Introduction
The study of free group outer automorphisms shares a lot with the theory of mapping class groups of surfaces. The relevant dictionary replaces compact surfaces with finite graphs (i.e., finite 1-dimensional CW-complexes) and surface homeomorphisms with graph homotopy equivalences. Although many results in surface theory have analogues in the free group setting, the situation for mapping classes tends to be comparatively well-behaved as one is working with surface homeomorphisms. On the other hand, an infinite order outer automorphism of a free group can never be represented by a homeomorphism of a graph. As a consequence, we have fewer tools at our disposal to study free group automorphisms. One such important missing tool is a complete analogue of Nielsen–Thurston theory.
Nielsen–Thurston theory
The main goal of the current project is defining an analogue of singular measured foliations in the free group setting. Let us start with a summary of the Nielsen–Thurston theory for surface homeomorphisms.
Suppose S is a compact surface with negative Euler characteristic, and let $f\colon S \to S$ be a homeomorphism. William Thurston proved [Reference Thurston21, Theorem 4] there is
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1. a canonical (potentially empty) union $\gamma $ of essential simple closed curves;
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2. a homeomorphism $g\colon S \to S$ isotopic to f that leaves a regular neighbourhood $N(\gamma )$ of the multicurve $\gamma $ invariant; and
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3. the restriction of g to permuted components of the complement $S \setminus N(\gamma )$ is either
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○ a pseudo-Anosov homeomorphism (i.e., has a canonical invariant measured singular foliation whose transverse measure is scaled by a stretch factor $\lambda> 1$ ); or
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○ a linear homeomorphism (i.e., reducible (possibly non-canonically) to a finite order homeomorphism) – Birman–Lubotzky–McCarthy later proved the latter reduction can be made canonical [Reference Birman, Lubotzky and McCarthy4, Theorem C], but that is not in the standard expositions of Thurston’s result.
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Thus, up to isotopy, we may assume a given surface homeomorphism $S \to S$ has a canonical decomposition into pseudo-Anosov components and their complements, and each orbit $S'$ of pseudo-Anosov components has an invariant measured singular foliation whose transverse measure is scaled by some stretch factor $\lambda _{S'}> 1$ . Lifting this decomposition to the universal cover $\widetilde {S}$ gives a canonical partition of the cover into lifts of the foliation’s leaves and the unfoliated complementary regions.
One may now endow this universal cover with a pseudo-metric using the lift of the transverse measure of the foliation. Morgan–Shalen proved [Reference Morgan and Shalen18, Section 1] that the corresponding metric space is an $\mathbb R$ -tree, or simply tree as we shall call it in this paper, and the action of $\pi _1(S)$ on the universal cover by deck transformations induces an action on this tree by isometries, or simply isometric action. Finally, any lift of the surface homeomorphism induces an expanding ‘dilation’ of the tree: the tree decomposes into finitely many orbits of subtrees, and the restriction of the induced map to each subtree is an expanding homothety – the expansion factor may vary with the subtree’s orbit.
The minimal isometric $\pi _1(S)$ -action on the tree has two interesting properties:
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○ arc (pointwise) stabilizers are trivial; and
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○ an element of $\pi _1(S)$ is elliptic if and only if its conjugacy class is represented by a closed curve in the complement of the pseudo-Anosov components.
Moreover, the tree’s equivariant dilation class is canonical by construction.
Our motivation and main theorem
When we started this project, we wanted to prove a direct free group analogue of this statement about the canonical limit tree – the dilation requirement on the expanding homeomorphism would have been relaxed and complements of pseudo-Anosov components replaced by polynomially growing subgroups. This proved to be extremely elusive, and we now doubt such a metric analogue always exists!
Fortunately, we did manage to prove a variation to this analogy. Let us return to the limit $\mathbb R$ -tree T given by the canonical decomposition of the universal cover $\widetilde {S}$ and the transverse measure. Closed arcs in T determine a real pretree structure; a real pretree is essentially what you get when you try to define an $\mathbb R$ -tree without a metric or topology (see Section 2.2). The isometric $\pi _1(S)$ -action on the $\mathbb R$ -tree determines a rigid $\pi _1(S)$ -action on its real pretree. Rigid actions are commonly known as non-nesting actions in the literature [Reference Levitt12], but we find the name ‘rigid’ more evocative.
There is no simple way to capture the ‘expanding’ nature of the homotheties without a metric on the real pretree. We introduce the term (free group)-expanding to describe the induced pretree-automorphisms (see Section 2.2). With this new terminology, the free group analogue for the real pretree statement becomes the following:
Main Theorem (Theorem 3.3).
If $\phi \colon F \to F$ is an automorphism of a finitely generated free group, then there is:
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1. a minimal rigid F-action on a real pretree T with trivial arc stabilizers;
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2. a $\phi $ -equivariant F-expanding pretree-automorphism $f\colon {T} \to {T}$ ; and
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3. an element in F is T-elliptic if and only if it grows polynomially under $[\phi ]$ -iteration.
