Proof outline.

Section 1 of [Reference Bestvina and Handel3] describes an algorithm that takes a topological representative of ${\phi \colon F \to F}$ on a free splitting of F with trivial vertex stabilizers as input and finds either an irreducible train track $\tau \colon T \to T$ on a free splitting with trivial vertex stabilizers or a topological representative on a nondegenerate free splitting with nontrivial vertex stabilizers. However, trivial vertex stabilizers are not crucial to the algorithm, and it can be adapted into one that takes a topological representative on a nondegenerate free splitting with vertex stabilizers $\mathcal G$ as input and finds either an irreducible train track on a free splitting with the same stabilizers or a topological representative on a nondegenerate free splitting whose vertex stabilizers properly carry $\mathcal G$ .

Starting with an initial input of a topological representative on a nondegenerate free splitting with vertex stabilizers $\mathcal G$ , the modified algorithm can then be repeatedly applied to its own outputs until it finds an irreducible train track on some nondegenerate free splitting whose vertex stabilizers carry $\mathcal G$ . This process will stop after at most $(c(F)-c(\mathcal G))$ repetitions. Finally, we note that F being ‘connected’ was not crucial to the algorithm and F can be safely replaced with a system $\mathcal F$ .

Proof. If $\lambda (\tau )=1$ , then $\tau $ is a simplicial automorphism and all immersed paths in $\mathcal T$ are $\tau $ -legal. So we may assume $\lambda (\tau )>1$ . Choose an oriented edge $e_i$ in a component $T_i \subset \mathcal T$ . Irreducibility of train track $\tau $ and $\lambda (\tau )>1$ imply we can find distinct translates $x_1 \cdot e_i, x_2 \cdot e_i$ in the oriented edge-path $\tau ^n(e_i)$ for some large n. The $\mathcal T$ -loxodromic element $x_2x_1^{-1}$ has a $\tau $ -legal axis in $\mathcal T$ : a fundamental domain for the axis is the $\tau $ -legal subpath of $\tau ^{n}(e_i)$ joining the midpoints of the chosen two translates of $e_i$ .

Proof. Suppose $(\Gamma _*, \mathcal G)$ is a free splitting of $\mathcal F$ . Let $\mathcal V$ be a set of orbit representatives for vertices in the free splitting with nontrivial stabilizers. For each $v \in \mathcal V$ , let $\mathcal D_v$ be a set of orbit representatives for half-edges originating from v and $\overline T_v$ the metric completion of the $\mathcal T_{\mathcal V}$ -component corresponding to $G_v$ .

Suppose $(p_d \in \overline T_v : v \in \mathcal V, d \in \mathcal D_v)$ is a choice of attaching points in $\overline T_{\mathcal V}$ .

(graph of actions) To simplify the discussion, we will pretend $\mathcal F=F$ is connected for the moment. Let E be the complement of the F-orbit of $\mathcal V$ in $(\Gamma , \mathcal G)$ ; essentially, we just want the half-edges in $\mathcal D_v$ to now have distinct origins. E inherits a free F-action from the free splitting. Set ; F acts on $\mathcal V^*$ by left-multiplication on the first factor. The equivariant blow-up $T^* = T^*(p_d : v \in \mathcal V, d \in \mathcal D_v)$ of $(\Gamma , \mathcal G)$ with respect to the forest $\mathcal T_{\mathcal V}$ is defined through the quotient $\mathcal V^* \sqcup E \overset {\iota }\to T^*$ given by the following identifications:

1. 1. for each $v \in \mathcal V$ , $d \in \mathcal D_v$ and $x \in F$ , the origin of the half-edge $x \cdot d$ in E is identified with $(x, p_d)$ in $\mathcal V^*$ (i.e., the half-edge d is attached to $p_d$ equivariantly); and

2. 2. for each $v \in \mathcal V$ , $p \in \overline T_v$ , $s \in G_v$ and $x\in F$ , the point $(xs, p)$ in $\mathcal V^*$ is identified with $(x, s \cdot p)$ in $\mathcal V^*$ .

As the F-actions on $\mathcal V^*$ and E respect the identifications, the blow-up $T^*$ inherits an F-action by homeomorphisms of the quotient topology. For each $s \in G_v$ , the image $\iota ( s, \overline T_v)$ is a $G_v$ -equivariant copy of $\overline T_v$ since $(\Gamma , \mathcal G)$ has trivial edge stabilizers; in general, we would need each attaching point $p_d \in \overline T_v$ to be fixed by the stabilizer of d in $G_v$ .

