Limit pretrees for free group automorphisms: existence

1.1. Free splittings and topological representatives

A free splitting of F is a simplicial tree T (i.e., 1-dimensional contractible CW-complex) and a minimal (left) F-action by simplicial automorphisms with trivial edge stabilizers. We will also assume the simplicial tree T has no bivalent vertices. Choose a maximal set of orbit representatives $\mathcal V$ for the set of vertices of T whose half-edge neighbourhood is connected. If the simplicial tree T is not a singleton, then the collection of nontrivial stabilizers $\mathcal G$ of vertices in $\mathcal V$ is a proper free factor system of F by Bass–Serre theory [Reference Serre20]. After labelling the vertices of the finite quotient graph $\Gamma = F \backslash T$ with $\mathcal G$ , we get a graph of groups decomposition $(\Gamma , \mathcal G)$ of F with trivial edge groups. In fact, by the fundamental theorem of Bass–Serre theory, there is a one-to-one correspondence between free splittings of F and graph of groups decompositions of F with trivial edge groups. For free splittings, we may use $(\Gamma , \mathcal G)$ as a synonym for T.

A (relative) topological representative of an automorphism $\phi \colon F \to F$ on a free splitting T of F is a map $f\colon T \to T$ satisfying the following conditions:

• ○ cellular: f maps vertices to vertices and is injective on edges; and

• ○ $\phi $ -equivariant: $f(x \cdot p) = \phi (x) \cdot f(p)$ for all $x \in F$ and $p \in T$ .

By definition, a topological representative on T induces a cellular map $[f]\colon \Gamma \to \Gamma $ on the quotient graph $\Gamma $ .

Suppose $f\colon T \to T$ is a topological representative of an automorphism $\phi \colon F \to F$ , $v \in \mathcal V$ a vertex orbit representative, and $G_v \in \mathcal G$ the stabilizer of v. Then $f(v) = x_v \cdot w$ for some element $x_v \in F$ and vertex orbit representative $w \in \mathcal V$ . The stabilizer of $f(v)$ is $x_vG_w x_v^{-1}$ and, if $G_v$ is not trivial, then the restriction of $\phi $ to $G_v$ is an isomorphism $G_v \to x_vG_w x_v^{-1}$ . By post-composing with the inner automorphism $\mathrm {inn}(x_v^{-1})\colon F \to F$ that maps $y \mapsto x_v^{-1} y x_v$ , we get a homomorphism $\phi _v\colon G_v \to G_w$ . The outer class $[\phi _v]$ is independent of the chosen element $x_v \in F$ with $f(v) = x_v \cdot w$ (i.e., the homomorphism $\phi _v$ is unique up to post-composition with an inner automorphism of $G_w$ ). The collection $\{\phi _v~:~ v \in \mathcal V \}$ is denoted by $\left .\phi \right |{}_{\mathcal G}$ and called a restriction of ϕ to $\mathcal G$ .

For the rest of the paper, we shall restrict $\mathcal V$ to the subset consisting of vertices with nontrivial stabilizers. This is mostly a stylistic choice made to simplify the exposition. For instance, the restriction $\left .\phi \right |{}_{\mathcal G}$ will permute the nontrivial stabilizers in $\mathcal G$ and, under this assumption, can be considered an automorphism of $\mathcal G$ .

1.2. Free group systems and automorphisms

To formalize the last statement, we define a (countable) group system $\mathcal H$ to be a disjoint union $\bigsqcup _{i \in \mathcal I} H_i$ of countably many nontrivial countable groups; the latter are the components of $\mathcal H$ . The empty system is the group system with empty index set $\mathcal I$ . If all component groups $H_i$ have [property-?], then we shall call $\mathcal H$ a ‘[property-?] group system’. For instance, we will mainly work with subgroup systems and free group systems. In some ambient group, a subgroup system $\mathcal G$ carries another subgroup system $\mathcal G'$ if each $\mathcal G'$ -component is contained in a conjugate of some $\mathcal G$ -component.

