# Limit pretrees for free group automorphisms: existence

Jean Pierre Mutanguha<EMAIL_ADDRESS>Web
address:https://mutanguha.com
Max Planck Institute for Mathematics, Bonn, Germany

###### Abstract

To any free group automorphism, we associate a real pretree with several nice
properties. First, it has a rigid/non-nesting action of the free group with
trivial arc stabilizers. Secondly, there is an expanding pretree-automorphism
of the real pretree that represents the free group automorphism. Finally and
crucially, the loxodromic elements are exactly those whose (conjugacy class)
length grows exponentially under iteration of the automorphism; thus, the
action on the real pretree is able to detect the growth type of an element.

This construction extends the theory of metric trees that has been used to
study free group automorphisms. The new idea is that one can equivariantly
blow up an isometric action on a real tree with respect to other real trees
and get a rigid action on a treelike structure known as a real pretree.
Topology plays no role in this construction as all the work is done in the
language of pretrees.

## Introduction

The study of free group outer automorphisms shares a lot with the theory of
mapping class groups of surfaces. The relevant dictionary replaces compacts
surfaces with finite graphs (i.e finite 1-dimensional CW-complexes) and
surface homeomorphisms with graph homotopy equivalences. While 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 [Thu88, Theorem
4] there is:

  1. 1.

a canonical (potentially empty) collection of essential simple closed curves
$\gamma$;

  2. 2.

a homeomorphism $g\colon S\to S$ isotopic to $f$ that leaves a regular
neighbourhood $N(\gamma)$ of the multicurve $\gamma$ invariant; and

  3. 3.

the restriction of $g$ to permuted components of the complement $S\setminus
N(\gamma)$ is either:

     * •

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

     * •

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 [BLM83, Theorem C] but that is not in the
standard expositions of Thurston’s result.

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 $S^{\prime}$ of pseudo-Anosov components has an invariant
measured singular foliation whose transverse measure is scaled by some stretch
factor $\lambda_{S^{\prime}}>1$. 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 [MS91, 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:

  * •

arc (pointwise) stabilizers are trivial; and

  * •

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 II.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 [Lev98], 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
II.2). With this new terminology, the free group analogue for the real pretree
statement becomes:

######  Main Theorem (Theorem III.3).

If $\phi\colon F\to F$ is an automorphism of a finitely generated free group,
then there is:

  1. 1.

a minimal rigid $F$-action on a real pretree $T$ with trivial arc stabilizers;

  2. 2.

a $\phi$-equivariant $F$-expanding pretree-automorphism $f\colon{T}\to{T}$;
and

  3. 3.

an element in $F$ is $T$-elliptic if and only if it grows polynomially with
respect to $[\phi]$.

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”.

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 [Mut22]. This is of much interest as it would be a canonical
representation of a free group automorphism. For instance, it would allow 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
III.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!

Acknowledgments: I would like to thank Ursula Hamenstädt whose suggestions
helped me clarify the blow-up construction. I am grateful to the Max Planck
Institute for Mathematics for their continued support.

###### Contents

  1. I Preliminaries
    1. I.1 Free splittings and topological representatives
    2. I.2 Free group systems and automorphisms
    3. I.3 Train track theory
  2. II Trees and real pretrees
    1. II.1 Trees and index theory
    2. II.2 Pretrees and rigid actions
  3. Interlude — same trees, different views
  4. III Limit pretrees: exponentially growing automorphisms
  5. IV Blow-ups: simple, naïve, and ideal
    1. IV.1 Simple patchwork
    2. IV.2 Naïve stitching
    3. IV.3 Ideal stitching
  6. Epilogue — are all limits the same?
  7. A Index theory for small rigid actions
  8. B Solutions to some exercises

##  I 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.

###  I.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
[Ser77]. 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 $x_{v}G_{w}x_{v}^{-1}$ and, if $G_{v}$ is not trivial,
then the restriction of $\phi$ to $G_{v}$ is an isomorphism $G_{v}\to
x_{v}G_{w}x_{v}^{-1}$. By post-composing with the inner automorphism
$\mathrm{inn}(x_{v}^{-1})\colon F\to F$ that maps $y\mapsto x_{v}^{-1}yx_{v}$,
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 $\phi$ 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
$\mathcal{G}$.

###  I.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
$\mathcal{G}^{\prime}$ if each $\mathcal{G}^{\prime}$-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
$\mathbb{Z}_{\geq-1}\cup\\{\infty\\}$. For a (possibly empty) free group
system $\mathcal{H}$,

$c(\mathcal{H})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{i\in\mathcal{I}}c(H_{i})\quad\in~{}\mathbb{Z}_{\geq
0}\cup\\{\infty\\}.$

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
$[\psi]$-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 $\psi$ 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
$f_{i}\colon T_{i}\to T_{\sigma\cdot i}$.

###  I.3 Train track theory

We conclude the chapter with some preliminary results from Bestvina–Handel’s
theory of train tracks [BH92, 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 $[e],[e^{\prime}]$ in the quotient graph
$\Gamma_{*}=\mathcal{F}\backslash\mathcal{T}$, there is an integer $n\geq 1$
for which $[f^{n}(e^{\prime})]$ contains $[e]$. To any topological
representative $f$, we can associate a nonnegative integer square matrix
$A(f)$ and $f$ 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 $\lambda(f)\geq 1$ 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 I.1 (cf. [BH92, 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
$\mathcal{G}$.

###### Proof outline.

Section 1 of [BH92] 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
a reduced edge-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 I.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 edge-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 two translates of $x_{1}\cdot
e_{i},x_{2}\cdot e_{i}$ in the oriented edge-path $\tau^{n}(e_{i})$ for some
large $n$. The $\mathcal{T}$-loxodromic element $x_{2}x_{1}^{-1}$ has a
$\tau$-legal axis in $\mathcal{T}$: the fundamental domain for the axis is the
$\tau$-legal subpath of $\tau^{n}(e_{i})$ joining the midpoints of the chosen
two translates of $e_{i}$. ∎

##  II Trees and real pretrees

Expanding irreducible train tracks are very useful for understanding the
dynamics of an automorphism since the iterates of edges all expanding 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 length (after reduction) remains 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 III.2).

###  II.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 $\lambda$-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 its 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

$\|\iota\|=\inf_{p\in T}d_{T}(p,\iota(p)).$

Any isometric $H$-action on a tree has a nonnegative function $\|\cdot\|\colon
H\to\mathbb{R}_{\geq 0}$ 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 $G_{p}\leq F$ 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:

$i[p]\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=c(G_{p})-1~{}+~{}\\#\mathrm{dir}[p]\quad\in~{}\mathbb{Z}_{\geq
0}\cup\\{\infty\\}.$

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

$i(F\backslash
T)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{[p]\,\in\,F\backslash
T}i[p].$

One of our main tools will be Gaboriau–Levitt’s index inequality [GL95,
Theorem III.2]:

$i(F\backslash T)<c(F).$

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.

###  II.2 Pretrees and rigid actions

Brian Bowditch’s paper [Bow99] 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
$[\cdot,\cdot]_{S}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=[\cdot,\cdot]\cap
S$.

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 [Bow99, 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 $T^{\prime}$ have pretree structures $[\cdot,\cdot]$ and
$[\cdot,\cdot]^{\prime}$ respectively. A set-bijection $f\colon T\to
T^{\prime}$ is a pretree-isomorphism if $[f(p),f(q)]^{\prime}=f([p,q])$ for
all $p,q\in T$. An injection $f\colon T\to T^{\prime}$ 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 of $f\colon T\to T$, 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 [Bow99, 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. 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 [Lev98]. 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.

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 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 minimal rigid $F$-action with
trivial arc stabilizers. 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 translate $f((p,q_{d^{\prime}}])$ for some orbit $G_{p}\cdot
d^{\prime}$ of directions at $p$. By the index inequality, there are finitely
many $G_{p}$-orbits of directions at $p$; so for some $n\geq 1$, each
$(p,q_{d}]$ properly embeds in a 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 $n\geq 1$, 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.

Figure 1: A tree with one “branch point” and (countably) infinitely many edges
attached.