The blow-up $T^*$ with the quotient topology has a natural projection $\pi \colon T^* \to (\Gamma , \mathcal G)$ whose point-preimages are connected: single point or a copy of $\overline T_v$ . This implies $T^*$ is uniquely arcwise connected. Since the vertex set of $(\Gamma , \mathcal G)$ is a discrete subspace with finitely many F-orbits, any closed arc in $T^*$ decomposes into finitely many subarcs that are in $\iota (E)$ or $\iota (\mathcal V^*)$ . Thus, closed arcs in $T^*$ inherit lengths from $(\Gamma _*, \mathcal G)$ and $\overline {\mathcal T}_{\mathcal V}$ ; this path metric makes $T^*$ a (metric) tree. By the equivariant construction, the tree $T^*$ inherits an isometric F-action with trivial arc stabilizers and an equivariant copy of $\mathcal T_{\mathcal V}$ .

The F-action on $T^*$ is what Levitt calls a graph of actions [Reference Levitt11, p. 32] – the underlying simplicial action is the free splitting $(\Gamma , \mathcal G)$ . Different choices $(p_d : v \in \mathcal V, d \in \mathcal D_v)$ of attaching points may produce drastically different graphs of actions.

Returning to the general case where $\mathcal F$ is possibly disconnected, we can apply the blow-up construction componentwise to get a forest $\mathcal T^*$ with an isometric $\mathcal F$ -action.

We have yet to use the $\psi $ -equivariant simplicial automorphism $\tau \colon (\Gamma _*,\mathcal G) \to (\Gamma _*, \mathcal G)$ or the $\left .\psi \right |{}_{\mathcal G}$ -equivariant expanding $\lambda $ -homothety $h_{\mathcal V} \colon \mathcal T_{\mathcal V} \to \mathcal T_{\mathcal V}$ .

(projective limit) Suppose we have a naïve blow-up $\mathcal T^*$ (made from an arbitrary choice of attaching points), and let $\| \cdot \|^*$ be its translation distance function. Since $\tau $ was a simplicial automorphism and $h_{\mathcal V}$ is a $\left .\psi \right |{}_{\mathcal G}$ -equivariant $\lambda $ -homothety, we get $\lambda ^{-n} \|\psi ^n(\cdot )\|^* \to \| \cdot \|$ , where the limit function $\| \cdot \|$ is the translation distance function for the required $\mathcal F$ -action.

For the second proof, we get the ideal attaching points without taking projective limits.

(ideal stitching) As the simplicial automorphism $\tau $ permutes the (finitely many) orbits of vertices and half-edges, it induces permutations $\beta \in \mathrm {Sym}(\mathcal V)$ and $\partial \in \mathrm {Sym}(\bigcup _{v\in \mathcal V} \mathcal D_v)$ . The maps $\tau $ and $h_{\mathcal V}$ induce a $\psi $ -equivariant PL-map $f^*\colon {\mathcal T^*} \to {\mathcal T^*}$ :

• ○ let $\nu \colon \mathcal V^* \to \mathcal \iota (\mathcal V^*)$ be given by , where $( x_v \in \mathcal F: v\in \mathcal V)$ was the implicit choice used to define the restriction $\left .\psi \right |{}_{\mathcal G}$ , and hence $h_{\mathcal V}$ and its extension $\bar h_{\mathcal V} \colon \overline {\mathcal T}_{\mathcal V} \to \overline {\mathcal T}_{\mathcal V}$ . Observe that $\nu (xs, p) = \nu (x, s\cdot p)$ when $s \in G_v$ (exercise).

  Let $\epsilon \colon E \to \mathcal T^*$ be given by the following:

• ○ for each half-edge e in E not in the $\mathcal F $ -orbit of $\bigcup _{v\in \mathcal V} \mathcal D_v$ , $\epsilon $ maps e onto $\iota (\tau (e)) \subset E$ along an orientation-preserving isometry.

• ○ for each $v \in \mathcal V$ , $d \in \mathcal D_v$ , and element x in the $\mathcal F$ -component that contains $G_v$ , $\epsilon $ maps $x \cdot d$ onto the concatenation of $\iota ( \psi (x)x_v , [\bar h_v(p_d), s_{\partial \cdot d} \cdot p_{\partial \cdot v}]_{\partial \cdot v} )$ and $\iota (\psi (x) \cdot \tau (d))$ along an orientation-preserving linear map, where $ \tau (d) = x_v s_{\partial \cdot d} \cdot \partial (d)$ – $s_{\partial \cdot d} \in G_{\beta \cdot v}$ is unique as edge stabilizers are trivial. Notice that the origin of $\epsilon ( x \cdot d)$ is $\nu (x,p_d)$ .

