such that $B\leq xAx^{-1}$ and, when $\mathcal{A}$ and $\mathcal{B}$ are free factor systems, we shall denoted it by $\mathcal{B}\preceq\mathcal{A}$. For a conjugacy class of elements $[g]$ in $F$, we shall also say $\boldsymbol{\mathcal{A}}$ carries $\boldsymbol{[g]}$ if there is a component $A\in\mathcal{A}$ and element $x\in F$ such that $g\in xAx^{-1}$. If $\mathcal{A}=\{A\}$ (or $\mathcal{B}=\{B\}$) is a singleton, then we will write, “$A$ carries $[g]\,/\,\mathcal{B}$ (or $B$).” One can easily verify that the $\preceq$-relation on free factor systems is a preorder, i.e., it is reflexive and transitive. So it determines an equivalence relation on the set of free factor systems and a partial order on the set of equivalence classes. Free factor systems will always be considered up to this equivalence relation. In particular, we can replace components in a system with conjugates whenever convenient.

Two group homomorphisms $h_{1},h_{2}:A\to B$ are equivalent if there is an inner automorphism $i_{b}:B\to B$ such that $h_{2}=i_{b}\circ h_{1}$ and this is denoted by $[h_{1}]=[h_{2}]$. Outer endomorphisms of $F$ are the equivalence classes on the set of endomorphisms of $F$. For instance, properties such as being irreducible are not just properties of an endomorphism $\phi$ but also its outer class $[\phi]$.

Let $\phi:F\to F$ be an endomorphism. We say a subgroup system $\mathcal{A}=\{A_{1},\ldots,A_{l}\}$ is $\boldsymbol{[\phi]}$-invariant if $\mathcal{A}$ carries the subgroup system $\phi(\mathcal{A})=\{\phi(A_{1}),\ldots,\phi(A_{l})\}$, i.e, there exists a set of elements $\{x_{1},\ldots,x_{l}\}\subset F$ and a function $\sigma:\{1,\ldots,l\}\to\{1,\ldots,l\}$ such that $\phi(A_{i})\leq x_{i}A_{\sigma(i)}x_{i}^{-1}$ for all $i$. A $[\phi]$-invariant subgroup system $\mathcal{A}$ is $\boldsymbol{[\phi]}$-fixed if its subgroups are permuted up to conjugacy, i.e., $\sigma$ is a permutation and $[\phi(A_{i})]=[A_{\sigma(i)}]$ for all $i$. When $\phi$ is an automorphism, all $[\phi]$-invariant free factor systems are $[\phi]$-fixed. When $\phi$ is injective, then, for any $k\geq 1$ and free factor system $\mathcal{A}$ of $F$, $\phi^{k}(\mathcal{A})$ is a free factor system of $\phi^{k}(F)$; conversely, any free factor system of $\phi^{k}(F)$ gives us a free factor system of $F$ via the isomorphism $\phi^{k}:F\to\phi^{k}(F)$. Finally, note that when $\mathcal{A}$ is a $[\phi]$-fixed free factor system but $\phi$ is not surjective, $\phi^{k}(\mathcal{A})$ will typically not be a free factor system of $F$ (See the remark above). A subgroup $A\leq F$ is eventually $\boldsymbol{[\phi]}$-periodic if $[\phi^{m}(A)]=[\phi^{n}(A)]$ for some $m>n\geq 1$, and it is $\boldsymbol{[\phi]}$-periodic if $[\phi^{m}(A)]=[A]$ for some $m\geq 1$.

The endomorphism $\phi$ is reducible if it has a nontrivial proper invariant free factor system and irreducible otherwise. One immediate consequence of Stallings folds [29] is the injectivity of irreducible endomorphisms (Observation below). So we can drop the injectivity hypothesis when specializing results to the irreducible case.