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 [Ser77]
and it is the quickest way to define free splittings. As a simple tree, the
running example can be defined as: $V=\mathbb{Z}_{\geq 0}$ and
$E=\\{\,\\{0,n\\}~{}:~{}n\geq 1\,\\}$. 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 paranthetical definition I gave at the
start of Chapter I. 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 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 $\\{v\\}\sqcup\left(\mathbb{Z}_{\geq
1}\times[0,1]\right)$ with the equivalence relation generated by $v\sim(n,0)$
for all $n\geq 1$. 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 sapces, 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 $(x_{n})_{n\geq 1}$ converges to $\pi(v)$
in the CW/quotient topology, while the sets $\\{y_{n}\\}_{n\geq 1}$ and
$\\{z_{n}\\}_{n\geq 1}$ 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:

$d_{T}(\pi(n,a),\pi(m,b))=\begin{cases}|a-b|&\text{if }m=n\\\
a+b&\text{otherwise}.\end{cases}$

The metric topology on $T$ is strictly coarser than the CW-topology as it
contains fewer open sets. This time the sequences $(x_{n})_{n\geq 1}$ and
$(y_{n})_{n\geq 1}$ now both converges to $\pi(v)$ in the metric topology. On
the other hand, the set $\\{z_{n}\\}_{n\geq 1}$ 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 [Bow99] 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 $(x_{n})_{n\geq 1}$, $(y_{n})_{n\geq 1}$,
and $(z_{n})_{n\geq 1}$ 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

$d_{T}^{\prime}(\pi(m,a),\pi(n,b))=\begin{cases}\frac{1}{m}|a-b|&\text{if
}m=n\\\ \frac{a}{m}+\frac{b}{n}&\text{otherwise}.\end{cases}$

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; 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!

##  III 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 I.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 III.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
$\left(\sqrt[n]{\|\psi^{n}(x)\|_{\Lambda*}}\right)_{n\geq 1}$ 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:

  1. 1.

$\lambda\left(\tau^{(i)}\right)=1$ for all $i=1,\ldots,k$;

  2. 2.

for any $x\in\mathcal{F}$, the sequence
$\left(\|\psi^{n}(x)\|_{\Lambda*}\right)_{n\geq 1}$ is bounded by a polynomial
in $n$; and

  3. 3.

$\underset{n\to\infty}{\liminf}\sqrt[n]{\|\psi^{n}(x)\|_{\Lambda*}}\leq 1$ for
all $x\in\mathcal{F}$.

An automorphism is polynomially growing rel. $\mathcal{G}$ if these conditions
hold; otherwise, it is exponentially growing rel. $\mathcal{G}$.

###### 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
$\underline{\lambda_{x}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\underset{n\to\infty}{\liminf}\sqrt[n]{\|\psi^{n}(x)\|_{\Lambda*}}$
in $\mathbb{R}_{\geq 0}$. 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 $\underline{\lambda_{x}}\geq
1$ 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, i.e.

$\frac{1}{K}\|\cdot\|_{\Lambda_{*}}\leq\|\cdot\|_{\Omega_{*}}\leq
K\|\cdot\|_{\Lambda_{*}}\text{\,(pointwise) for some constant }K\geq 1.$

This implies $\underline{\lambda_{x}}$ is independent of the free splitting
$(\Lambda_{*},\mathcal{G})$ and concludes the first part.

$(\ref{cond - growth.tt}\implies\ref{cond - growth.seq})$: 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
$(\Gamma_{*}^{(k)},\mathcal{G})$ 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
$(\Gamma_{*}^{(i-1)},\mathcal{F}_{i+1})$ with respect to
$(\Omega_{*}^{(i)},\mathcal{G})$. As $\tau^{(i-1)}$ is a simplicial
automorphism, all “top-stratum” edges gain a predetermined prefix and suffix
in $(\Omega_{*}^{(i+1)},\mathcal{G})$ 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.

$(\ref{cond - growth.seq}\implies\ref{cond - growth.exprate})$: Polynomials
are sub-exponential.

$(\ref{cond - growth.exprate}\implies\ref{cond - growth.tt})$: 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 I.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,
$\|\cdot\|_{\Gamma_{*}^{(i)}}\leq\|\cdot\|_{\Omega_{*}^{(i)}}$ pointwise, and
$\|\cdot\|_{\Omega_{*}^{(i)}}$ is the restriction of
$\|\cdot\|_{\Omega_{*}^{(1)}}$. So $\underline{\lambda_{x}}\geq\lambda>1$. ∎

For any element $x$ in $\mathcal{F}$, define
$\underline{\lambda}(x;\psi;\mathcal{G})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\underset{n\to\infty}{\liminf}\sqrt[n]{\|\psi^{n}(x)\|_{\mathcal{G}}}$.
We say $x\in\mathcal{F}$ grows exponentially rel. $\mathcal{G}$ if
$\underline{\lambda}(x;\psi,\mathcal{G})>1$; otherwise, it grows polynomially
rel. $\mathcal{G}$. The “rel. $\mathcal{G}$” will be omitted when
$\mathcal{G}$ is empty. Suppose $\mathcal{H}$ is a $[\psi]$-invariant subgroup
system of finite typethat 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 III.1 converges (see [BH92, Remark 1.8]). Gilbert
Levitt gave a finer classification for an element’s growth rates [Lev09,
Theorem 6.2].

The next proposition appears in Levitt–Lustig’s paper [LL00, 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 III.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 $i\geq 1$; 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 I.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

$\|\cdot\|^{(k)}\colon\mathcal{F}_{k}\to\mathbb{R}_{\geq 0}\quad\text{by}\quad
x\mapsto\underset{n\to\infty}{\lim}\lambda^{-n}\|\psi^{n}(x)\|_{\Gamma_{*}^{(k)}}.$

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 [CM87, 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
[GLL98, 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
$\mathcal{F}_{k}$ [CM87, 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,
$k\geq 2$ and, for some $i\leq k$, 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 IV.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

$\|\cdot\|_{\mathcal{Y}}\leq K\|\cdot\|_{\Lambda_{*}}\text{\,pointwise for
some constant }K\geq 1.$

Therefore,
$\lambda^{n}\|\cdot\|_{\mathcal{Y}}=\|\psi^{n}(\cdot)\|_{\mathcal{Y}}\leq
K\|\psi^{n}(\cdot)\|_{\Lambda_{*}}$. 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 III.1
and III.2: the blow-up of an expanding real pretree rel. an expanding forest.

######  Theorem III.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:

  1. 1.

a minimal rigid $F$-action on a real pretree ${T}$ with trivial arc
stabilizers;

  2. 2.

a $\phi$-equivariant $F$-expanding pretree-automorphism $f\colon{T}\to{T}$;
and

  3. 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 III.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 I. 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 III.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 [Lev09].

Set
${T}^{(1)}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}={Y}^{(1)}$
with its canonical pretree structure and
$f^{(1)}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=h^{(1)}$. So
there is a minimal rigid $F$-action on a real pretree ${T}^{(i-1)}$ with
trivial arc stabilizers (for some $i\geq 2$), 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 IV.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:

  1. 1.

a minimal rigid $F$-action on ${T}^{(i)}$ with trivial arc stabilizers;

  2. 2.

a $\phi$-equivariant $F$-expanding pretree-automorphism
$f^{(i)}\colon{T}^{(i)}\to{T}^{(i)}$; and

  3. 3.

$x\in F$ is ${T}^{(i)}$-loxodromic if and only if it is
${\mathcal{Y}}^{(j)}$-loxodromic for some $j\leq i$.

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: 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 III.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 $\left(\|\psi^{n}(x)\|_{\Lambda*}\right)_{n\geq 1}$ 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 III.3, by Condition 3. As any
restriction $\left.\phi\right|_{\mathcal{H}}$ is polynomially growing rel.
$\mathcal{G}$, we are done by Proposition III.1$(\ref{cond -
growth.exprate}{\Rightarrow}\ref{cond - growth.seq})$. ∎

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 III.5.

Suppose $\phi\colon F\to F$ is a free group automorphism with a limit pretree
$T$. Then the following are equivalent:

  1. 1.

$\phi$ is atoroidal;

  2. 2.

all nontrivial elements in $F$ grow exponentially; and

  3. 3.

the $F$-action on $T$ is free.

###### Proof.

$(\ref{char - atoroidal}\implies\ref{char - purelyexp})$: Set
$\mathcal{G}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\emptyset$.
If $F$ has a nontrivial element that grows polynomially, then the canonical
$[\phi]$-invariant subgroup system $\mathcal{H}$ of finite type given by
Theorem III.3 is not empty and any restriction
$\left.\phi\right|_{\mathcal{H}}$ is polynomially growing. By Proposition
III.1$(\ref{cond - growth.exprate}{\Rightarrow}\ref{cond - growth.tt})$,
conjugacy classes of nontrivial elements in the lowest stratum of
$\left.\phi\right|_{\mathcal{H}}$ are $[\phi]$-periodic.