By definition, $\nu \sqcup \epsilon \colon \mathcal V^* \sqcup E \to \mathcal T^*$ factors through $\iota $ to induce a PL-map $f^*$ .

The PL-map $f^*$ is injective if and only if $\bar h_v(p_d) = s_{\partial \cdot d} \cdot p_{\partial \cdot d}$ for all $v \in \mathcal V$ , $d \in \mathcal D_v$ . This finite system of equations with unknowns $( p_d \in \overline T_v : v \in \mathcal V, d \in \mathcal D_v )$ is restated as

Note that minimality of $\mathcal T_{\mathcal V}$ implies $h_{\mathcal V}$ is surjective and hence invertible. So $h_{v,d}\colon T_{\beta \cdot v} \to T_v$ , the composition of an isometry with a contracting homothety, is a contracting homothety. As $\bigcup _{v\in \mathcal V} \mathcal D_v$ is finite and $\overline {\mathcal T}_{\mathcal V}$ is complete, our system of equations has a unique solution by the contraction mapping theorem (i.e., there is a unique tuple $\vec p = (p_d \in \overline T_v : v \in \mathcal V, d \in \mathcal D_v)$ with $p_d = h_{v,d}(p_{\partial \cdot d})$ for all $v \in \mathcal V$ , $d \in \mathcal D_v$ ).

Let $\mathcal T^* = \mathcal T^*(\vec p\,)$ be the blow-up given by this unique solution. Then the homeomorphism $f^*$ is isometric on permuted components of the simplicial part $\iota (E)$ and an expanding $\lambda $ -homothety on components of the non-simplicial part $\iota (\mathcal V^*)$ .

(simplicial collapse) To finish the proof, we collapse the simplicial edges to get an isometric $\mathcal F$ -action on a forest with trivial arc stabilizers. Let $\mathcal T$ be the characteristic subforest for $\mathcal F$ (i.e., the $\mathcal F$ -action on $\mathcal T$ is minimal). Since we only collapsed the simplicial part, the PL-map $f^*$ induces a $\psi $ -equivariant expanding $\lambda $ -homothety $h\colon \mathcal T \to \mathcal T$ .

(uniqueness) Suppose some $\mathcal F$ -forest $\mathcal T'$ decomposed as a graph of actions with underlying simplicial action $(\Gamma _*, \mathcal G)$ and vertex actions $\mathcal T_{\mathcal V}$ and admitted a homothety $h'$ that acts by $\tau $ on $(\Gamma _*, \mathcal G)$ and $h_{\mathcal V}$ on $\mathcal T_{\mathcal V}$ . This forest must arise from the above construction and the equivariant isometric identification with $\mathcal T$ follows from uniqueness of the solution $\vec p$ .

Sketch of proof.

$\mathcal T[P]$ -loxodromics are precisely $\mathcal T$ -loxodromics as no point in P is fixed by a $\mathcal T$ -loxodromic element of $\mathcal G$ . Also, observe that for all elements g in $\mathcal G$ , the $\mathcal T[P]$ -directions at $\mathrm {Fix}_{\mathcal T[P]}(g)$ are in one-to-one equivariant bijection with the $\mathcal T$ -directions at $\mathrm {Fix}_{\mathcal T}(g)$ . So the induced action on $\mathcal T[P]$ is rigid. Finally, any arc of $\mathcal T[P]$ contains an arc of $\mathcal T$ ; if the latter have finite stabilizers, then the rigid action on $\mathcal T[P]$ has finite arc stabilizers.

Proof of real pretree’s existence.

Following the discussion preceding the theorem’s statement, let $T_v~(v \in \mathcal V)$ be the component of ${\mathcal T}_{\mathcal V}$ corresponding to $G_v$ .