The nontrivial point stabilizers are finitely generated. Moreover, these subgroups are proper and have rank strictly less than that of F if and only if $[\phi ]$ is exponentially growing.
Remark. The third condition can be restated in terms of loxodromic elements. The actual theorem will be stated and proven in a ‘relative setting’. The growth types of elements and automorphisms are defined in Chapter 3.
We suspect that the action’s rigidity can be strengthened: perhaps, the induced action on the pretree completion is a convergence action. To complete the analogy with the surface setting, we discuss in the epilogue the extent to which a limit pretree produced by this theorem is canonical – this is proven in the sequel [Reference Mutanguha19]. This is of much interest as it is a canonical representation of a free group automorphism. For instance, it allows us to classify automorphisms in terms of the limit pretree’s index (see Appendix A).
In the long run, we would also like to find interesting canonical representations of polynomially growing automorphisms – any pretree of Theorem 3.3 is a point for these automorphisms.
Proof outline for the main theorem. Using irreducible train tracks, construct an isometric action on a limit tree with trivial arc stabilizers. This action admits an expanding homothety that represents the given free group automorphism. In particular, polynomially growing elements are elliptic; however, it is possible for some exponentially growing elements to be elliptic as well. By considering restrictions of the automorphism to point stabilizers, we get a hierarchy of isometric actions on trees. Most importantly, elements grow polynomially if and only if they are elliptic in each tree in the hierarchy.
The key step describes how to combine this hierarchy of trees, through blow-ups, into one treelike structure – a real pretree. While intuitive, the details of this construction get a bit technical. If done appropriately, this new structure will admit an F-expanding pretree-automorphism. Since we have a hierarchy of expanding homotheties, the contraction mapping theorem implies the blow-up construction can indeed be done appropriately!
1. Preliminaries
In this paper, F will denote a nontrivial free group of finite rank. Note that subscripts will never indicate the rank but will instead be mostly used to index a collection of free groups.
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:
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○ cellular: f maps vertices to vertices and is injective on edges; and
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○ $\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:
Theorem 1.1 (cf. [Reference Bestvina and Handel3, Theorem 1.7]).
If $\psi \colon \mathcal F \to \mathcal F$ is an automorphism of a free group system $\mathcal F$ and $\mathcal G$ a $[\psi ]$ -invariant proper free factor system of $\mathcal F$ , then there is an irreducible train track $\tau \colon \mathcal T \to \mathcal T$ for $\psi $ defined on some nondegenerate free splitting $\mathcal T$ of $\mathcal F$ whose vertex stabilizers carry $\mathcal G$ .
See also Rylee Lyman’s [Reference Lyman16, Theorem A] for a very general version of this theorem.
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 $\mathcal F$ .
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).
Proposition 1.2. If $\psi \colon \mathcal F \to \mathcal F$ is an automorphism and $\tau \colon \mathcal T \to \mathcal T$ an irreducible train track for $\psi $ , then each nondegenerate component of $\mathcal T$ contains a $\tau $ -legal axis of some loxodromic element of $\mathcal F$ .
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 $e_i$ .
2. Trees and real pretrees
Expanding irreducible train tracks are very useful for understanding the dynamics of an automorphism since the iterates of edges all expand by the same factor under the eigenmetric. Unfortunately, since the train track necessarily fails to be injective near some vertices, not all paths will similarly expand: for instance, there are so-called ‘periodic Nielsen paths’ whose lengths (after reduction) remain uniformly bounded under iteration.
One way to get around this is to promote the expanding irreducible train track to an expanding homothety; however, this promotion requires leaving the category of simplicial trees and working in the metric category instead (Proposition 3.2).
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
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]:
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 $p,q,r \in T$ ,
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1. (symmetric) $[p,q] = [q,p]$ contains $\{p, q\}$ ;
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2. (thin) $[p,r] \subset [p,q] \cup [q,r]$ ; and
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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.
Interlude – same trees, different views
This interlude is meant to describe the different perspectives on ‘simplicial trees’. As it was borne out of my own failure to appreciate the differences before starting this project, I am writing the interlude from a personal point of view.
What is a simplicial tree? Figure 1 is visual representation of a simplicial tree that will be the running example for the interlude.
The most elementary definition describes it as a combinatorial object. A simple graph is a pair $(V, E)$ , where V is the vertex set and the edge set E is a collection of size $2$ subsets of V. I could use this to define reduced paths and cycles; then a simple tree would be a path-connected cycle-free simple graph. For example, this was the language used by Serre in [Reference Serre20], and it is the quickest way to define free splittings. As a simple tree, the running example can be defined as: $V = \mathbb Z_{\ge 0}$ and $E = \{\,\{ 0, n\}~:~n \ge 1\,\}$ . This definition is unique up to a simplicial automorphism: a set-bijection of vertex sets that induces a set-bijection of the edge sets.