The complexity of a (possibly trivial) free group H is $c(H) = 2 \cdot \mathrm {rank}(H) - 1$ , which takes values in $\mathbb Z_{\ge -1} \cup \{ \infty \}$ . For a (possibly empty) free group system $\mathcal H$ ,

A group system has finite type if the index set is finite and components are finitely generated. In particular, a free group system has finite type exactly when its complexity is finite. In this paper, $\mathcal F$ will denote a nonempty free group system of finite type.

The collection of nontrivial vertex stabilizers $\mathcal G$ for a free splitting of F can and will be viewed as a subgroup (or rather, free factor) system of finite type (even if empty). Similarly, we define a free splitting $\mathcal T$ of $\mathcal F$ to be a ‘disjoint union’ of free splittings of the components of $\mathcal F$ . A free splitting $\mathcal T$ is degenerate if all component simplicial trees $T_i$ are singletons. By passing to the (representatives of) nontrivial vertex stabilizers in a nondegenerate free splitting, we can inductively form a descending chain of free factor systems with strictly decreasing complexity. Starting with F, the length of such a chain is at most $c(F) + 1$ . We will mostly state and prove the results in terms of free group systems to facilitate induction on complexity.

An automorphism $\psi \colon \mathcal H \to \mathcal H$ of a group system $\mathcal H$ is a disjoint union of isomorphisms $\psi _i\colon H_i \to H_{\sigma \cdot i}~(i \in \mathcal I)$ where $\sigma \in \mathrm {Sym}(\mathcal I)$ . A subgroup system $\mathcal G = \bigsqcup _{j \in \mathcal J} G_j$ of $\mathcal H$ is [ψ]-invariant if $\psi (G_j) = h_j \, G_{\alpha \cdot j} \, h_j^{-1}$ for all $j \in \mathcal J$ , for some choice $( h_j \in \mathcal H : j \in \mathcal J)$ and $\alpha \in \mathrm {Sym}(\mathcal J)$ . The collection of isomorphisms $\left .\psi \right |{}_{\mathcal G} = \{ \mathrm {inn}(h_j^{-1}) \circ \left .\psi \right |{}_{G_j} \colon G_j \to G_{\alpha \cdot j} \}$ will be called a restriction of ψ to $\mathcal G$ ; note that $\left .\psi \right |{}_{\mathcal G}$ involves an implicit choice $( h_j : j \in \mathcal J)$ .

Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism and $\mathcal T$ a free splitting of $\mathcal F$ . A topological representative of $\psi $ on $\mathcal T$ is a disjoint union of $\psi _i$ -equivariant cellular maps $f_i\colon T_i \to T_{\sigma \cdot i}$ .

1.3. Train track theory

We conclude the chapter with some preliminary results from Bestvina–Handel’s theory of train tracks [Reference Bestvina and Handel3, Section 1]. A (relative) train track for an automorphism $\psi \colon \mathcal F \to \mathcal F$ is a topological representative of $\psi $ whose iterates are all topological representatives as well; equivalently, a train track is a topological representative whose restrictions of iterates to any edge are all injective. A topological representative $f \colon \mathcal T \to \mathcal T$ is irreducible if for any pair of edges $[e], [e']$ in the quotient graph $\Gamma _* = \mathcal F \backslash \mathcal T$ , there is an integer $n \ge 1$ for which $[f^n(e')]$ contains $[e]$ . To any topological representative f, we can associate a nonnegative integer square matrix $A(f)$ : rows and columns are indexed by the (unoriented) edges $[e], [e']$ resp. and matrix entry is the number of times $[e]$ appears in the path $[f(e')]$ ; then f is irreducible if and only if $A(f)$ is irreducible.

Let $f\colon \mathcal T \to \mathcal T$ be an irreducible topological representative. By Perron-Frobenius theory, the matrix $A(f)$ has a unique eigenvalue $\lambda (f) \ge 1$ with a positive eigenvector $\nu (f)$ ; it follows from the theory that f is a simplicial automorphism if $\lambda (f) = 1$ . Using $\nu (f)$ , we can equip $\mathcal T$ with an invariant metric (i.e., $\mathcal F$ acts by isometries with respect to this metric); furthermore, after applying an equivariant isotopy, the restriction of f to any edge will be a $\lambda (f)$ -homothety. The metric on $\mathcal T$ will be referred to as the eigenmetric. We have set the stage for the foundational theorem due to Bestvina–Handel:

See also Rylee Lyman’s [Reference Lyman16, Theorem A] for a very general version of this theorem.