Assume $\vec p = ( p_d \in \widehat T_v: v \in \mathcal V, d \in \mathcal D_v)$ is a choice of points where each point $p_d$ is fixed by the direction stabilizer $\mathrm {Stab}_{G_v}(d)$ ; these points will be referred to as the attaching points. The real pretrees T and $T_v[\,\vec p\,]~(\text {for } v \in \mathcal V)$ have pretree structures denoted $[\cdot , \cdot ]$ and $[\cdot , \cdot ]_v$ , respectively. The equivariant blow-up $T^* = T^*(\vec p\,)$ of T with respect to ${\mathcal T}_{\mathcal V}[\,\vec p\,]$ is defined through the quotient

$$\begin{align*}\iota \colon T \setminus (H \cdot \mathcal V)~\sqcup~ H \times {\mathcal T}_{\mathcal V}[\,\vec p\,] \longrightarrow T^*,\end{align*}$$

given by these identifications:

• ○ for each $v \in \mathcal V$ , $p \in T_v[\,\vec p\,]$ , $s \in G_v$ , and $x \in H$ , identify $(xs, p)$ with $(x, s \cdot p)$ .

Just as in the simplicial setting, $T^*$ inherits an H-action – at least by set-bijections. By construction, $T_v[\,\vec p \,]$ is equivariantly identified with $\iota (G_v \times T_v[\,\vec p \,])$ for $v \in \mathcal V$ , and equivariantly collapsing the translates of $\mathcal T_{\mathcal V}[\,\vec p\,]$ produces an equivariant surjection $\pi \colon T^* \to T$ . This almost establishes Condition 2: we need to define a pretree structure that is H-invariant and projects to the pretree structure of T under $\pi $ .

For a pretree structure, we need a function $[\cdot , \cdot ]^*\colon T^* \times T^* \to \mathcal P(T^*)$ . Let $p^*, q^* \in T^*$ .

Case 1: π(p ^*) = π(q ^*) ∈ H ⋅ V . Choose $y \in H$ so that $\pi (p^*) = y \cdot v$ with $v \in \mathcal V$ . Then $p^* = \iota (y, p)$ and $q^* = \iota (y, q)$ for some $p,q \in T_v[\,\vec p\,]$ . Define

Let us quickly check that $[p^*, q^*]^*$ is independent of our choice of $y \in H$ . If $s \in G_v$ , then $\pi (p^*) = ys \cdot v$ , $p^* = \iota (ys, p')$ , and $q^* = \iota (ys, q')$ where $s\cdot p' = p$ and $s\cdot q' = q$ by $\iota $ ’s definition. So $\iota (ys , [p', q']_v) = \iota ( y, s \cdot [p', q']_v) = \iota ( y, [p,q]_v).$

In this case, we are equivariantly extending the pretree-structure of $\mathcal T_{\mathcal V}[\,\vec p\,]$ to its orbit in $T^*$ (i.e., the image $\iota (H \times \mathcal T_{\mathcal V}[\,\vec p\,])$ ).

Case 2: both π(p ^*) = p, π(q ^*) = q are not in H ⋅ V . Consider the closed interval $[p,q] \subset T$ . Then

$$\begin{align*}[p,q] = [p,q] \setminus (H \cdot \mathcal V)~\cup \bigcup_{v \in \mathcal V, k \ge 1} \{ y_{v,k} \cdot v \}, \end{align*}$$

for some (possibly finite or empty) subset $\{ y_{v,k} \}_{v \in \mathcal V, k \ge 1}$ of H. For each $v \in \mathcal V$ and $k \ge 1$ , the interval $[p, y_{v,k} \cdot v)$ is contained in the direction $y_{v,k} s_{v, k_{-}} \cdot d(v,k_{-})$ at $y_{v,k} \cdot v$ for some $s_{v, k_{-}} \in G_v$ and $d(v,k_{-}) \in \mathcal D_v$ . The element $s_{v, k_{-}}$ is unique up to right-multiplication by $\mathrm {Stab}_{G_v}(d(v,k_{-}))$ . So the point $s_{v, k_{-}}\cdot p_{d(v,k_{-})}$ is independent of the choice of $s_{v,k_{-}}$ since the attaching point $p_{d(v,k_{-})}$ is fixed by $\mathrm {Stab}_{G_v}(d(v,k_{-}))$ . Similarly, $(y_{v,k} \cdot v, q]$ is in $y_{v,k} s_{v, k_{+}} \cdot d(v, k_{+})$ at $y_{v,k} \cdot v$ .

Define

Again, $[p^*, q^*]^*$ is independent of the choice of $y_{v,k}$ . In this case (and the rest), we are equivariantly ‘attaching’ directions d to points $p_d$ to get the pretree $T^*(\vec p\,)$ .