This perspective is especially useful if you are interested in algorithmic questions. The downside is that it can get quite cumbersome to describe maps between simple trees that send edges to paths. Instead, it helps to work with topological spaces where the language of maps and their deformations is already well-established.
The first topological definition: a cellular tree is a 1-dimensional contractible CW-complex. This was the parenthetical definition I gave at the start of Chapter 1. This definition assumes you already know what a CW-complex is. For our purposes, a 1-dimension CW-complex is a quotient of the disjoint union of a discrete space (vertex set) and copies of closed intervals $[0,1]$ with an equivalence relation identifying each endpoint of the closed intervals with some vertex. The complex is endowed with the quotient topology, and I will skip defining contractibility. If you were a pedantic reader, then you may have noticed that I abused terminology while defining free splittings: I define a simplicial tree as a ‘cellular tree’, yet I require that the free group act by ‘simplicial automorphism’; I never exactly explained what a simplicial automorphism of a CW-complex is! The moral of the story is that I have prioritized brevity. Anyway, as a cellular tree, the running example is defined as the quotient of $\{ v \} \sqcup \left ( \mathbb Z_{\ge 1} \times [0,1]\right )$ with the equivalence relation generated by $v \sim (n, 0)$ for all $n \ge 1$ . The set of equivalence classes (without the topology) will be denoted T, and let $\pi $ be the quotient function onto T.
Now that I am dealing with topological spaces, I can discuss concepts like convergence or compactness. Let $x_n = (1, \frac {1}{n}), y_n = (n, \frac {1}{n}),$ and $z_n = (n, 1)$ . Then the sequence $(x_n)_{n\ge 1}$ converges to $\pi (v)$ in the CW/quotient topology, while the sets $\{ y_n \}_{n \ge 1}$ and $\{ z_n \}_{n \ge 1}$ have no limit points; moreover, this topology is not metrizable.
The metric definition: a discrete (metric) tree is a (metric) tree that is the convex hull of its singular points (i.e., number of directions at the point is not 2), and the subspace of these points is discrete. Metric spaces have a topology generated by the basis of open balls. In practice, this topology is secondary, and it is usually more convenient to work with the metric directly. To view T as a discrete tree, equip it with the combinatorial convex metric:
The metric topology on T is strictly coarser than the CW-topology as it contains fewer open sets. This time, the sequences $(x_n)_{n\ge 1}$ and $(y_n)_{n\ge 1}$ both converge to $\pi (v)$ in the metric topology. However, the set $\{ z_n \}_{n \ge 1}$ still has no limit points. So the metric topology is not compact.
As you shall see in the next chapter, the metric setting helps us understand what an automorphism does under iteration, especially at the ‘limit’. This starts with equipping free splittings (simple or cellular trees) with the eigenmetric of an irreducible train track and taking the projective-limit of iterating the train track to get a useful and probably nondiscrete tree.
The final topological definition: a (separable) ‘visual’ tree is a connected subspace of a dendrite (i.e., compact separable metric tree) that is the convex hull of its own singular points and whose branch points cannot accumulate to a branch point along one direction. I could remove the separability condition by replacing ‘dendrite’ with ‘dendron’, but that would necessitate a definition for dendrons. There is no metric characterization for dendrons as there is for dendrites, and, unfortunately, their purely topological definition is beyond the scope of this paper; see [Reference Bowditch5] for details. A pretree characterization for dendrons can be the observers’ topology on a complete real pretree. As a visual tree, the running example is the observers’ topology on the cellular or metric tree T. The observers’ topology is even coarser than the metric topology, and it makes the sequences $(x_n)_{n\ge 1}$ , $(y_n)_{n\ge 1}$ and $(z_n)_{n\ge 1}$ all converge to $\pi (v)$ . In fact, T with the observers’ topology is compact – it is a dendrite! As a visual tree, the topology is metrizable, and one compatible convex metric is given by
The underlying thesis of this project is that the ‘metric category’ might not be right for defining blow-ups of nondiscrete trees. It is not clear to me that it is even possible in general! However, defining blow-ups in the ‘visual category’ is rather natural. In fact, we avoid topology altogether and carry out the construction in the ‘pretree category’!
Finally, a topological-combinatorial hybrid definition: a simple pretree is a pretree whose closed intervals are finite sets; and a simple real pretree is real pretree that is the convex hull of its singular points, and these points form a simple pretree. The simple pretree definition is simply paraphrasing that of a simple tree, while the simple real pretree definition simultaneously captures the combinatorial nature of trees and allows topological approaches. Simple real pretrees have two canonical topologies: the coarser one is the observers’ topology, and the finer one is more-or-less the CW-topology.
So what is a simplicial tree? Well, it could be a simple tree, cellular tree, discrete tree, visual tree or simple real pretree; the answer depends on what I need it for. For instance, if I want to discuss ‘simplicial actions’ on $\mathbb R$ , then only the first two definitions are applicable – $\mathbb R$ has no singular points!