$(\ref{char - purelyexp}\implies\ref{char - atoroidal})$: Any nontrivial
element in a $[\phi]$-periodic conjugacy class has polynomial (in fact
“constant”) growth by definition.

$(\ref{char - purelyexp}\iff\ref{char - freeaction})$: This is Theorem
III.3(3). ∎

##  IV Blow-ups: simple, naïve, and ideal

We now define the equivariant blow-ups that were the key steps in the proofs
of Proposition III.2 and Theorem III.3. Although we present the constructions
as two separate ideas (simplicial and non-simplicial), the underlying
principle is the same:

  1. 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 $T$;

  2. 2.

then define a naïve equivariant blow-up of $T$ with respect to
$\mathcal{T}_{\mathcal{V}}$;

  3. 3.

there is a lot of freedom in the naïve construction but most choices will not
be useful;

  4. 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;

  5. 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 lifts is
$f_{\mathcal{V}}$ — in a way, $f^{*}$ is formed by stitching $f$ and
$f_{\mathcal{V}}$ together;

  6. 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.

###  IV.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 IV.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:

  1. 1.

a $\psi$-equivariant simplicial automorphism
$\tau\colon(\Gamma_{*},\mathcal{G})\to(\Gamma_{*},\mathcal{G})$,

  2. 2.

a minimal isometric $\mathcal{G}$-action on a forest
$\mathcal{T}_{\mathcal{V}}$ with trivial arc stabilizers; and

  3. 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:

  1. 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

  2. 2.

a unique $\psi$-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 $\mathcal{T}_{\mathcal{V}}$.

We state and prove the theorem in terms of homotheties due to the specific
needs in Proposition III.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
$\mathcal{V}^{*}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=F\times\overline{\mathcal{T}}_{\mathcal{V}}$;
$F$ 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 [Lev94, 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
$\tau\colon(\Gamma_{*},\mathcal{G})\to(\Gamma_{*},\mathcal{G})$ nor 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 unique $\psi$-equivariant PL-map
$f^{*}\colon{\mathcal{T}^{*}}\to{\mathcal{T}^{*}}$:

  * •

let $\nu\colon\mathcal{V}^{*}\to\mathcal{\iota}(\mathcal{V}^{*})$ be given by
$\nu(x,p)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\iota(\psi(x)x_{v},\bar{h}_{v}(p))$,
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:

  * •

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 unique 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:

$p_{d}=h_{v,d}(p_{\partial\cdot d})~{}\text{ for all
}v\in\mathcal{V},d\in\mathcal{D}_{v},\quad\text{where
}h_{v,d}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}={\bar{h}}_{v}^{\,-1}\circ
s_{\partial\cdot d}\colon T_{\beta\cdot v}\to T_{v}.$

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
$h\colon\mathcal{T}\to\mathcal{T}$. ∎

###  IV.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 $T$. If the
action is minimal, then $T$ must be short. Recall that the pretree completion
$\widehat{T}$ is itself real in this case. Given any subset
$P\subset\widehat{T}$, we define $T[P]$ to be the union in $\widehat{T}$ of
$T$ and the $G$-orbit of $P$. Generally, assume a countable group system
$\mathcal{G}=\bigsqcup_{i\in\mathcal{I}}G_{i}$ acts minimally on real pretrees
$\mathcal{T}_{\mathcal{I}}$. Then for any subset
$P\subset\widehat{\mathcal{T}}$, we can similarly define $\mathcal{T}[P]$ and
let $T_{i}[P]$ be the component of $\mathcal{T}[P]$ indexed by
$i\in\mathcal{I}$. We now make a simple but crucial observation:

######  Lemma IV.2.

Let $\mathcal{G}$ be a countable group system that admits a minimal rigid
action on real pretrees $\mathcal{T}$. If a subset
$P\subset\widehat{\mathcal{T}}$ contains no points fixed by a
$\mathcal{T}$-loxodromic element of $\mathcal{G}$, then the induced
$\mathcal{G}$-action on the real pretrees $\mathcal{T}[P]$ is rigid.

Additionally, if the rigid action on $\mathcal{T}$ has finite arc stabilizers,
then so does the rigid action on $\mathcal{T}[P]$.

###### Sketch of proof.

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

For a given countable group $H$ and rigid $H$-action on a real pretree $T$
with finite arc stabilizers, fix a set of orbit representatives
$\mathcal{V}\subset T$ for branch points with infinite stabilizers
$\mathcal{G}$. By countability of $H$ and finiteness of arc stabilizers, the
set $\mathcal{V}$ is countable. For each $v\in\mathcal{V}$, consider the
$G_{v}$-action on the directions at $v$ and fix a set of orbit representatives
$\mathcal{D}_{v}$ for these directions. For a direction $d$ at $v$, we let
$\mathrm{Stab}_{G_{v}}(d)$ denote the (setwise) stabilizer of $d$ in $G_{v}$.

######  Proposition IV.3.

Suppose a real pretree ${T}$ has a rigid $H$-action with finite arc
stabilizers, $\mathcal{G}$ represents the infinite ${T}$-point stabilizers,
and there is a minimal rigid $\mathcal{G}$-action on real pretrees
${\mathcal{T}}_{\mathcal{V}}$.

If $\vec{p}=(p_{d}\in\widehat{T}_{v}:v\in\mathcal{V},d\in\mathcal{D}_{v})$ is
a choice of points and each point $p_{d}$ is fixed by the direction stabilizer
$\mathrm{Stab}_{G_{v}}(d)$, then there is:

  1. 1.

an $H$-action on a real pretree $T^{*}=T^{*}(\vec{p}\,)$ by pretree-
automorphisms;

  2. 2.

equivariantly collapsing the copies of
${\mathcal{T}}_{\mathcal{V}}[\,\vec{p}\,]$ in $T^{*}$ recovers the real
pretree ${T}$;

If no attaching point $p_{d}$ in $\vec{p}$ is fixed by a
${\mathcal{T}}_{\mathcal{V}}$-loxodromic element in $\mathcal{G}$, then

  1. 3.

the $H$-action on $T^{*}$ is rigid;

  2. 4.

the characteristic convex subset of $T^{*}$ for $\mathcal{G}$ is an
equivariant copy of ${\mathcal{T}}_{\mathcal{V}}$; and

  3. 5.

an element in $H$ is $T^{*}$-loxodromic if and only if it is ${T}$\- or
${\mathcal{T}}_{\mathcal{V}}$-loxodromic.

Moreover, if the $\mathcal{G}$-action on $\mathcal{T}_{\mathcal{V}}$ has
finite arc stabilizers, then so does the $H$-action on $T^{*}$. If the rigid
$H$-action on $T$ is minimal, then so is the rigid $H$-action on $T^{*}$.

For the sake of compartmentalizing the proposition’s long proof, we break it
into two parts.

###### Proof of real pretree’s existence.

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

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

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

given by these identifications:

  * •

for each $v\in\mathcal{V}$, $p\in T_{v}[\,\vec{p}\,]$, $s\in G_{v}$, and $x\in
H$, identify $(xs,p)$ with $(x,s\cdot p)$.

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

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

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

$[p^{*},q^{*}]^{*}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\iota(y,[p,q]_{v}).$

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

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

Case 2: $\pi(p^{*})=p,\pi(q^{*})=q\notin H\cdot\mathcal{V}$. Consider the
closed interval $[p,q]\subset T$. Then

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

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

Define

$[p^{*},q^{*}]^{*}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\iota{\left([p,q]\setminus(H\cdot\mathcal{V})~{}\cup\bigcup_{v\in\mathcal{V},k\geq
1}\\{y_{v,k}\\}\times[s_{v,k_{-}}\cdot p_{d(v,k_{-})},\,s_{v,k_{+}}\cdot
p_{d(v,k_{+})}]_{v}\right)}.$

Again, $[p^{*},q^{*}]^{*}$ is independent of the choice of $y_{v,k}$. In this
case (and the rest), we are equivariantly “attaching” directions $d$ to points
$p_{d}$ to get the pretree $T^{*}(\vec{p}\,)$.