Case 3: π(p ^*) = p ∉ H ⋅ V, π(q ^*) ∈ H ⋅ V (or vice-versa). Choose an element $y_\omega \in H$ so that $\pi (q^*) = y_\omega \cdot w$ with $w \in \mathcal V$ . Then $q^* = \iota (y_\omega , q)$ for some $q \in T_w$ . Consider the closed interval $[p,y_\omega \cdot w] \subset T$ . Then

$$\begin{align*}[p, y_\omega \cdot w] = [p, y_\omega \cdot w] \setminus (H \cdot \mathcal V)~\cup \{ y_\omega \cdot w \} \cup \bigcup_{\substack{v \in \mathcal V, k \ge 1 \\ y_{v,k} \cdot v \neq y_\omega \cdot w}} \{ y_{v,k} \cdot v \}. \end{align*}$$

As in the previous case, for each $v \in \mathcal V$ and $k \ge 1$ , the interval $[p, y_\omega \cdot w]$ determines directions $y_{v,k} s_{v,k_\pm } \cdot d(v,k_\pm )$ at the interior point $y_{v,k} \cdot v$ . Additionally, the interval determines some direction $y_\omega s_\omega \cdot d(\omega )$ at $y_\omega \cdot w$ .

Define $[p^*,q^*]^*$ to be the $\iota $ -image of

$$\begin{align*}[p,y_\omega \cdot w] \setminus (H \cdot \mathcal V) ~\cup \left(\{ y_\omega \} \times [s_\omega \cdot p_{d(\omega)},\,q]_w \right) \cup \bigcup_{\substack{v \in \mathcal V, k \ge 1 \\ y_{v,k} \cdot v \neq y_\omega \cdot w}} \{ y_{v,k} \} \times [s_{v,k_{-}} \cdot p_{d(v,k_{-})},\,s_{v,k_{+}} \cdot p_{d(v,k_{+})}]_v. \end{align*}$$

Case 4: π(p ^*), π(q ^*) ∈ H ⋅ V with π(p ^*) ≠ π(q ^*). Choose $y_\alpha , y_\omega \in H$ so that $\pi (p^*) = y_\alpha \cdot a$ and $\pi (q^*) = y_\omega \cdot w$ with $a, w \in \mathcal V$ . The rest is just like Case 3 except that this time, $[y_\alpha \cdot a, y_\omega \cdot w]$ determines directions $y_\alpha s_\alpha \cdot d(\alpha )$ and $y_\omega s_\omega \cdot d(\omega )$ at $y_\alpha \cdot a$ and $y_\omega \cdot w$ , respectively. We omit the redundant details.

By construction, $\pi ([p^*,q^*]^*)=[\pi (p^*), \pi (q^*)]$ . From this, we can directly verify that

1. 1. $[\cdot , \cdot ]^*$ satisfies the pretree axioms (see Appendix B for details).

2. 2. H acts on $T^*$ by pretree-automorphisms – that is, for all $x \in H$ and $p^*,q^* \in T^*$ , we have $[x \cdot p^*, x \cdot q^*]^* = x \cdot [p^*,q^*]^*$ (exercise).

Remember that $(T, [\cdot , \cdot ])$ and $(T_v[\,\vec p\,], [\cdot , \cdot ]_v)~(v \in \mathcal V)$ were real pretrees. Let us now check that the pretree $(T^*, [\cdot , \cdot ]^*)$ is real (i.e., any closed interval $I^* = [p^*, q^*]^*$ is pretree-isomorphic to a closed interval of $\mathbb R$ ). By the definition of $[\cdot , \cdot ]^*$ , $I^*$ is formed by removing countably many points from a closed interval $I \subset T$ and replacing them with closed intervals pretree-isomorphic to closed intervals of ${\mathcal T}_{\mathcal V}[\,\vec p\,]$ -components. Note that I is pretree-isomorphic to a closed interval of $\mathbb R$ since T is a real pretree. For the same reason, closed intervals of ${\mathcal T}_{\mathcal V}[\,\vec p\,]$ -components are pretree-isomorphic to closed intervals of $\mathbb R$ . Similar to the Cantor function, we can use the given pretree-isomorphisms (and the axiom of choice) to construct an isomorphism between $I^*$ and a closed interval of $\mathbb R$ . We leave the details as an exercise. So $T^*$ is a real pretree.

Proof that conditions are satisfied....