3. Limit pretrees: exponentially growing automorphisms
Returning to free group automorphisms, we can now use translation distances in free splittings (with invariant combinatorial metrics) to classify free group automorphisms.
Fix an automorphism $\psi \colon \mathcal F \to \mathcal F$ and a $[\psi ]$ -invariant proper free factor system $\mathcal G$ of $\mathcal F$ . By Theorem 1.1, there is an irreducible train track $\tau ^{(1)}$ for $\psi $ on a free splitting $(\Gamma _*^{(1)}, \mathcal F_2)$ of $\mathcal F_1 = \mathcal F$ , where $\mathcal F_2$ carries $\mathcal G$ . The train track $\tau ^{(1)}$ determines a restriction $\left .\psi \right |{}_{\mathcal F_2}$ ; applying the theorem repeatedly, we get a finite descending sequence of irreducible train tracks $\tau ^{(i)}$ for restrictions $\left .\psi \right |{}_{\mathcal F_{i}}$ on free splittings $(\Gamma _*^{(i)}, \mathcal F_{i+1})$ of $\mathcal F_{i}$ , and the final train track $\tau ^{(k)}$ in the sequence is defined rel. $\mathcal G$ (i.e., $\mathcal F_{k+1} = \mathcal G$ ). The next proposition collects standard facts that immediately follow from this train track theory; we sketch the proof mainly to highlight the study of automorphisms through the blow-up of a free splitting rel. another free splitting.
Proposition 3.1. Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism, $\mathcal G$ a $[\psi ]$ -invariant proper free factor system of $\mathcal F$ , and $(\Lambda _*, \mathcal G)$ a free splitting of $\mathcal F$ . For any element x in $\mathcal F$ , the limit inferior of the sequence $\left (\sqrt [n]{\|\psi ^n(x)\|_{\Lambda *}} \right )_{n \ge 1}$ is finite and independent of $(\Lambda _*, \mathcal G)$ .
Let $\left (\tau ^{(i)}\right )_{i=1}^k$ be a descending sequence of irreducible train tracks with $\tau ^{(k)}$ defined rel. $\mathcal G$ . Then the following are equivalent:
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1. $\lambda \left (\tau ^{(i)}\right ) = 1$ for all $i = 1, \ldots , k$ ;
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2. for any $x \in \mathcal F$ , the sequence $\left (\|\psi ^n(x)\|_{\Lambda *} \right )_{n \ge 1}$ is bounded by a polynomial in n; and
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3. $\underset {n\to \infty }\liminf \sqrt [n]{\|\psi ^n(x)\|_{\Lambda *}} \le 1$ for all $x \in \mathcal F$ .
An automorphism is polynomially growing rel. 𝒢 if these conditions hold; otherwise, it is exponentially growing rel. 𝒢.
Sketch of proof.
The automorphism $\psi $ has a Lipschitz topological representative on the free splitting $(\Lambda _*, \mathcal G)$ of $\mathcal F$ . So for any element x in $\mathcal F$ , we can set in $\mathbb R_{\ge 0}$ . By $[\psi ]$ -invariance of $\mathcal G$ , $\underline {\lambda _x} = 0$ if $\| x \|_{\Lambda _*} = 0$ . Since $0$ is isolated in the image of $\| \cdot \|_{\Lambda _*}$ , we get $\underline {\lambda _x} \ge 1$ if $\| x \|_{\Lambda _*}> 0$ . Any pair of free splittings $(\Lambda _*, \mathcal G)$ and $(\Omega _*, \mathcal G)$ of $\mathcal F$ have equivariant Lipschitz maps between them, and so their translation distance functions are comparable; that is,
This implies $\underline {\lambda _x}$ is independent of the free splitting $(\Lambda _*, \mathcal G)$ and concludes the first part.
$(1 \implies 2)$ : Suppose $\lambda (\tau ^{(i)}) = 1$ for all $i = 1, \ldots , k$ . So the train tracks $\tau ^{(i)}$ are simplicial automorphisms; in particular, all edge-paths in $(\Omega _*^{(k)}, \mathcal G) = (\Gamma _*^{(k)}, \mathcal G)$ have constant growth. For induction, assume all edge-paths in $(\Omega _*^{(i)}, \mathcal G)$ have at most degree $(k-i)$ polynomial growth. Let $(\Omega _*^{(i-1)}, \mathcal G)$ be a blow-up of $(\Gamma _*^{(i-1)},\mathcal F_{i})$ with respect to $(\Omega _*^{(i)}, \mathcal G)$ . As $\tau ^{(i-1)}$ is a simplicial automorphism, each ‘top-stratum’ edge gains a predetermined prefix and suffix in $(\Omega _*^{(i)}, \mathcal G)$ under iteration; these prefixes and suffixes have at most degree $(k-i)$ polynomial growth. So edge-paths in $(\Omega _*^{(i-1)}, \mathcal G)$ have at most degree $(k-i+1)$ polynomial growth. By induction and comparability again, edge-paths in $(\Lambda _*, \mathcal G)$ have at most degree $(k-1)$ polynomial growth.