Case 3: $\pi(p^{*})=p\notin H\cdot\mathcal{V},\pi(q^{*})\in
H\cdot\mathcal{V}$. Choose $y_{\omega}\in H$ so that
$\pi(q^{*})=y_{\omega}\cdot w$ with $w\in\mathcal{V}$. Then
$q^{*}=\iota(y_{\omega},q)$ for some $q\in T_{w}$. Consider the closed
interval $[p,y_{\omega}\cdot w]\subset T$. Then

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

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

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

$[p,y_{\omega}\cdot
w]\setminus(H\cdot\mathcal{V})~{}\cup\left(\\{y_{\omega}\\}\times[s_{\omega}\cdot
p_{d(\omega)},\,q]_{w}\right)\cup\bigcup_{\begin{subarray}{c}v\in\mathcal{V},k\geq
1\\\ y_{v,k}\cdot v\neq y_{\omega}\cdot
w\end{subarray}}\\{y_{v,k}\\}\times[s_{v,k_{-}}\cdot
p_{d(v,k_{-})},\,s_{v,k_{+}}\cdot p_{d(v,k_{+})}]_{v}.$

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

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

  1. 1.

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

  2. 2.

$H$ acts on $T^{*}$ by pretree-automorphisms, i.e. for all $x\in H$ and
$p^{*},q^{*}\in T^{*}$, we have $[x\cdot p^{*},x\cdot
q^{*}]^{*}=x\cdot[p^{*},q^{*}]^{*}$ (exercise).

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

######  Remark.

Although we invoked the axiom of (countable) choice to conclude that the blow-
up $T^{*}$ is real, this can be avoided in the setting we are most intersted
in: $H=F$ and the action is minimal. In this case, Gaboriau–Lustig’s index
theory implies there are only finitely many orbits of branch points and hence
only finitely many choice to be made.

###### Proof that conditions are satisfied….

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

• Condition 3, rigidity of $H$-action on $T^{*}$:

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

First, we show that the fixed-point set $\mathrm{Fix}_{T^{*}}(x)$ is convex.
By the equivariance of $\pi$ and rigidity of the $H$-action on $T$,
$\mathrm{Fix}_{T}(x)$ is nonempty and convex. As $x$ acts as a pretree-
automorphism, it must also fix the limit points of
$\iota(\mathrm{Fix}_{T}(x)\setminus\left(H\cdot\mathcal{V}\right))$ in
$\iota(H\times\mathcal{T}_{\mathcal{V}}[\,\vec{p}\,])$. By Lemma IV.2, the
$\mathcal{G}$-action on $\mathcal{T}_{\mathcal{V}}[\,\vec{p}\,]$ is rigid and
so $\mathrm{Fix}_{T^{*}}(x)$ is convex: this used the assumption $\vec{p}$ was
a choice of points not fixed by $\mathcal{T}_{\mathcal{V}}$-loxodromic
elements in $\mathcal{G}$. Thus each $T^{*}$-direction $d$ at
$\mathrm{Fix}_{T^{*}}(x)$ has a unique attaching point $p^{*}$ in
$\mathrm{Fix}_{T^{*}}(x)$.

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

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

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

• Condition 5, characterizing $T^{*}$-loxodromics — or equivalently,
$T^{*}$-elliptics:

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

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

• Finiteness of arc stabilizers:

Assume the rigid $\mathcal{G}$-action on $\mathcal{T}_{\mathcal{V}}$ has
finite arc stabilizers. Let $A^{*}\subset T^{*}$ be an arc, then
$A\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\pi(A^{*})\subset T$
is either a singleton or an arc. If $A$ is an arc, then, by the equivariance
of $\pi$ and rigidity of actions (Condition 3), the stabilizer of $A^{*}$ is
the stabilizer of $A$; so it is finite. If $A=\\{p\\}$ is a singleton, then
$p=y\cdot v$ for some $y\in H$ and $v\in\mathcal{V}$. Thus
$A^{*}=\iota(y,A_{v})$ for some arc $A_{v}\subset T_{v}[\,\vec{p}\,]$ and its
stabilizer in $H$ is conjugate to the stabilizer of $A_{v}$ in $G_{v}$;
therefore, it is finite by Lemma IV.2.

• Minimality:

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

###  IV.3 Ideal stitching

The crucial step in proving Theorem IV.1 was using the contraction mapping
theorem to find the right blow-up that induced a homothety. We use the same
idea for the next theorem, which is the heart of the paper.

For a minimal rigid $F$-action on a real pretree with trivial arc stabilizers,
let $\mathcal{V}$ be a set of orbit representatives of branch points with
nontrivial stabilizers $G_{v}$. The point stabilizer system
$\mathcal{G}=\bigsqcup_{v\in\mathcal{V}}G_{v}$ has finite type by
Gaboriau–Levitt’s index inequality.

######  Theorem IV.4.

Suppose a real pretree $T$ has a minimal rigid $F$-action with trivial arc
stabilizers, and let $\phi\colon F\to F$ be an automorphism.

Assume $f\colon T\to T$ is a $\phi$-equivariant pretree-automorphism and
$\mathcal{G}$ represents the nontrivial $T$-point stabilizers with:

  1. 1.

a minimal isometric $\mathcal{G}$-action on a forest
${\mathcal{Y}}_{\mathcal{V}}$ with trivial arc stabilizers; and

  2. 2.

a $\left.\phi\right|_{\mathcal{G}}$-equivariant expanding homothety
$h_{\mathcal{V}}\colon{\mathcal{Y}}_{\mathcal{V}}\to{\mathcal{Y}}_{\mathcal{V}}$.

Then there is:

  1. 1.

a minimal rigid $F$-action on a unique real pretree $T^{*}$ with trivial arc
stabilizers, where

  2. 2.

the characteristic convex subset of $T^{*}$ for $\mathcal{G}$ is an
equivariant copy of ${\mathcal{Y}}_{\mathcal{V}}$;

  3. 3.

an element in $F$ is $T^{*}$-loxodromic if and only if it is $T$\- or
${\mathcal{Y}}_{\mathcal{V}}$-loxodromic; and

  4. 4.

$f^{*}\colon T^{*}\to T^{*}$ is the unique $\phi$-equivariant pretree-
automorphism induced by $f$ and $h_{\mathcal{V}}$.

Moreover, if $f$ is $F$-expanding, then so is $f^{*}$.

###### Proof.

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

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

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

  * •

for $p\in{T}\setminus(F\cdot\mathcal{V})$, define
$f^{*}(\iota(p))\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\iota(f(p))$;
and

  * •

for $(x,p)\in F\times{\mathcal{Y}}_{\mathcal{V}}[\,\vec{p}\,]$, define
$f^{*}(\iota(x,p))\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\iota(\psi(x)x_{v},\hat{h}_{v}(p))$,
where the implicit choice $(x_{v}\in F:v\in\mathcal{V})$ was used to define
the restriction $\left.\phi\right|_{\mathcal{G}}$, and hence $h_{\mathcal{V}}$
and its extension
$\hat{h}_{\mathcal{V}}\colon\mathcal{Y}_{\mathcal{V}}[\,\vec{p}\,]\to\mathcal{Y}_{\mathcal{V}}[\,\vec{p}\,]$.

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

As $\bigcup_{v\in\mathcal{V}}\mathcal{D}_{v}$ is finite and $h_{\mathcal{V}}$
is an expanding homothety, the system of equations has a unique solution
$\vec{p}=(p_{d}\in\overline{Y}_{v}:v\in\mathcal{V},d\in\mathcal{D}_{v})$ in
the metric completion of $\mathcal{Y}_{\mathcal{V}}$; these points are not
fixed by any ${\mathcal{Y}}_{\mathcal{V}}$-loxodromic element in
$\mathcal{G}$. Let $\mathcal{T}^{*}=\mathcal{T}^{*}(\vec{p}\,)$ be the unique
real pretree given by this solution. Then the unique induced function $f^{*}$
is a $\psi$-equivariant pretree-automorphism, as required for Condition 4.

Finally, assume $f$ is $F$-expanding. For all $x\in F$ and $n\geq 1$, we need
to show the composition
$\gamma^{*}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=x\circ(f^{*})^{n}$
is either: 1) elliptic with a unique fixed point and it expands at this fixed
point; or 2) loxodromic with an axis that is not shared with any
$T^{*}$-loxodromic element in $F$.

Let $\gamma\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=x\circ
f^{n}\colon T\to T$, then we can use the $F$-expanding assumption on $f$.