For the topological perspective, graphs are 1-dimensional CW-complexes and a graph map $f:X\to X^{\prime}$ will be a continuous map of graphs that sends vertices to vertices and any edge to a vertex or immersed path. An edge $e$ of $X$ is pretrivial if $f(e)$ is a vertex. For a graph map $f:\Gamma\to\Gamma^{\prime}$ of finite graphs, let $K$ be the maximum of the (combinatorial) length of the edge-path $f(e)$ as $e$ varies over all the edges of $\Gamma$. Then $f$ is $K$-Lipschitz, a fact that will be used throughout the paper. Generally, $K(f)$ will denote some convenient Lipschitz constant for $f$ rather than the infimum. For the moment, $X$ is used for arbitrary graphs but, for most of the paper, $\Gamma$ will be used for finite connected noncontractible graphs and $T$ for infinite simply-connected graphs (simplicial trees). A core graph is a graph with no proper deformation retract.

A direction at a vertex $v\in X$ is a half-edge attached to the vertex. A vertex is bivalent if it has exactly two directions. The set of directions at vertex $v$ is denoted by $T_{v}X$. branch points are vertices with at least three directions and natural edges are maximal edge-paths whose interior vertices are bivalent. We will say a graph map is natural if it maps branch points to branch points and any natural edge to a branch point or immersed path. If the graph map $f:X\to X^{\prime}$ has no pretrivial edges, then the restriction to initial segments induces the derivative map at $v$, $df_{v}:T_{v}X\to T_{f(v)}X^{\prime}$. The graph map $f$ is an immersion if it is locally injective, i.e., it has no pretrivial edges and the derivative maps $df_{v}$ are injective for all vertices $v$; note that immersions are natural. An immersion $f$ is expanding if the length of $f^{n}(e)$ is unbounded as $n\to\infty$ for every edge $e$ of $X$.

With immersions defined, we preface the observation with a summary of Stallings’ folding theorem. Let $R$ be a rose whose edges are indexed by a basis $\{a_{1},\ldots,a_{n}\}$ of $F$ and let $f:R\to R$ be the map where $f(a_{i})$ is the immersed edge-path in $R$ labelled by $\phi(a_{i})$. Stallings [29] showed that $f$ factors as $\iota\circ f_{l}\circ\cdots\circ f_{1}$ where each $f_{i}$ is a fold and $\iota$ is an immersion. We will use this factorization again to prove bounded cancellation in Lemma 1.4 and Proposition 3.1.

Let $\mathcal{A}=\{A_{1},\ldots,A_{l}\}$ be a nontrivial subgroup system of $F$, then an $\boldsymbol{\mathcal{A}}$-marked graph $(\Gamma_{*},\alpha_{*})$ is a collection of graphs $\Gamma_{*}=\{\Gamma_{1},\ldots,\Gamma_{l}\}$ where the finite connected core graphs $\Gamma_{i}$ are indexed by markings, i.e., isomorphisms $\alpha_{i}:A_{i}\to\pi_{1}(\Gamma_{i})$. When $\mathcal{A}=\{A\}$ is a singleton, we may write $A$-marked graph in place of $\mathcal{A}$-marked graph. A marked graph $(\Gamma,\alpha)$ is an $F$-marked graph. For any marked graph $(\Gamma,\alpha)$ and any conjugacy class of a nontrivial element $g\in F$, denoted by $[g]$, we define its length with respect to $\boldsymbol{\alpha}$, $\lVert g\rVert_{\alpha}$, to be the (combinatorial) length of the immersed loop in $\Gamma$ representing $[g]$.

More generally, we want to consider pairs $\mathcal{A}\preceq\mathcal{B}$ of free factor systems of $F$. For each component $B_{i}\in\mathcal{B}$, let $\mathcal{A}_{i}$ be either the nonempty maximal subset of $\mathcal{A}$ carried by $B_{i}$ or the trivial system if no such subset exists. Typically, we shall replace components of $\mathcal{A}$ with conjugates so that $\mathcal{A}$ is also a free factor system of $\mathcal{B}$, i.e., each $\mathcal{A}_{i}$ is a free factor system of $B_{i}$ (next remark below). A $\boldsymbol{(\mathcal{B},\mathcal{A})}$-forest $T_{*}$ is a simplicial forest of $(B_{i},\mathcal{A}_{i})$-trees $T_{i}$, i.e., a collection of simplicial trees $T_{*}=\{T_{1},\ldots,T_{k}\}$ where each tree $T_{i}$ is equipped with a minimal simplicial $B_{i}$-action whose edge stabilizers are trivial and point stabilizers are trivial or conjugates (in $B_{i}$) of $\mathcal{A}_{i}$-components. We note that a $(B,\{B\})$-tree is a point also known as a degenerate tree.