$(2 \,{\implies}\, 3)$ : Polynomials are sub-exponential.
$(3 \,{\implies}\, 1)$ : Suppose $\lambda = \lambda \left (\tau ^{(i)}\right )> 1$ for some i. Equip $(\Gamma _*^{(i)}, \mathcal F_{i+1})$ with the eigenmetric $d_i$ , and let $x \in \mathcal F_i$ be a $\tau ^{(i)}$ -legal element (Proposition 1.2). The choice of metric and legality of x imply $\|\psi ^n(x)\|_{d_i} = \lambda ^n \cdot \| x \|_{d_i}> 0$ . Let $(\Omega _*^{(1)}, \mathcal G)$ be the free splitting constructed above. Then $\| \cdot \|_{d_i}$ and $\| \cdot \|_{\Gamma _*^{(i)}}$ are comparable, $\|\cdot \|_{\Gamma _*^{(i)}} \le \|\cdot \|_{\Omega _*^{(i)}}$ pointwise, and $\| \cdot \|_{\Omega _*^{(i)}}$ is the restriction of $\| \cdot \|_{\Omega _*^{(1)}}$ . So $\underline {\lambda _x} \ge \lambda> 1$ .
For any element x in $\mathcal F$ , define . We say $x \in \mathcal F$ grows exponentially rel. 𝒢 if $\underline \lambda (x;\psi ,\mathcal G)> 1$ ; otherwise, it grows polynomially rel. 𝒢. The ‘rel. $\mathcal G$ ’ will be omitted when $\mathcal G$ is empty. Suppose $\mathcal H$ is a $[\psi ]$ -invariant subgroup system of finite type that carries $\mathcal G$ . By passing to the characteristic subforest for $\mathcal H$ in a free splitting $(\Gamma _*,\mathcal G)$ of $\mathcal F$ , we see that $\underline \lambda (x;\left .\phi \right |{}_{\mathcal H},\mathcal G) = \underline \lambda (x;\psi ,\mathcal G)$ for elements x in $\mathcal H$ . We use this observation whenever we pass to invariant subgroup systems of finite type.
Remark. It was known since the introduction of train tracks that the first sequence in the statement of Proposition 3.1 converges (see [Reference Bestvina and Handel3, Remark 1.8]). Gilbert Levitt gave a finer classification for an element’s growth rate [Reference Levitt13, Theorem 6.2].
The next proposition appears in Levitt–Lustig’s paper [Reference Levitt and Lustig14, Proposition 3.2]. Our proof takes a slightly different approach using the blow-up of a free splitting rel. expanding forest. This is done to highlight ideas crucial to the proof of the main theorem.
Proposition 3.2. Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism and $\mathcal G$ a $[\psi ]$ -invariant proper free factor system of $\mathcal F$ . If $\psi \colon \mathcal F \to \mathcal F$ is exponentially growing rel. $\mathcal G$ , then there is a minimal isometric $\mathcal F$ -action on a nondegenerate forest $\mathcal Y$ with trivial arc stabilizers for which $\mathcal G$ is elliptic, and a $\psi $ -equivariant expanding homothety $h\colon \mathcal Y \to \mathcal Y$ .
In particular, $\mathcal Y$ -loxodromic elements in $\mathcal F$ grow exponentially rel. $\mathcal G$ .
Proof (modulo a black box).
Suppose $\psi \colon \mathcal F \to \mathcal F$ is exponentially growing rel. $\mathcal G$ , and set $\mathcal F_1 = \mathcal F$ . Then there is a finite sequence of irreducible train tracks $\tau ^{(i)}$ for restrictions $\left .\psi \right |{}_{\mathcal F_{i}}$ on nondegenerate free splittings $(\Gamma _*^{(i)}, \mathcal F_{i+1})$ of $\mathcal F_{i}$ with $\lambda (\tau ^{(i)})> 1$ for some $i \ge 1$ ; we equip the free splittings with eigenmetrics $d_i$ . After truncating the sequence if necessary, assume $\lambda (\tau ^{(i)}) = 1$ for $i = 1, \ldots , k-1$ and $\lambda = \lambda (\tau ^{(k)})> 1$ . By Proposition 1.2, there is a $\tau ^{(k)}$ -legal element $x_0$ in $\mathcal F_k$ .