Now assume none of the attaching points $\vec p$ are fixed by a ${\mathcal T}_{\mathcal V}$ -loxodromic element in $\mathcal G$ .

$\bullet $ Condition 3, rigidity of H-action on $T^*$ :

Suppose $x \in H$ has a nonempty fixed-point set $\mathrm {Fix}_{T^*}(x) \subset T^*$ . We need to show that x fixes no $T^*$ -direction at $\mathrm {Fix}_{T^*}(x)$ .

First, we show that the fixed-point set $\mathrm {Fix}_{T^*}(x)$ is convex. By the equivariance of $\pi $ and rigidity of the H-action on T, $\mathrm {Fix}_T(x)$ is nonempty and convex. As x acts as a pretree-automorphism, it must also fix the limit points of $\iota (\mathrm {Fix}_{T}(x) \setminus \left ( H\cdot \mathcal V\right ))$ in $\iota (H \times \mathcal T_{\mathcal V}[\,\vec p\,])$ . By Lemma 4.2, the $\mathcal G$ -action on $\mathcal T_{\mathcal V}[\,\vec p\,]$ is rigid and so $\mathrm {Fix}_{T^*}(x)$ is convex: this used the assumption $\vec p$ was a choice of points not fixed by $\mathcal T_{\mathcal V}$ -loxodromic elements in $\mathcal G$ . Thus, each $T^*$ -direction d at $\mathrm {Fix}_{T^*}(x)$ has a unique attaching point $p^*$ in $\mathrm {Fix}_{T^*}(x)$ .

Finally, pick a $T^*$ -direction d at $\mathrm {Fix}_{T^*}(x)$ , and let $p^* \in \mathrm {Fix}_{T^*}(x)$ be its attaching point. If $\pi (p^*) \notin H\cdot \mathcal V$ , then d corresponds to a T-direction at $\mathrm {Fix}_T(x)$ and the latter is not fixed by x as the H-action on T is rigid. If $\pi (p^*) = y \cdot v$ for some $y \in H$ and $v \in \mathcal V$ , then d corresponds to either a $\iota (y, T_{v})$ -direction at $\iota (y, T_{v}) \cap \mathrm {Fix}_{T^*}(x)$ or a T-direction at $\mathrm {Fix}_T(x)$ (possible when $p^* \in \iota (y, \vec p\,)$ ). Again, in either case, the direction is not fixed due to rigidity of actions. As d was arbitrary, we are done: x fixes no $T^*$ -direction at $\mathrm {Fix}_{T^*}(x)$ .

$\bullet $ Condition 4, characteristic convex subset of $T^*$ for $\mathcal G$ :

Fix a point $v \in \mathcal V$ . We need to show that the characteristic convex subset of $T^*$ for $G_v$ is $\iota (G_v \times T_v)$ . Since the rigid H-action on T has finite arc stabilizers and the T-elliptic subgroup $G_v$ is infinite, the characteristic convex subset of T for $G_v$ is the singleton $\{ v \}$ . Thus, by the equivariance of $\pi $ , a characteristic convex subset of $T^*$ for $G_v$ is contained in $\pi ^{-1}(v) = \iota (G_v \times T_v[\,\vec p\,])$ . But $\pi ^{-1}(v)$ and ${T}_v[\,\vec p\,]$ are equivariantly pretree-isomorphic (Condition 2). The characteristic convex subset of $T_v[\,\vec p\,]$ for $G_v$ is $T_v$ since we assumed the $G_v$ -action on $T_v$ is minimal; therefore, the characteristic convex subset of $T^*$ for $G_v$ is $\iota (G_v \times T_v)$ , as needed.

$\bullet $ Condition 5, characterizing $T^*$ -loxodromics – or equivalently, $T^*$ -elliptics:

Suppose x is T- and ${\mathcal T}_{\mathcal V}$ -elliptic (i.e., after replacing it with a conjugate if necessary, $x \in G_v$ for some $v \in \mathcal V$ and is also $T_v$ -elliptic). So $\iota (x, \mathrm {Fix}_{T_v}(x)) \subset \mathrm {Fix}_{T^*}(x)$ and x is $T^*$ -elliptic.