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

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

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

$s_{v}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=y^{-1}x\psi^{n}(y)\psi^{n-1}(x_{v})\cdots
x_{\beta^{n-1}\cdot v}\text{ is in }G_{v},$

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

Let
$\gamma_{v}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=s_{v}\circ
h_{\beta^{n-1}\cdot v}\circ\cdots\circ h_{v}$. As $h_{\mathcal{V}}$ is an
expanding homothety, the extension of the composition $\gamma_{v}$ to the
metric completion has a unique fixed point $q_{v}\in\overline{Y}_{v}$ that is
repelling. If $q_{v}\in Y_{v}[\,\vec{p}\,]$, then $\gamma^{*}$ is elliptic
with a unique fixed point $\iota(y,q_{v})$ and it expands at $\iota(y,q_{v})$.
Otherwise, $q_{v}\notin Y_{v}[\,\vec{p}\,]$ and $\gamma^{*}$ is loxodromic. As
$\gamma$ is elliptic, $\gamma^{*}$ does not share its axis (in $T^{*}$) with
any $T$-loxodromic element in $F$. As $\iota(y,q_{v})$ is an end of the axis
for $\gamma^{*}$ and $q_{v}$ is in the metric completion $\overline{Y}_{v}$,
$\gamma^{*}$ cannot share its axis with a conjugate (in $F$) of a
$Y_{v}$-loxodromic element of $G_{v}$. By Condition 3, $\gamma^{*}$ does not
share an axis with any $T^{*}$-loxodromic element in $F$ and we are done. ∎

## Epilogue — are all limits the same?

As a preview for the sequel [Mut22], we end the paper discussing whether the
real pretree of Theorem III.3 is canonical. There are two ways “canonical” can
be interpreted:

  1. 1.

the real pretree produced by the proof of the theorem does not depend on any
choices made in the proof — in a sense, the real pretree was predetermined;

  2. 2.

limit pretrees (i.e. pretrees satisfying the conclusion of the theorem) are
equivariantly pretree-isomorphic.

A priori, the second requirement seems stronger as, presumably, there might be
limit pretrees that do not arise from the blow-up construction of the theorem.
While the jury is still out on the second interpretation, we prove the first
one to be true in the sequel!

A careful rereading of Theorem III.3’s proof reveals that Theorem I.1 is the
only source of indeterminacy: both blow-up constructions in Theorems IV.1 and
IV.4 use uniquely determined attaching points! In other words, once the
hierarchy of expanding forests (produced by Proposition III.2) is fixed, the
real pretree given at the end of the proof is already determined.
Unfortunately, the irreducible train tracks (produced by Theorem I.1) used in
Proposition III.2 do depend on choices; in general, there are no “canonical”
train tracks and our main motivation was to address this issue.

To this end, we also suspect that Theorem III.3 can be strengthened by
requiring the real pretree’s completion to admit an equivariant surjection
from the Cantor set boundary of the free group $\partial F$ that is continuous
with respect to the observers’ topology. First and foremost, this would
immediately allow us to upgrade the rigid action on the real pretree to a
convergence action on the completion. Secondly, this would imply the
completion of the limit pretree is dual to certain (relative) laminations in
the free group (rel. the maximal elliptic subgroup system). Using the dynamics
of the automorphism acting on $\partial F$, we might be able to show that
these laminations, and hence their dual pretrees, are unique. Alternatively,
Bestvina–Feighn–Handel defined topmost attracting laminations [BFH00, Section
6] that could perhaps be used to prove a canonical version of Proposition
III.2, thereby making the limit pretrees of Theorem III.3 canonical. This
alternative is the approach taken in the sequel.

Fortunately, there is already precedent for these approaches. When the free
group outer automorphism is irreducible with infinite order,
Bestvina–Feighn–Handel proved that a limit tree produced by the proof of
Proposition III.2 is unique up to equivariant homothety [BFH97, Lemma 3.4]. In
particular, the limit tree is independent of the chosen train track used to
define it. In fact, the limit trees are canonical in a stronger sense: all
trees satisfying the conclusion of Proposition III.2 are equivariantly
homothetic. Importantly, they do not assume beforehand that these trees are
projective limits of iterating a topological representative. Later,
Levitt–Lustig proved the corresponding observers’ compactifications of these
trees admit canonical equivariant quotient maps from the boundary $\partial F$
[LL03].

For an irreducible atoroidal automorphism, Kapovich–Lustig [KL15, Proposition
4.5] recently showed that the dynamical lamination dual to the limit tree is
exactly the geometric lamination defined by Mahan Mj (formerly Mitra) [Mit97,
p. 388] using the hyperbolic geometry of the corresponding mapping torus
$F\rtimes\mathbb{Z}$.

When we drop irreducibility but keep the atoroidal assumption, there still are
promising results. Following works of Mj and Bowditch, Elizabeth Field has
recently used the hyperbolicity of the mapping torus to construct a canonical
dendrite that is dual to Mj’s geometric lamination [Fie20]. We note that the
index of the dendrite’s interior is an invariant of the atoroidal outer
automorphism. It might follow from the construction that Field’s dendrite is a
completion of a limit pretree. On the other hand, Uyanik has proven that
atoroidal automorphisms act with generalized north-south dynamics on the
projective space of currents [Uya19, Theorem 1.4]. As an extension of
Kapovich–Lustig’s result, the geometric lamination should be the support of
the repelling simplex of projective currents!

Finally, when we drop the atoroidal assumption, it seems reasonable to
conjecture that there are generalizations of Field’s and Uyanik’s results that
use relative hyperbolicity and relative currents respectively, and these can
be unified.

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##  Appendix A Index theory for small rigid actions

In this appendix, we give Gaboriau–Levitt’s proof of the index inequality.
Although the inequality was originally stated for isometric actions, their
proof only used the rigidity of actions — the metric plays no other important
role. Thus, we sketch this general version of the index theory.

A rigid $F$-action on a real pretree $T$ is small if arc stabilizers are
cyclic. Fix a minimal small rigid $F$-action on a real pretree $T$. For each
orbit of points $[p]\in F\backslash T$, let $G_{p}\leq F$ be the stabilizer
for $p\in T$ and $\\#\mathrm{dir}_{1}[p]$ denote the number of $G_{p}$-orbits
of directions at $p$ with trivial stabilizers. Recall that the complexity of
$G_{p}$ is $c(G_{p})=2\cdot\mathrm{rank}(G_{p})-1$. The index at $[p]$ is:

$i[p]\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=c(G_{p})-1+\\#\mathrm{dir}_{1}[p]\quad\in~{}\mathbb{Z}_{\geq
0}\cup\\{\infty\\}.$

Nonnegativity follows from minimality of the action ($F$ is not trivial). The
index is:

$i(F\backslash
T)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{[p]\in
F\backslash T}i[p]\quad\in~{}\mathbb{Z}_{\geq 0}\cup\\{\infty\\}.$

Gaboriau–Levitt proved the following:

$i(F\backslash T)<c(F).$

### • The real pretree associated to a rigid system

A real pretree is finite if it is the convex hull of a finite subset. A
function $j\colon T\to T^{\prime}$ of real pretrees is a morphism if every
closed $T$-interval is a finite union of closed subintervals that are pretree-
embedded into $T^{\prime}$ by $j$.

Let $K$ be a finite real pretree and $x_{i}\colon A_{i}\to B_{i}~{}(1\leq
i\leq n)$ be pretree-isomorphisms of finite real pretrees in $K$.
Alternatively, $x_{i}\colon K\to K$ are partial pretree-automorphisms whose
domains are convex and complete. Denote this system by $\mathcal{K}$. A system
$\mathcal{K}=(K,\\{x_{i}\\})$ is nontrivial if no domain
$A_{i}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{dom}(x_{i})$
is empty.

Let $F_{\mathcal{K}}$ be free group generated by the set
$\\{x_{i}\\}_{i=1}^{n}$. Note that any element $x\in F_{\mathcal{K}}$ (i.e.
reduced word in $x_{i}^{\pm 1}$) determines a partial pretree-automorphism
$x\colon K\to K$ with a possibly empty domain. The length of $x\in
F_{\mathcal{K}}$ as a reduced word in $x_{i}^{\pm}$ is denoted
$|x|_{\mathcal{K}}$.

######  Theorem A.1 (cf. [GL95, Theorem I.1]).

For a nontrivial system $\mathcal{K}=(K,\\{x_{i}\\})$, there is:

  1. 1.

an $F_{\mathcal{K}}$-action on a real pretree $T_{\mathcal{K}}$ by pretree-
automorphisms;

  2. 2.

a pretree-embedding $\iota\colon K\to T_{\mathcal{K}}$ with
$x_{i}\cdot\iota(a)=\iota(x_{i}(a))$ for $a\in A_{i}~{}(1\leq i\leq n)$; and

  3. 3.

the $F_{\mathcal{K}}$-orbit of $\iota(K)$ covers $T_{\mathcal{K}}$, i.e.
$T_{\mathcal{K}}=F_{\mathcal{K}}\cdot\iota(K)$.