Let $f\colon \mathcal T \to \mathcal T$ be a topological representative. We say an immersed path $\sigma $ in $\mathcal T$ is f-legal if restrictions of f-iterates to $\sigma $ are all injective. Thus, a train track is a topological representative $\tau $ whose edges are $\tau $ -legal. Besides knowing edges (and their forward iterates) are legal, it is useful to have legal elements (i.e., loxodromic elements with legal axes).

2.1. Trees and index theory

A (metric) tree is a 0-hyperbolic geodesic metric (nonempty) space – 0-hyperbolic means the union of any two sides of a geodesic triangle contains the third side of the triangle; this is also known as an $\mathbb R$ -tree. More generally, a forest $\mathcal T$ is a disjoint union $\bigsqcup _{i \in \mathcal I} T_i$ of trees. Let T be a tree; a direction at a point $p \in T$ is a component of the complement $T \setminus \{ p \}$ . A branch point of T is a point with at least three directions.

A $\lambda $ -homothety $f\colon T \to T$ is expanding if $\lambda> 1$ . Being expanding is invariant under iteration and composition with an isometry. It follows from the contraction mapping theorem that an expanding $\lambda $ -homothety of a complete tree has a unique fixed point and it is a repellor. Generally, a λ-homothety $f\colon \mathcal T \to \mathcal T$ of a forest $\mathcal T = \bigsqcup _{i \in \mathcal I} T_i$ is a disjoint union of $\lambda $ -homotheties $f_i\colon T_i \to T_{\sigma \cdot i}~(i\in \mathcal I)$ , where $\sigma \in \mathrm {Sym}(\mathcal I)$ .

An isometry of a tree is either elliptic if it fixes a point or loxodromic if it has a unique minimal invariant subtree isometric to $\mathbb R$ , known as the axis, on which it acts by a nontrivial translation. We will mostly only care about isometric actions on trees with finite arc stabilizers (i.e., actions by isometries where the stabilizers of nondegenerate intervals are finite). When the acting group is torsion-free, we can equivalently say isometric actions with trivial arc stabilizers.

Suppose $d_T$ is the metric on T; an isometry $\iota \colon T \to T$ has translation distance given by

$$\begin{align*}\| \iota \| = \inf_{p \in T} d_T(p, \iota(p)).\end{align*}$$

Any isometric H-action on a tree has a nonnegative function $\|\cdot \|\colon H \to \mathbb R_{\ge 0}$ given by the translation distances. For a group system $\mathcal H = \bigsqcup _{i \in \mathcal I} H_i$ and a forest $\mathcal T = \bigsqcup _{i \in \mathcal I} T_i$ , an isometric $\mathcal H$ -action on $\mathcal T$ is a disjoint union of isometric $H_i$ -actions on $T_i$ .

A characteristic subtree for an isometric action on a tree T is a minimal invariant subtree in T. An isometric action on a tree T is minimal if T is the characteristic subtree.

Suppose F acts minimally on a nondegenerate tree T by isometries with trivial arc stabilizers. We now introduce the index theory for such actions. For each F-orbit of points $[p] \in F \backslash T$ , let $G_p \le F$ denote the stabilizer of $p \in T$ and $\#\mathrm {dir}[p]$ denote the number of $G_p$ -orbits of directions at p. The (local) index at $[p]$ is

The (global) index of $F \backslash T$ is

One of our main tools will be Gaboriau–Levitt’s index inequality [Reference Gaboriau and Levitt8, Theorem III.2]:

$$\begin{align*}i(F \backslash T) < c(F).\end{align*}$$

For example, the index of any free splitting of F is $c(F)-1$ . In Appendix A, we will sketch Gaboriau–Levitt’s proof in a metric-agnostic setting. Corollaries of the inequality: there are finitely many F-orbits of directions at branch points and the point stabilizers subgroup system is represented by a system that has strictly lower complexity than F. For a free group system of finite type, the index of its isometric action on a forest is the sum of indices for each component, and the index inequality still holds.