Conversely, suppose x is T- or $\mathcal T_{\mathcal V}$ -loxodromic. By the equivariance of $\pi $ , $\pi (\mathrm {Fix}_{T^*}(x))$ is contained in $\mathrm {Fix}_T(x)$ . If x is T-loxodromic (i.e., $\mathrm {Fix}_T(x) = \emptyset $ ), then it is also $T^*$ -loxodromic. Suppose x is T-elliptic but $\mathcal T_{\mathcal V}$ -loxodromic. Then $\mathrm {Fix}_{T}(x)$ is a singleton since the rigid H-action on T has finite arc stabilizers. After replacing it with a conjugate if necessary, assume $x \in G_v$ and is $T_v$ -loxodromic for some $v\in \mathcal V$ . As $\mathrm {Fix}_{T^*}(x)$ is contained in $\pi ^{-1}(v)$ and the pretrees $\pi ^{-1}(v)$ and $T_v[\,\vec p\,]$ are equivariantly identified, we get $\mathrm {Fix}_{T^*}(x)$ is $\iota (x, \mathrm {Fix}_{T_v[\,\vec p\,]}(x))$ . But x is $T_v[\,\vec p\,]$ -loxodromic since it is $T_v$ -loxodromic, and it fixes none of the attaching points $\vec p$ by assumption; therefore, x is $T^*$ -loxodromic.

$\bullet $ Finiteness of arc stabilizers:

Assume the rigid $\mathcal G$ -action on $\mathcal T_{\mathcal V}$ has finite arc stabilizers. Let $A^* \subset T^*$ be an arc; then is either a singleton or an arc. If A is an arc, then, by the equivariance of $\pi $ and rigidity of actions (Condition 3), the stabilizer of $A^*$ is the stabilizer of A; so it is finite. If $A = \{ p \}$ is a singleton, then $p = y \cdot v$ for some $y\in H$ and $v \in \mathcal V$ . Thus, $A^* = \iota (y, A_v)$ for some arc $A_v \subset T_v[\,\vec p\,]$ and its stabilizer in H is conjugate to the stabilizer of $A_v$ in $G_v$ ; therefore, it is finite by Lemma 4.2.

$\bullet $ Minimality:

Assume the rigid H-action on T is minimal. By the equivariance of $\pi $ , any characteristic convex subset of $T^*$ for H must map onto T under the projection $\pi $ . In particular, any characteristic convex subset will contain $\iota (T \setminus (H \cdot \mathcal V))$ and the attaching points $\iota (H \times \vec p\,)$ . By Condition 4, any characteristic convex subset will contain $\iota (H \times \mathcal T_{\mathcal V})$ ; therefore, $T^*$ is the characteristic convex subset for H (i.e., the H-action on $T^*$ is minimal).

Proof. For each $v\in \mathcal V$ , fix a set of F-orbit representatives $\mathcal D_v$ for ${T}$ -directions at v. The set $\mathcal D_v$ is finite as well, again by the index inequality. Since the $\phi $ -equivariant pretree-automorphism f permutes the F-orbits of ${T}$ -branch points and ${T}$ -directions, it induces permutations $\beta \in \mathrm {Sym}(\mathcal V)$ and $\partial \in \mathrm {Sym}(\bigcup _{v \in \mathcal V} \mathcal D_v)$ with $\partial (d) \in \mathcal D_{\beta \cdot v}$ if $d \in \mathcal D_v$ .

By viewing $\mathcal Y_{\mathcal V}$ as real pretrees, the minimal isometric $\mathcal G$ -action is also a minimal rigid $\mathcal G$ -action. For any choice $\vec p = ( p_d \in \widehat Y_v: v \in \mathcal V, d \in \mathcal D_v)$ of points not fixed by a ${\mathcal Y}_{\mathcal V}$ -loxodromic element in $\mathcal G$ , we can apply Proposition 4.3 to construct a real pretree $T^* = T^*(\vec p \,)$ that satisfies Conditions 1–3. To find the choice $\vec p$ that implies the remaining Condition 4, we must use the pretree-automorphism $f\colon T \to T$ and expanding homothety $ h_{\mathcal V}\colon {\mathcal Y}_{\mathcal V} \to {\mathcal Y}_{\mathcal V}$ .

The maps f and $h_{\mathcal V}$ induce a $\phi $ -equivariance set-bijection $f^*\colon T^* \to T^*$ :

• ○ for $p \in {T} \setminus (F \cdot \mathcal V)$ , define ; and

• ○ for $(x,p) \in F \times {\mathcal Y}_{\mathcal V}[\, \vec p\,]$ , define , where the implicit choice $( x_v \in F: v\in \mathcal V)$ was used to define the restriction $\left .\phi \right |{}_{\mathcal G}$ , and hence $h_{\mathcal V}$ and its extension $\hat h_{\mathcal V}\colon \mathcal Y_{\mathcal V}[\,\vec p\,] \to \mathcal Y_{\mathcal V}[\,\vec p\,]$ .