As $\tau ^{(k)}$ is $\lambda $ -Lipschitz, we can define a limit function
Culler–Morgan proved $\|\cdot \|^{(k)}$ is a translation distance function for a minimal isometric $\mathcal F_{k}$ -action on a forest $\mathcal T^{(k)}$ with cyclic arc stabilizers [Reference Culler and Morgan6, Theorem 5.3]; the forest is not degenerate as ${\|x_0\|^{(k)} = \|x_0\|_{\Gamma _v^{(k)}}> 0}$ . Lustig proved the $\mathcal F_{k}$ -action on $\mathcal T^{(k)}$ has trivial arc stabilizers [Reference Gaboriau, Levitt and Lustig9, Appendix]. There is a $\left .\psi \right |{}_{\mathcal F_{k}}$ -equivariant $\lambda $ -homothety $h^{(k)}\colon \mathcal T^{(k)} \to \mathcal T^{(k)}$ since $\|\psi (x) \|^{(k)} = \lambda \|x\|^{(k)}$ for all elements x in $\mathcal F_{k}$ [Reference Culler and Morgan6, Theorem 3.7]. By construction, the $[\psi ]$ -invariant proper free factor system $\mathcal G$ is $\mathcal T^{(k)}$ -elliptic.
The main step of the proof is a construction that allows us to ‘merge’ the higher polynomial strata and the exponential stratum at k. If $k = 1$ , then set $\mathcal Y = \mathcal T^{(k)}$ , $h = h^{(k)}$ , and we are done. Otherwise, $k \ge 2$ and, for some $i \le k$ , there is a minimal isometric $\mathcal F_{i}$ -action on a nondegenerate forest $\mathcal T^{(i)}$ with trivial arc stabilizers and an equivariant copy of $\mathcal T^{(k)}$ , and a $\left .\psi \right |{}_{\mathcal F_{i}}$ -equivariant $\lambda $ -homothety $h^{(i)}\colon \mathcal T^{(i)} \to \mathcal T^{(i)}$ .
In the next chapter (simple patchwork 4.1, black box), we construct a unique forest $\mathcal T^{(i-1)}$ that is ‘an equivariant blow-up and simplicial collapse’ of $(\Gamma _*^{(i-1)}, \mathcal F_{i})$ with respect to $\tau ^{(i-1)}$ and $h^{(i)}$ . In particular, there is a minimal isometric $\mathcal F_{i-1}$ -action on $\mathcal T^{(i-1)}$ with trivial arc stabilizers and an equivariant copy of $\mathcal T^{(i)}$ , and a $\left .\psi \right |{}_{\mathcal F_{i-1}}$ -equivariant $\lambda $ -homothety $h^{(i-1)}$ on $\mathcal T^{(i-1)}$ induced by $\tau ^{(i-1)}$ and $h^{(i)}$ . By induction, we have a minimal isometric $\mathcal F$ -action on a forest $\mathcal Y = \mathcal T^{(1)}$ with trivial arc stabilizers, an equivariant copy of $\mathcal T^{(k)}$ and a $\psi $ -equivariant $\lambda $ -homothety $h=h^{(1)}$ on $\mathcal Y$ .
For the last part of the proposition, choose a free splitting $(\Lambda _*, \mathcal G)$ for $\mathcal F$ . Let $\|\cdot \|_{\Lambda _*}$ and $\| \cdot \|_{\mathcal Y}$ be the translation distance functions for $(\Lambda _*, \mathcal G)$ and $\mathcal Y$ , respectively. Since $\mathcal G$ is $\mathcal Y$ -elliptic, there is an equivariant Lipschitz map $(\Lambda _*, \mathcal G) \to \mathcal Y$ and
Therefore, $\lambda ^n \| \cdot \|_{\mathcal Y} = \|\psi ^n(\cdot )\|_{\mathcal Y} \le K \|\psi ^n(\cdot )\|_{\Lambda _*}$ . So $\mathcal Y$ -loxodromic elements in $\mathcal F$ grow exponentially rel. $\mathcal G$ .
The main theorem of this paper associates to a free group automorphism a real pretree with a minimal rigid F-action whose loxodromic elements are precisely the elements that grow exponentially. This time, we need a more delicate idea than the ones highlighted in the proofs of Propositions 3.1 and 3.2: the blow-up of an expanding real pretree rel. an expanding forest.
Theorem 3.3. If $\phi \colon F \to F$ is an automorphism and $\mathcal G$ a $[\phi ]$ -invariant proper free factor system of F, then there is
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1. a minimal rigid F-action on a real pretree ${T}$ with trivial arc stabilizers;
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2. a $\phi $ -equivariant F-expanding pretree-automorphism $f\colon {T} \to {T}$ ; and
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3. an element in F is ${T}$ -loxodromic if and only if it grows exponentially rel. $\mathcal G$ .
The T-point stabilizers give a canonical $[\phi ]$ -invariant subgroup system $\mathcal H$ of finite type, and any restriction $\left .\phi \right |{}_{\mathcal H}$ is polynomially growing rel. $\mathcal G$ (if $\mathcal H$ is not empty). This system has strictly lower complexity than F if and only if $\phi $ is exponentially growing rel. $\mathcal G$ .
A (forward) limit pretree of a free group automorphism is a real pretree satisfying the theorem’s conclusion.
Proof (modulo a black box).