If another pretree $F_{\mathcal{K}}$-action on a real pretree $T^{\prime}$
satisfies Condition 2 with $\iota^{\prime}\colon K\to T^{\prime}$, then there
is a unique equivariant morphism $j\colon T_{\mathcal{K}}\to T^{\prime}$ with
$j(\iota(a))=\iota^{\prime}(a)$ for $a\in K$.

The last part states that the pair of the pretree $T_{\mathcal{K}}$ and
$F_{\mathcal{K}}$-action is universal with respect to Condition 2. The real
pretree $T_{\mathcal{K}}$ (along with its $F_{\mathcal{K}}$-action and
pretree-embedding $\iota\colon K\to T_{\mathcal{K}}$) will be referred to as
the system’s associated real pretree.

###### Sketch of proof.

The set $T_{\mathcal{K}}$ is defined through the quotient

$\pi\colon F_{\mathcal{K}}\times K\to T_{\mathcal{K}},$

given by the identifications:

  * •

for each $a\in A_{i}$ and $x\in F_{\mathcal{K}}$, $(xx_{i},a)$ is identified
with $(x,x_{i}(a))$.

$T_{\mathcal{K}}$ inherits a set $F_{\mathcal{K}}$-action from the left-
multiplication on the first factor of $F_{\mathcal{K}}\times K$. We need to
define a pretree structure on $T_{\mathcal{K}}$.

Let the length of $(x,a)\in F_{\mathcal{K}}\times K$ be $|x|_{\mathcal{K}}$.
Every point $p\in T_{\mathcal{K}}$ has a unique shortest element
$\delta(p)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(x,a)\in\pi^{-1}(p)$.
Suppose $p,q\in T_{\mathcal{K}}$ have representatives $\delta(p)=(x,a)$ and
$\delta(q)=(y,b)$.

Case 1: $x=y$. Define
$[p,q]\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\pi(x,[a,b]_{K})$.

Case 2: $|y^{-1}x|_{\mathcal{K}}=m\geq 1$. Let $y^{-1}x\in F_{\mathcal{K}}$ be
the reduced word $y_{m}\cdots y_{1}$ in $x_{i}^{\pm}$. Set $c_{1}$ to be the
projection of $a$ to the domain $\mathrm{dom}(y_{1})\neq\emptyset$. Assume
$c_{j}\in\mathrm{dom}(y_{j})$ for $1\leq j<m$, and set $c_{j+1}$ to be the
projection of $y_{j}(c_{j})$ to $\mathrm{dom}(y_{j+1})\neq\emptyset$. Define

$[p,q]\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\pi(x,[a,c_{1}]_{K})\cup\left(\bigcup_{j=1}^{m-1}\pi(xy_{1}^{-1}\cdots
y_{j}^{-1},[y_{j}(c_{j}),c_{j+1}]_{K})\right)\cup\pi(y,[y_{m}(c_{m}),b]_{K}).$

We leave it as an exercise to check that $(T_{\mathcal{K}},[\cdot,\cdot])$ is
a real pretree and $F_{\mathcal{K}}$ acts on $T_{\mathcal{K}}$ by pretree-
automorphisms. This concludes Theorem A.1(1).

Let $1\in F_{\mathcal{K}}$ be the identity element. The map $\iota\colon K\to
T_{\mathcal{K}}$ given by $a\mapsto\pi(1,a)$ is a pretree-embedding (see Lemma
A.2 below). If $a\in A_{i}$, then by construction

$x_{i}\cdot\iota(a)=\pi(x_{i},a)=\pi(1,x_{i}(a))=\iota(x_{i}(a)).$

Also by construction, $F_{\mathcal{K}}\cdot\iota(K)=\pi(F_{\mathcal{K}}\times
K)=T_{\mathcal{K}}$. This gives us Theorem A.1(2-3).

Finally, we also leave the proof of the universal property as an exercise: use
the fact that any closed interval of $T_{\mathcal{K}}$ decomposes (by
construction) into finitely many closed subintervals, each contained in a
translate of $\iota(K)$. ∎

######  Lemma A.2 (cf. [GL95, Proposition I.4]).

Let $\mathcal{K}$ be a nontrivial system and $T_{\mathcal{K}}$ be the
associated real pretree. For $y\in F_{\mathcal{K}}$ and $a,b\in K$,

$y\cdot\iota(a)=\iota(b)\text{ if and only if }y(a)=b.$

###### Sketch of proof.

The identifications on $F_{\mathcal{K}}\times K$ used to define
$T_{\mathcal{K}}$ are symmetric but not transitive. Since the reduced word in
$x_{i}^{\pm}$ representing an element of $F_{\mathcal{K}}$ is unique, the
identifications generate the following equivalence relation: for $a,b\in K$
and $x,y\in F_{\mathcal{K}}$, $(xy,a)\sim(x,b)$ if and only if $y(a)=b$. ∎

######  Corollary A.3 (cf. [GL95, Proposition I.5]).

Let $\mathcal{K}$ be a nontrivial system and $T_{\mathcal{K}}$ be the
associated real pretree. For any nontrivial $y\in F_{\mathcal{K}}$ and any
morphism $j\colon T_{\mathcal{K}}\to T^{\prime}$ as in the last part of
Theorem A.1, the restriction of $j$ to the fixed-point set
$\mathrm{Fix}_{T_{\mathcal{K}}}(y)$ is injective.

###### Sketch of proof.

As $j$ is equivariant, $\iota^{\prime}$ is injective, and
$j(\iota(a))=\iota^{\prime}(a)$ for $a\in K$, it is enough to show that
$\mathrm{Fix}_{T_{\mathcal{K}}}(y)$ is contained in some translate of
$\iota(K)$. Without loss of generality, we assume $y\in F_{\mathcal{K}}$ is a
cyclically reduced nonempty word in $x_{i}^{\pm}$ and show
$\mathrm{Fix}_{T_{\mathcal{K}}}(y)$ is contained in $\iota(K)$ — or rather, we
prove the contrapositive.

Let $p\in T_{\mathcal{K}}$ be fixed by a nontrivial $y\in F_{\mathcal{K}}$ and
assume $p\notin\iota(K)$, i.e. $\delta(p)=(x,a)$ has positive length. Since
$(yx,a)$ and $(x,a)$ represent $p$, we get $x^{-1}yx(a)=a$ by Lemma A.2. Let
$z\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=x^{-1}yx\in
F_{\mathcal{K}}$ be the reduced nonempty word $z_{l}\cdots z_{1}$ in
$x_{i}^{\pm}$. Then $(xz_{1}^{-1},z_{1}(a))$ and $(xz_{l},z_{l}^{-1}(a))$ also
represent $p$. By minimality of $\delta(p)=(x,a)$, the element $x\in
F_{\mathcal{K}}$ as a reduced word in $x_{i}^{\pm}$ cannot end with $z_{1}$
nor $z_{l}^{-1}$. So $y=xzx^{-1}$ as a reduced word in $x_{i}^{\pm}$ is not
cyclically reduced. ∎

A partial pretree-automorphism is rigid if either its fixed-point set is empty
or it fixes no direction at its fixed-point set. A nontrivial system
$\mathcal{K}$ is rigid if every element of $F_{\mathcal{K}}$ determines a
rigid partial pretree-automorphism of $K$.

######  Lemma A.4.

If $\mathcal{K}$ is a rigid system, then the pretree $F_{\mathcal{K}}$-action
on the associated real pretree $T_{\mathcal{K}}$ is rigid.

###### Proof.

Suppose $x\in F_{\mathcal{K}}$ has fixed points in $T_{\mathcal{K}}$ and let
$d$ be an arbitrary $T_{\mathcal{K}}$-direction at
$\mathrm{Fix}_{T_{\mathcal{K}}}(x)$. Let
$p_{d}\in\mathrm{Fix}_{T_{\mathcal{K}}}(x)$ be an attaching point for $d$. By
construction, some nondegenerate interval $[p_{d},q]\subset\\{p_{d}\\}\cup d$
is contained in a translate $y\cdot\iota(K)$ for some $y\in F_{\mathcal{K}}$.
Let $p_{d}=y\cdot\iota(a_{d})$ and $q=y\cdot\iota(b)$ for some $a_{d},b\in K$.