When $\mathcal{A}$ is the trivial system, an $(F,\mathcal{A})$-tree is a free minimal $F$-tree $T$. In that case, the quotient $F\backslash T$ is a marked graph. When $\mathcal{A}$ is a nontrivial proper free factor system, the quotient of an $(F,\mathcal{A})$-tree is a graph of groups decomposition of $F$ with trivial edge groups and $\mathcal{A}$ as the nontrivial vertex groups [28]; such decompositions are sometimes known as free splittings of $\boldsymbol{F}$ and they will be our relative analogues for marked graphs. Any given $(F,\mathcal{A})$-tree endowed with the combinatorial metric has an associated length function $l_{T}:F\to\mathbb{R}$. Precisely, any isometry of a simplicial tree is either elliptic (fixes a point) or loxodromic (preserves an axis of least displacement). If $g\in F$ is elliptic, then $l_{T}(g)=0$; otherwise, $l_{T}(g)>0$ is the translation distance of $g$ acting on its axis. When $\mathcal{A}$ is trivial and $F\backslash T$ is the marked graph $(\Gamma,\alpha)$, then $l_{T}(\cdot)=\|\cdot\|_{\alpha}$ as functions $F\to\mathbb{R}$. The minimal subtree $T(H)\subset T$ of a nontrivial subgroup $H\leq F$ is the smallest subtree with a minimal induced $H$-action, i.e., the union of all fixed points and axes of nontrivial elements in $H$.

Suppose $\psi:F\to F^{\prime}$ is an injective homomorphism. Then it determines a contravariant preimage function $\psi^{-1}$ from the poset of free factor systems of $F^{\prime}$ to the poset of free factor systems of $F$. One way to define this function is through trees. Let $\mathcal{A}^{\prime}$ be a free factor system of $F^{\prime}$, $T$ be any $(F^{\prime},\mathcal{A}^{\prime})$-tree, and $T(\psi(F))\subset T$ be the minimal subtree of $\psi(F)\leq F^{\prime}$. In particular, the quotient $\psi(F)\backslash T(\psi(F))$ is a free splitting of $\psi(F)$ whose nontrivial vertex groups form a nontrivial free factor system $\mathcal{A}^{\prime\prime}$ of $\psi(F)$. If $T(\psi(F))$ has a free $\psi(F)$-action, then this condition is independent of the choice of $(F^{\prime},\mathcal{A}^{\prime})$-tree $T$ and we set $\psi^{-1}\cdot\mathcal{A}^{\prime}$ to be the trivial system. Otherwise, we have a nontrivial free factor system $\mathcal{A}^{\prime\prime}$ of $\psi(F)$ and a corresponding nontrivial free factor system of $F$ via the isomorphism $\psi:F\to\psi(F)$; the equivalence class of the latter free factor system is independent of the $(F^{\prime},\mathcal{A}^{\prime})$-tree $T$ and we will denote it by $\psi^{-1}\cdot\mathcal{A}^{\prime}$.

Let $T$ and $T^{\prime}$ be $(F,\mathcal{A})$- and $(F^{\prime},\mathcal{A}^{\prime})$-trees respectively and $\psi:F\to F^{\prime}$ be an injective homomorphism such that $\mathcal{A}^{\prime}$ carries $\psi(\mathcal{A})$ or, equivalently, $\psi^{-1}\cdot\mathcal{A}^{\prime}\succeq\mathcal{A}$. This carrying condition ensures that elliptic elements in $F$ (with respect to $T$) have elliptic $\psi$-images (with respect to $T^{\prime}$). A tree map $f:T\to T^{\prime}$ is $\boldsymbol{\psi}$-equivariant if $f(g\cdot x)=\psi(g)\cdot f(x)$ for all $g\in F$ and $x\in T$. If we require $\psi^{-1}\cdot\mathcal{A}^{\prime}=\mathcal{A}$, then loxodromic elements in $F$ have loxodromic $\psi$-images; this is the relative analogue of a $\pi_{1}$-injective graph map.