2.2. Pretrees and rigid actions

Brian Bowditch’s paper [Reference Bowditch5] is a good survey about pretrees and other ‘treelike’ structures. A pretree is a (nonempty) set T and a function $[\cdot ,\cdot ]\colon T \times T \to \mathcal P(T)$ (i.e., an association of a subset $[p,q] \subset T$ to each pair of points $p,q \in T$ ) that satisfies the pretree axioms: for all $p,q,r \in T$ ,

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

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

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

The subsets $[p,q]$ from the definition will be refered to as closed intervals and should be thought of as encoding a ‘betweenness’ relation on T. Define the open interval $(p,q)$ to be the subset $[p,q] \setminus \{p,q\}$ . Similarly, define half-open intervals $[p,q) = (q,p] = [p,q] \setminus \{q\}$ . Naturally, the real line $\mathbb R$ is a pretree; a tree has a (canonical) pretree structure where closed intervals are the closed geodesic segments; also, any subset $S \subset T$ of a pretree inherits a pretree structure given by .

A direction at $p \in T$ is a maximal subset $D_p \subset T$ for which $p \notin [q,r]$ for all $q, r \in D_p$ . As we did for trees, a branch point for a pretree is a point with at least three directions. The observers’ topology on a pretree is the canonical topology generated by the subbasis of directions; Bowditch calls it the order topology [Reference Bowditch5, Section 7]. Generally, a direction at a nonempty subset $S \subset T$ is a maximal $D_S \subset T$ for which $[q,r] \cap S = \emptyset $ for all $q,r \in D_S$ .

Suppose T and $T'$ have pretree structures $[\cdot , \cdot ]$ and $[\cdot , \cdot ]'$ , respectively. A set-bijection $f\colon T\to T'$ is a pretree-isomorphism if $[f(p), f(q)]' = f([p,q])$ for all $p,q \in T$ . An injection $f\colon T \to T'$ is a pretree-embedding if it is a pretree-isomorphism onto the image $f(T)$ with the inherited pretree structure; we will only need pretree-embeddings for Appendix A. Note that pretree-isomorphisms induce homeomorphisms of the observers’ topologies. There is another canonical topology finer than the observers’ topology (see the interlude chapter); we will not need either canonical topologies for the results of the paper.

For a pretree-automorphism $f \colon T \to T$ , the fixed-point set $\mathrm {Fix}_T(f)$ is the subset of points in T fixed by f. A pretree-automorphism f is rigid if either $\mathrm {Fix}_T(f)$ is empty or f fixes no direction at $\mathrm {Fix}_T(f)$ . An isometry of a (subset of a) tree is a rigid pretree-automorphism of its (inherited) pretree structure. A subset $C \subset T$ is convex if $[p,q] \subset C$ for all $p,q \in C$ . The fixed-point set $\mathrm {Fix}_T(f)$ of a rigid pretree-automorphism is convex.

A pretree is real if every closed interval $[p,q]$ is pretree-isomorphic to a closed interval of $\mathbb R$ . The pretree structure of a tree is real. A nondegenerate convex subset $A \subset T$ is an arc if any $x,y,z \in A$ simultaneously lie in some closed interval. A pretree is complete if every arc is an interval. Notably, any real pretree T embeds in a canonical complete pretree $\widehat T$ known as the pretree completion [Reference Bowditch5, Lemma 7.14]. A real pretree is short if every arc is pretree-isomorphic to an arc in $\mathbb R$ . The real pretree structure of a tree is short. For any short real pretree T, the pretree completion $\widehat T$ is real [Reference Bowditch5, Lemma 7.15]. For example, the pretree $\mathbb R$ is not complete (it is an arc but not an interval), but it is short by definition; the pretree completion of $\mathbb R$ is the extended real line, which has two additional points $\{ \pm \infty \}$ , may be denoted $[-\infty , \infty ]$ , and is pretree-isomorphic to a closed interval of $\mathbb R$ . The long line is the prototype of a real pretree whose pretree completion, the extended long line, is not real: it is a closed interval not pretree-isomorphic to a closed interval of $\mathbb R$ .