For any $p^{\prime}=y\cdot\iota(a^{\prime})\in[p_{d},q]$, Lemma A.2 implies
$x\cdot p^{\prime}=p^{\prime}$ if and only if
$y^{-1}xy(a^{\prime})=a^{\prime}$. In particular,
$a_{d}\in\mathrm{Fix}_{K}(y^{-1}xy)$. Since no point in $d$, and hence
$(p_{d},q]=y\cdot\iota((a_{d},b]_{K})$, is fixed by $x$, no point in
$(a_{d},b]_{K}$ is fixed by the partial pretree-automorphism $y^{-1}xy$. So
$(a_{d},b]_{K}$ determines a $K$-direction $d_{K}$ at
$\mathrm{Fix}_{K}(y^{-1}xy)$.

By rigidity of the system, $y^{-1}xy$ (partially applied to $K$) does not fix
$d_{K}$; therefore, the element $x\in F_{\mathcal{K}}$ (acting on
$T_{\mathcal{K}}$) does not fix $d$ (Lemma A.2) and we are done. ∎

Assume the system $\mathcal{K}$ is rigid and $T_{\mathcal{K}}$ is the
associated real pretree. Let $S\subset K$ be the finite subset of vertices
(i.e. branch points or endpoints) in $K$, $A_{i}$, and $B_{i}~{}(1\leq i\leq
n)$.

######  Lemma A.5 (cf. [GL95, Proposition I.8]).

If $p\in T_{\mathcal{K}}$ is a branch point, then the $F_{\mathcal{K}}$-orbit
of $p$ contains a point of $\iota(S)$ and there are finitely many
$G_{p}$-orbits of directions at $p$.

###### Sketch of proof.

Suppose $[p,q]\cap\iota(K)=\\{p\\}$ and let $p=\iota(a)$ with $a\in K$. Let
$y\in F_{\mathcal{K}}$ be the shortest reduced word $y_{m}\cdots y_{1}$ in
$x_{i}^{\pm}$ whose translate $y\cdot\iota(K)$ covers a nondegenerate
subinterval $[p,p^{\prime}]$. So
$[p,p^{\prime}]=y\cdot\iota([b,b^{\prime}]_{K})$ for some $b,b^{\prime}\in K$
and $y(b)=a$ (Lemma A.2). Minimality of $y$ and
$[p,p^{\prime}]\cap\iota(K)=\\{p\\}$ imply
$[b,b^{\prime}]_{K}\cap\mathrm{dom}(y_{1})=\\{b\\}$. Then $b\in S$.

Deduce that the $F_{\mathcal{K}}$-orbit of any branch point in
$T_{\mathcal{K}}$ intersects $\iota(S)$. Similarly, the
$F_{\mathcal{K}}$-orbit of any direction at a branch point contains an “end”
of $\iota(K\setminus S)$; therefore, the number of $G_{p}$-orbits of
directions at $p$ is at most the number of ends of
$\iota(K)\setminus\left(\iota(S)\cap(F_{\mathcal{K}}\cdot p)\right)$. ∎

### • Geometric actions

Fix a basis $x_{1},\ldots,x_{n}$ for the free group $F$. Let $T$ be a real
pretree with a rigid $F$-action. For any finite real pretree $K\subset T$, we
get a system $\mathcal{K}=(K,\\{x_{i}\\})$ by letting $x_{i}\in F$ partially
act on $K$ as the partial pretree-automorphism given by restricting the action
of $x_{i}$ on $T$ to the domain $\left(x_{i}^{-1}\cdot K\right)\cap K$. The
free groups $F_{\mathcal{K}}$ and $F$ will be identified as they have the
“same” basis $\\{x_{i}\\}$.

Choose a finite real pretree $K\subset T$ so that the system $\mathcal{K}$ is
nontrivial, in which case it will be a rigid system. By Theorem A.1 and Lemma
A.4, the rigid system $\mathcal{K}$ has an associated real pretree, denoted
$T_{K}$, with a rigid $F$-action. By Lemma A.2, the rigid $F$-action on
$T_{K}$ is small if the rigid $F$-action on $T$ was small.

Now suppose the $F$-action on (a nondegenerate) $T$ is minimal and pick a
basepoint $p_{0}\in T$. By minimality of the rigid $F$-action on $T$, the
point $p_{0}$ is in the $T$-axis for some loxodromic element $x\in F$. For any
integer $m$, define $K_{m}$ to be the convex hull of the partial orbit
$\\{y\cdot p_{0}:|y|\leq m\\}$ and let $T_{m}$ be the real pretree associated
to $(K_{m},\\{x_{i}\\})$. Choose any $m\geq|x|$, then the interval
$[x^{-1}\cdot p_{0},p_{0}]\subset K_{m}$ is contained in the domain of
$x\colon K_{m}\to K_{m}$ (as the composition of partial pretree-automorphisms
$x_{i}\colon K_{m}\to K_{m}$). Since $x([x^{-1}\cdot
p_{0},p_{0}])=[p_{0},x\cdot p_{0}]$, we deduce $x$ is $T_{m}$-loxodromic and
its axis in $T_{m}$ contains $\iota_{m}(p_{0})$. Consequently, the rigid
$F$-action on $T_{m}$ is minimal.

Using the universal property in Theorem A.1, the chain $K_{1}\subset
K_{2}\subset\cdots\subset T$ gives us a direct system of equivariant morphisms
$T_{1}\to T_{2}\to\cdots\to T$. The real pretree $T$ is a strong limit of this
direct system: each closed interval $I\subset T_{1}$ has an image under the
morphism $T_{1}\to T_{m}$ (for large enough $m$) that is embedded into $T$ by
the morphism $T_{m}\to T$. For the proof, pick a closed interval $I\subset
T_{1}$; its image under the morphism $j_{1}\colon T_{1}\to T$ is a finite real
pretree, which is contained in $K_{m}$ for large $m$; therefore,
$\iota_{m}(j_{1}(I))\subset\iota_{m}(K_{m})$, the image of $I$ under $T_{1}\to
T_{m}$, is embedded into $T$ (with image $j_{1}(I)$) by $T_{m}\to T$ as the
latter restricts to a pretree-embedding of $\iota_{m}(K_{m})$.

######  Lemma A.6 (cf. [GL95, Proposition II.1]).

Let $T$ be a real pretree with a minimal rigid $F$-action. The following
conditions are equivalent:

  1. 1.

$T$ is equivariantly pretree-isomorphic to $T_{\mathcal{K}}$ for some rigid
system $\mathcal{K}=(K,\\{x_{i}\\})$; and

  2. 2.

$T$ can only be a strong limit in a trivial way: if $T$ is a strong limit of a
direct system of equivariant morphisms $T_{1}\to T_{2}\to\cdots\to T$ with
minimal rigid $F$-actions on $T_{j}$, then $T_{m}\to T$ is an equivariant
pretree-isomorphism for some $m\geq 1$.

A real pretree $T$ with a minimal rigid $F$-action is geometric if these
conditions hold. By the second characterization, “geometricity” will be
independent of the choice of basis for $F$.

###### Sketch of proof.

$(\ref{alem:geometric-action:trivial}\implies\ref{alem:geometric-
action:geometric})$: By the preceding discussion, $T$ is a strong limit of a
direct system of equivariant morphisms $T_{1}\to T_{2}\to\cdots\to T$ and the
rigid $F$-action on $T_{j}$ is minimal for large enough $j$. Since we are
assuming strong limits are trivial, we have $T_{m}\to T$ is an equivariant
pretree-isomorphism for some large $m$. Recall that $T_{m}$ is associated to a
rigid system given by a finite real pretree $K_{m}\subset T$ and we are done.

$(\ref{alem:geometric-action:geometric}\implies\ref{alem:geometric-
action:trivial})$: Let $\rho\colon T_{\mathcal{K}}\to T$ be an equivariant
pretree-isomorphism for some rigid system $\mathcal{K}=(K,\\{x_{i}\\})$. In
particular, $T$ satisfies the universal property in Theorem A.1 with respect
to the pretree-embedding $\rho\circ\iota\colon K\to T$. We need to show that
an arbitary strong limit $T_{1}\to T_{2}\to\cdots\to T$ of minimal rigid
actions is trivial. As a strong limit of minimal actions, we can lift the
union of $\rho(\iota(K))\subset T$ and its translates
$x_{i}\cdot\rho(\iota(K))\subset T~{}(1\leq i\leq n)$ to a pretree-isomorphic
copy in $T_{m}$ for some large enough $m$. By the universal property, there is
a unique equivariant morphism $T\to T_{m}$. So the equivariant morphism
$T_{m}\to T$ (from the strong limit) must be a pretree-isomorphism and hence
the strong limit is trivial. ∎

### • Counting branch points

We are now ready to prove the index inequality.