Suppose $\mathcal{A}$ is a nontrivial $[\phi]$-invariant free factor system of $F$ for some injective endomorphism $\phi:F\to F$; so there is a function $\sigma:\{1,\ldots,l\}\to\{1,\ldots,l\}$ and inner automorphisms $i_{x_{i}}:F\to F$ such that the restrictions $\phi_{i}=\left.\left(i_{x_{i}}\circ\phi\right)\right|_{A_{i}}$ are homomorphisms $\phi_{i}:A_{i}\to A_{\sigma(i)}$. The collection $\{\phi_{i}\}$ is a restriction of $\boldsymbol{\phi}$ to $\boldsymbol{\mathcal{A}}$, which we denote by $\left.\phi\right|_{\mathcal{A}}:\mathcal{A}\to\mathcal{A}$. A weak representative for $\boldsymbol{[\phi_{i}]}$ is a graph map $f_{i}:\Gamma_{i}\to\Gamma_{\sigma(i)}$ from an $A_{i}\,$-marked graph $(\Gamma_{i},\alpha_{i})$ to an $A_{\sigma(i)}\,$-marked graph $(\Gamma_{\sigma(i)},\alpha_{\sigma(i)})$ such that $[\pi_{1}(f_{i})\circ\alpha_{i}]=[\alpha_{\sigma(i)}\circ\phi_{i}]$ for all $i\in\{1,\ldots,l\}$. A weak representative for $\boldsymbol{\left[\left.\phi\right|_{\mathcal{A}}\,\right]}$ is a graph map $f_{*}:\Gamma_{*}\to\Gamma_{*}$ of an $\mathcal{A}$-marked graph $(\Gamma_{*},\alpha_{*})$ that is a disjoint union of weak representatives for $[\phi_{i}]$. A natural representative is a weak representative that is also natural. A topological representative is a weak representative that has no pretrivial edges and whose underlying graph $\Gamma_{*}$ has no bivalent vertices.

Recall $\mathcal{A}\preceq\mathcal{B}$. Suppose $\phi^{-1}\cdot\mathcal{A}=\mathcal{A}$, $\mathcal{B}$ is $[\phi]$-invariant, and let $\sigma^{\prime}:\{1,\ldots k\}\to\{1,\ldots k\}$ be the function corresponding to the $[\phi]$-invariance of $\mathcal{B}$ and $\left.\phi\right|_{\mathcal{B}}=\{\phi_{i}:B_{i}\to B_{\sigma^{\prime}(i)}\}$ be a restriction of $\phi$. An $\boldsymbol{\mathcal{A}}$-relative weak representative for $\boldsymbol{\left.\phi\right|_{\mathcal{B}}}$ is a forest map $f_{*}:T_{*}\to T_{*}$ of a $(\mathcal{B},\mathcal{A})$-forest $T_{*}$ that is a disjoint union of $\phi_{i}$-equivariant tree maps $f_{i}:T_{i}\to T_{\sigma^{\prime}(i)}$. An $\boldsymbol{\mathcal{A}}$-relative natural representative is an ${\mathcal{A}}$-relative weak representative that is also natural. An $\boldsymbol{\mathcal{A}}$-relative representative is an ${\mathcal{A}}$-relative weak representative with no pretrivial edges and whose underlying forest has no bivalent vertices. A relative representative is minimal if it has no orbit-closed invariant subforests whose components are bounded. An (expanding resp.) $\boldsymbol{\mathcal{A}}$-relative immersion for $\boldsymbol{\left.\phi\right|_{\mathcal{B}}}$ is an $\mathcal{A}$-relative representative for $\left.\phi\right|_{\mathcal{B}}$ that is also an (expanding resp.) immersion.

In the first part of the paper, we show that any injective endomorphism of $F$ has a canonical invariant free factor system $\mathcal{A}$ and an expanding $\mathcal{A}$-relative immersion $f$ such that $\mathcal{A}$ is eventually mapped into a canonical fixed free factor system. In the second part of the paper, we use the canonical systems and the expanding relative immersion to prove the main theorem.