If $\phi $ is polynomially growing rel. $\mathcal G$ , then let ${T}$ be the degenerate real pretree (i.e., a singleton) with a trivial F-action; Conditions 1–3 hold automatically. For the rest of the proof, we assume that $\phi $ is exponentially growing rel. $\mathcal G$ .
Let $Y^{(1)}$ be a nondegenerate tree for $\phi $ given by Proposition 3.2. By Gaboriau–Levitt’s index inequality, $i(F \backslash Y^{(1)}) < c(F)$ , we can define $\mathcal V$ to be a finite set of representatives for the F-orbits of branch points v with nontrivial stabilizers $G_v$ . The index inequality implies the subgroup system $\mathcal G^{(2)} = \bigsqcup _{v \in \mathcal V} G_v$ has complexity $c(\mathcal G^{(2)}) < c(F)$ as the tree $Y^{(1)}$ is not degenerate and the F-action is minimal. The system is $[\phi ]$ -invariant and has a restriction automorphism $\left .\phi \right |{}_{\mathcal G^{(2)}}\colon \mathcal G^{(2)} \to \mathcal G^{(2)}$ that is unique up to post-composition with inner automorphisms of $\mathcal G^{(2)}$ -components – just as we showed for free splittings in Chapter 1. If $\mathcal G^{(2)}$ is not empty and the restriction $\left .\phi \right |{}_{\mathcal G^{(2)}}$ is exponentially growing rel. $\mathcal G$ , then we can repeatedly apply Proposition 3.2 to the restrictions.
The complexities of the point stabilizer systems are strictly descending, and in the end, we get a finite sequence of nondegenerate forests $\mathcal Y^{(i)}$ , minimal isometric $\mathcal G^{(i)}$ -actions with trivial arc stabilizers and $\left .\phi \right |{}_{\mathcal G^{(i)}}$ -equivariant expanding homotheties $h^{(i)} \colon \mathcal Y^{(i)} \to \mathcal Y^{(i)}$ . An element in F grows exponentially rel. $\mathcal G$ if and only if it is conjugate to a $\mathcal Y^{(i)}$ -loxodromic element in some system $\mathcal G^{(i)}$ . An almost identical argument is in the preliminaries of [Reference Levitt13].
Set with its canonical pretree structure and . So there is a minimal rigid F-action on a real pretree ${T}^{(i-1)}$ with trivial arc stabilizers (for some $i\ge 2$ ), a $\phi $ -equivariant F-expanding pretree-automorphism $f^{(i-1)}$ and the ${T}^{(i-1)}$ -point stabilizers are represented by $\mathcal G^{(i)}$ .
The novel construction in the paper (ideal stitching 4.4, black box) defines a unique real pretree ${T}^{(i)}$ that is an equivariant blow-up of $T^{(i-1)}$ with respect to $f^{(i-1)}$ and $h^{(i)}$ . In particular, there is
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1. a minimal rigid F-action on ${T}^{(i)}$ with trivial arc stabilizers;
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2. a $\phi $ -equivariant F-expanding pretree-automorphism $f^{(i)} \colon {T}^{(i)} \to {T}^{(i)}$ ; and
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3. $x \in F$ is ${T}^{(i)}$ -loxodromic if and only if it is ${\mathcal Y}^{(j)}$ -loxodromic for some $j \le i$ .
By induction, we may assume we have a minimal rigid F-action on a real pretree T with trivial arc stabilizers; a $\phi $ -equivariant F-expanding pretree-automorphism $f\colon T \to T$ ; and an element of F is T-loxodromic if and only if it is loxodromic in some forest $\mathcal Y^{(i)}$ . By construction of the sequence of forests, the last condition translates to the following: an element in F is ${T}$ -loxodromic if and only if it grows exponentially rel. $\mathcal G$ , as required.
Thus, the T-point stabilizers are represented by a canonical $[\phi ]$ -invariant subgroup system $\mathcal H$ . Since $\phi $ is exponentially growing rel. $\mathcal G$ , T-loxodromic elements exist and T is not degenerate. By the index inequality, we get $c(\mathcal H) < c(\mathcal F)$ .
We can now give the more natural characterization of polynomially growing elements:
Corollary 3.4. Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism, $\mathcal G$ a $[\psi ]$ -invariant proper free factor system of $\mathcal F$ , and $(\Lambda _*, \mathcal G)$ a free splitting of $\mathcal F$ . An element x in $\mathcal F$ grows polynomially rel. $\mathcal G$ if and only if the sequence $\left (\|\psi ^n(x)\|_{\Lambda *} \right )_{n \ge 1}$ is bounded by a polynomial in n.
Proof. The reverse direction is immediate: polynomials are subexponential. Suppose a nontrivial element $x \in \mathcal F$ grows polynomially rel. $\mathcal G$ . Then it has a conjugate in $\mathcal H$ , the canonical $[\phi ]$ -invariant subgroup system of finite type given by Theorem 3.3(3). As any restriction $\left .\phi \right |{}_{\mathcal H}$ is polynomially growing rel. $\mathcal G$ , we are done by Proposition 3.1 $(3{\Rightarrow }2)$ .