A pretree-automorphism $f\colon T \to T$ of a real pretree is elliptic if it has a fixed point and loxodromic otherwise; in the latter case, there is a maximal arc $A \subset T$ , known as the axis, preserved by f. For an elliptic pretree-automorphism f of a real pretree T, the complement $T \setminus \mathrm {Fix}_T(f)$ is ‘open’ in the following sense: the complement $(p,q) \setminus \mathrm {Fix}_T(f)$ is a union of open intervals for any $p,q \in T$ ; in particular, a direction d at $\mathrm {Fix}_T(f)$ has an attaching point $p_d \in \mathrm {Fix}_T(f)$ . If f is also rigid, then the direction d has a unique attaching point due to convexity. In the literature, rigid pretree-automorphisms of real pretrees have been studied as non-nesting homeomorphisms [Reference Levitt12] – Levitt’s results are stated with $\mathbb R$ -trees, but the metrics are never used and so the results apply to pretrees as well. Again, we mostly only care about rigid actions on real pretrees with finite arc stabilizers (i.e., actions by rigid pretree-automorphisms where the (pointwise) stabilizers of arcs are finite).

A characteristic convex subset for a rigid action on a real pretree T is a minimal invariant nonempty convex subset of T. A rigid action on a real pretree T is minimal if T is the characteristic convex subset. Note that a real pretree T that admits a minimal rigid action by a countable group must be short: T will be a countable union of closed intervals; so its arcs are countable ascending unions of closed intervals and hence pretree-isomorphic to arcs in $\mathbb R$ .

Suppose we have a minimal rigid F-action on a nondegenerate real pretree T with trivial arc stabilizers. Then we can define the index of $F \backslash T$ exactly as we did for minimal isometric F-actions with trivial arc stabilizers. Gaboriau–Levitt’s index inequality still holds since their proof extends almost verbatim to this setting of minimal rigid actions with trivial arc stabilizers (see Appendix A for the sketch).

We now introduce a new term that will be our replacement for expanding homotheties in the real pretree setting. Let $\phi \colon F \to F$ be an automorphism and T a real pretree with a chosen minimal rigid F-action whose arc stabilizers are trivial. Recall that $f\colon T \to T$ is $\phi $ -equivariant if $f(x \cdot p) = \phi (x) \cdot f(p)$ for all $x \in F$ and $p \in T$ . A $\phi $ -equivariant pretree-automorphism $f\colon T \to T$ expands at $p \in \mathrm {Fix}_T(f)$ if each orbit $G_p \cdot d$ of directions at p contains some half-open interval $(p, q_d]$ and each $(p,q_d]$ properly embeds in a $G_p$ -translate of $f((p, q_{d'}])$ for some orbit $G_p \cdot d'$ of directions at p. By the index inequality, there are finitely many $G_p$ -orbits of directions at p; so for some $n \ge 1$ , each $(p, q_d]$ properly embeds in a $G_p$ -translate of $f^n((p, q_d])$ – this justifies the term ‘expanding’.

A $\phi $ -equivariant pretree-automorphism $f\colon T \to T$ is F-expanding if for any element $x \in F$ and integer $n \ge 1$ , the composition $x \circ f^n \colon T \to T$ 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. For the motivation behind this definition, note that $\phi $ -equivariant expanding homotheties are (by the contraction mapping theorem) F-expanding pretree-automorphism of the canonical pretree structure.

4.1. Simple patchwork

In this case, a blow-up is easy but tedious to define; ensuring the induced map is a homothety is the tricky bit.

We state and prove the theorem in terms of homotheties due to the specific needs in Proposition 3.2, but the argument actually holds if ‘expanding homothety’ is replaced with ‘expansions (or contractions)’. We only need a hypothesis that lets us use the contraction mapping theorem.

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 $h_{\mathcal V} \colon \mathcal T_{\mathcal V} \to \mathcal T_{\mathcal V}$ .

Remark. We give two related proofs of the conclusion to the theorem. The first proof is short, but it does not generalize to non-metric settings – we only sketch it. The second proof is thorough as it contains the ideas needed later for the main construction.

(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 $\vec p$ .

4.2. Naïve stitching

Due to the involved nature of the proofs when dealing with non-simplicial trees, the blow-up and stitching constructions are split into two sections; let us start with blow-ups.

Suppose a countable group G acts rigidly on a real pretree T. Recall that if the action is minimal, then T must be short and the pretree completion $\widehat T$ is itself real. Given any subset