For each representative $v \in \mathcal V$ and direction $d \in \mathcal D_v$ , let $s_{\partial \cdot d} \in G_{\beta \cdot v}$ be an element such that $f(d) = x_v s_{\partial \cdot d} \cdot \partial (d)$ ; the element $s_{\partial \cdot d}$ is unique (trivial arc stabilizers). The set-bijection $f^*$ is a pretree-automorphism if and only if $\hat h_v(p_d) = s_{\partial \cdot d} \cdot p_{\partial \cdot d}$ for all $v \in \mathcal V$ and $d \in \mathcal D_v$ (see Appendix B for details).

As $\bigcup _{v\in \mathcal V} \mathcal D_v$ is finite and $h_{\mathcal V}$ is an expanding homothety, the system of equations has a unique solution $ \vec p = (p_d \in \overline Y_v : v \in \mathcal V, d \in \mathcal D_v)$ in the metric completion of $\mathcal Y_{\mathcal V}$ ; these points are not fixed by any ${\mathcal Y}_{\mathcal V}$ -loxodromic element in $\mathcal G$ . Let $T^* = T^*(\vec p\,)$ be the real pretree given by this solution. Then the induced function $f^*$ is a $\psi $ -equivariant pretree-automorphism, as required for Condition 4. Any pair $(T', f')$ satisfying the same conclusion arises from the same construction, and uniqueness follows from uniqueness of $\vec p$ .

Finally, assume f is F-expanding. For all $x \in F$ and $n \ge 1$ , we need to show the composition is either 1) elliptic with a unique fixed point and it expands at this fixed point or 2) loxodromic with an axis that is not shared with any $T^*$ -loxodromic element in F.

Let ; then we can use the F-expanding assumption on f.

For the first case, assume $\gamma $ is loxodromic with an axis that is not shared with any T-loxodromic element in F. Then by construction of $T^*$ and Condition 3, $\gamma ^*$ is loxodromic with an axis that is not shared with any $T^*$ -loxodromic element in F. For the rest of the proof, we may assume $\gamma $ is elliptic with a unique fixed point $p \in T$ and it expands at p.

For the second case, suppose p has a trivial stabilizer. Then $\gamma ^*$ is elliptic with a unique fixed point $\iota (p) \in T^*$ , and it expands at $\iota (p)$ .

For the final case, suppose p has a nontrivial stabilizer. Then $p = y \cdot v$ for some $y \in F$ and $v \in \mathcal V$ . In particular, v is fixed by $\beta ^n \in \mathrm {Sym}(\mathcal V)$ since $p = y \cdot v$ is fixed by $\gamma = x \circ f^n$ . In fact, $y \cdot v = x \circ f^n(y \cdot v) = x \psi ^n(y)\psi ^{n-1}(x_v)\cdots x_{\beta ^{n-1} \cdot v} \cdot v$ implies

and $\gamma ^*(\iota (y, q)) = (y, s_v \cdot \hat h_{\beta ^{n-1} \cdot v} \circ \cdots \circ \hat h_v(q))$ for all $q \in Y_v[\,\vec p\,]$ .

Let . As $h_{\mathcal V}$ is an expanding homothety, the extension of the composition $\gamma _v$ to the metric completion has a unique fixed point $q_v \in \overline Y_v$ that is repelling. If $q_v \in Y_v[\,\vec p\,]$ , then $\gamma ^*$ is elliptic with a unique fixed point $\iota (y,q_v)$ and it expands at $\iota (y,q_v)$ . Otherwise, $q_v \notin Y_v[\,\vec p\,]$ and $\gamma ^*$ is loxodromic. As $\gamma $ is elliptic, $\gamma ^*$ does not share its axis (in $T^*$ ) with any T-loxodromic element in F. As $\iota (y, q_v)$ is an end of the axis for $\gamma ^*$ and $q_v$ is in the metric completion $\overline Y_v$ , $\gamma ^*$ cannot share its axis with a conjugate (in F) of a $Y_{v}$ -loxodromic element of $G_v$ . By Condition 3, $\gamma ^*$ does not share an axis with any $T^*$ -loxodromic element in F, and we are done.