We also give a dendrological characterization of purely exponentially growing automorphisms. An automorphism $\phi \colon F \to F$ is atoroidal if there are no $[\phi ]$ -periodic conjugacy classes of nontrivial elements in F.
Corollary 3.5. Suppose $\phi \colon F \to F$ is a free group automorphism with a limit pretree T. Then the following are equivalent:
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1. $\phi $ is atoroidal;
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2. all nontrivial elements in F grow exponentially; and
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3. the F-action on T is free.
Proof. $(1 \implies 2)$ : Set . If F has a nontrivial element that grows polynomially, then the canonical $[\phi ]$ -invariant subgroup system $\mathcal H$ of finite type given by Theorem 3.3 is not empty and any restriction $\left .\phi \right |{}_{\mathcal H}$ is polynomially growing. By Proposition 3.1 $(3{\Rightarrow }1)$ , conjugacy classes of nontrivial elements in the lowest stratum of $\left .\phi \right |{}_{\mathcal H}$ are $[\phi ]$ -periodic.
$(2 \implies 1)$ : Any nontrivial element in a $[\phi ]$ -periodic conjugacy class has polynomial (in fact, ‘constant’) growth by definition.
$(2 \iff 3)$ : This is Theorem 3.3(3).
4. Blow-ups: simple, naïve, and ideal
We now define the equivariant blow-ups that were the key steps in the proofs of Proposition 3.2 and Theorem 3.3. Although we present the constructions as two separate ideas (simplicial and non-simplicial), the underlying principle is the same:
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1. start with an F-action on a tree (or real pretree) T and a $\mathcal G$ -action on a forest $\mathcal T_{\mathcal V}$ where $\mathcal G$ are conjugacy representatives of the nontrivial point stabilizers of T;
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2. then define a naïve equivariant blow-up of T with respect to $\mathcal T_{\mathcal V}$ ;
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3. there is a lot of freedom in the naïve construction, but most choices will not be useful;
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4. let $f\colon T \to T$ and $f_{\mathcal V}\colon \mathcal T_{\mathcal V} \to \mathcal T_{\mathcal V}$ be $\phi $ - and $\left .\phi \right |{}_{\mathcal G}$ -equivariant homeomorphisms, respectively;
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5. consider the equivariant copies of $\mathcal T_{\mathcal V}$ in some blow-up tree (or real pretree) $T^*$ ; then f induces a $\phi $ -equivariant function $f^*\colon T^* \to T^*$ whose restriction to the copies is $f_{\mathcal V}$ – in a way, $f^*$ is formed by stitching f and $f_{\mathcal V}$ together;
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6. for most blow-ups, $f^*$ will not be a homeomorphism; but if $f_{\mathcal V}$ is expanding, then a unique fixed point theorem produces the ‘ideal’ blow-up $T^*$ whose corresponding map $f^*$ is a homeomorphism.
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.
Theorem 4.1. Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism of a free group system $\mathcal F$ and $(\Gamma _*, \mathcal G)$ a free splitting of $\mathcal F$ . Assume there is
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1. a $\psi $ -equivariant simplicial automorphism $\tau \colon (\Gamma _*, \mathcal G) \to (\Gamma _*, \mathcal G)$ ,
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2. a minimal isometric $\mathcal G$ -action on a forest $\mathcal T_{\mathcal V}$ with trivial arc stabilizers; and
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3. a $\left .\psi \right |{}_{\mathcal G}$ -equivariant expanding $\lambda $ -homothety $h_{\mathcal V}\colon \mathcal T_{\mathcal V} \to \mathcal T_{\mathcal V}$ .
Then there is
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1. a minimal isometric $\mathcal F$ -action on a forest $\mathcal T$ with trivial arc stabilizers, an equivariant copy of $\mathcal T_{\mathcal V}$ , and
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2. a $\psi $ -equivariant expanding $\lambda $ -homothety $h\colon \mathcal T \to \mathcal T$ induced by $\tau $ and $h_{\mathcal V}$ .
In fact, the $\mathcal F$ -action on $\mathcal T$ decomposes as a graph of actions whose underlying simplicial action is the free splitting $(\Gamma _*, \mathcal G)$ , and the vertex actions are the given $\mathcal G$ -action on $\mathcal T_{\mathcal V}$ . The homothety h acts by $\tau $ on the underlying simplicial action and $h_{\mathcal V}$ on the vertex actions. Any pair $(\mathcal T', h')$ satisfying this conclusion admits an equivariant isometry $\mathcal T' \to \mathcal T$ that conjugates $h'$ to h.
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:
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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
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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 $f^*\colon {\mathcal T^*} \to {\mathcal T^*}$ :
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○ 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:
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○ 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.
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○ 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