######  Theorem A.7 (cf. [GL95, Theorem III.2]).

For a minimal small rigid $F$-action on a real pretree $T$,

  1. 1.

$i(F\backslash T)=c(F)-1$ if $T$ is geometric, and

  2. 2.

$i(F\backslash T)<c(F)-1$ otherwise.

###### Proof of Part 1.

Suppose $\mathcal{K}=(K,\\{x_{i}\colon A_{i}\to B_{i}\\})$ is a rigid system
and the rigid $F$-action on the associated real pretree $T_{\mathcal{K}}$ is
small and minimal. For a finite real pretree $H$, the valence of $h\in H$ —
denoted $\nu_{H}(h)$ — is the number of directions at $h$. The valence satisfy
the identity: $\sum_{h\in H}\left(\nu_{H}(h)-2\right)=-2.$

Fix a point $p\in T_{\mathcal{K}}$. Then $p=x\cdot\iota(a)$ for some $x\in F$
and $a\in K$ by Theorem A.1(3). Define
$V_{p}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{\,b\in
K:y\cdot\iota(a)=\iota(b)\text{ for some }y\in F\,\\}$. The “Cayley graph”
$\mathcal{O}_{p}$ oriented labelled graph with vertex set $V_{p}$ and an
oriented labelled edge $b\overset{x_{i}}{\to}c$ if $x_{i}(b)=c$. The Cayley
graph is connected (Lemma A.2).

Each oriented labelled edge has a weight
$w{\left(b\overset{x_{i}}{\to}c\right)}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\nu_{A_{i}}(b)$.
The “blow-up” $\mathcal{O}_{p}^{\prime}$ is constructed by replacing: 1) each
vertex $b\in V_{p}$ with $\nu_{K}(b)$-many vertices corresponding to the
$K$-directions at $b$; and 2) each edge $b\overset{x_{i}}{\to}c$ with
$w{\left(b\overset{x_{i}}{\to}c\right)}$-many oriented labelled edges in the
“obvious way.” Let $\pi\colon\mathcal{O}_{p}^{\prime}\to\mathcal{O}_{p}$ be
the natural finite-to-one projection.

######  Lemma A.8 (cf. [GL95, Lemma III.5]).

Let $G_{p}\leq F$ be the stabilizer of $p\in T_{\mathcal{K}}$.

  1. 1.

$\pi_{1}(\mathcal{O}_{p})\cong G_{p}$ (natural group isomorphism).

  2. 2.

$\pi_{0}(\mathcal{O}_{p}^{\prime})\cong
G_{p}\backslash\pi_{0}(T_{\mathcal{K}}\setminus\\{p\\})$ (natural set-
bijection).

  3. 3.

for $\mathcal{O}_{d}\in\pi_{0}(\mathcal{O}_{p}^{\prime})$ corresponding to a
$T_{\mathcal{K}}$-direction $d$ at $p$,
$\pi_{1}(\mathcal{O}_{d})\cong\mathrm{Stab}_{G_{p}}(d)$.

###### Proof sketch for lemma.

There is a natural injective homomorphism $\pi_{1}(\mathcal{O}_{p},a)\to
G_{\iota(a)}$ given by “reading the oriented labels.” This homomorphism is
surjective (Lemma A.2) — as $G_{\iota(a)}=x^{-1}G_{p}x$, this concludes part
1.

Distinct components of $\mathcal{O}_{p}^{\prime}$ correspond to distinct
$F$-orbits of $T_{\mathcal{K}}$-directions at translates of $p$ by Lemma A.2.
So there is a natural injective set-function
$\pi_{0}(\mathcal{O}_{p}^{\prime})\to
G_{p}\backslash\pi_{0}(T_{\mathcal{K}}\setminus\\{p\\}).$ The proof of Lemma
A.5 shows that this function is surjective — this concludes part 2.

As in the first part, we have an injective homomorphism
$\pi_{1}(\mathcal{O}_{d},d_{b})\to\mathrm{Stab}_{G_{\iota(b)}}(d_{b})$, where
$b\in V_{p}$; in particular, $G_{\iota(b)}=x^{-1}G_{p}x$ and
$\mathrm{Stab}_{G_{\iota(b)}}(d_{b})$ is conjugate in $F$ to a stabilizer in
$G_{p}$ of a $T_{\mathcal{K}}$-direction at $p$. This natural homomorphism is
surjective (Lemma A.2) and we are done. ∎

Recall that $S\subset K$ consists of the vertices of $K$, $A_{i}$, and
$B_{i}$; it is a finite set. Let $\mathcal{G}\subset\mathcal{O}_{p}$ be a
finite connected subgraph containing all vertices in $S$ and all edges of
weight $\neq 2$. Set
$\mathcal{G}^{\prime}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\pi^{-1}(\mathcal{G})\subset\mathcal{O}_{p}^{\prime}$;
this is a finite subgraph as well since $\pi$ was finite-to-one.

By Lemma A.5 and Lemma A.8(2-3), $\mathcal{O}_{p}^{\prime}$ has finitely many
components, each with rank 0 or 1 (small rigid action). Enlarge $\mathcal{G}$
while maintaining finiteness and assume $\mathcal{G}^{\prime}$ supports the
fundamental group of each component of $\mathcal{O}_{p}^{\prime}$.

For any finite connected graph, we have the Euler characteristic identity:

$1-\mathrm{rank}(\pi_{1}(\,\cdot\,))=\\#V(\,\cdot\,)-\\#E(\,\cdot\,),$

where $V(\,\cdot\,)$ and $E(\,\cdot\,)$ denote the vertex and edge sets
respectively. Sum over the finitely many components of
$\mathcal{G}_{j}^{\prime}\subset\mathcal{G}^{\prime}$; we get

$\sum_{j}\left(1-\mathrm{rank}(\pi_{1}(\mathcal{G}_{j}^{\prime}))\right)=\sum_{b\in
V(\mathcal{G})}\nu_{K}(b)-\sum_{e\in E(\mathcal{G})}w(e).$

Subtract the double of the Euler characteristic identity applied to
$\mathcal{G}$:

$\displaystyle
2\cdot\mathrm{rank}(\pi_{1}(\mathcal{G}))-2+\sum_{j}\left(1-\mathrm{rank}(\pi_{1}(\mathcal{G}_{j}^{\prime}))\right)$
$\displaystyle=\sum_{b\in V(\mathcal{G})}(\nu_{K}(b)-2)-\sum_{e\in
E(\mathcal{G})}(w(e)-2)$ (1) $\displaystyle=\sum_{b\in
V_{p}}(\nu_{K}(b)-2)-\sum_{e\in E(\mathcal{O}_{p})}(w(e)-2).$

The last equality follows from the fact $\mathcal{G}$ was assumed to contain
all vertices and edges with nonzero summand contributions to the right-hand
side. Note that the right-hand side is finite and independent of
$\mathcal{G}$. The summands on the left-hand side are nonnegative; therefore,
$\pi_{1}(\mathcal{G})$ has uniformly bounded rank and
$\pi_{1}(\mathcal{O}_{p})$ is finitely generated.

Assume $\mathcal{G}$ supports the fundamental group of $\mathcal{O}_{p}$. Then
each component of $\mathcal{O}_{p}^{\prime}$ has at most one component of
$\mathcal{G}^{\prime}$. By Lemma A.8, Equation (1) can be rewritten as

$i[p]=\sum_{b\in V_{p}}(\nu_{K}(b)-2)-\sum_{i=1}^{n}\sum_{b\in V_{p}\cap
A_{i}}(\nu_{A_{i}}(b)-2).$

Summing over all $F$-orbits $[p]\in F\backslash T_{\mathcal{K}}$ and applying
the valence identity in the first paragraph of the proof, we get

$\displaystyle i(F\backslash T_{\mathcal{K}})$ $\displaystyle=\sum_{b\in
K}(\nu_{K}(b)-2)-\sum_{i=1}^{n}\sum_{b\in A_{i}}(\nu_{A_{i}}(b)-2)$
$\displaystyle=-2+2n$ $\displaystyle=c(F)-1.$

This concludes the geometric case of the theorem. ∎

###### Proof of Part 2.

Suppose $T$ is a nontrivial strong limit of a direct system of equivariant
morphisms $T_{1}\to T_{2}\to\cdots\to T$ where each $T_{i}$ is geometric.

Fix a branch point $p\in T$, then choose $g_{1},\ldots,g_{r}\in G_{p}$ that
freely generate a free factor of $G_{p}$ and directions $d_{1},\ldots,d_{s}$
at $p$ that have trivial stabilizers and are in distinct $G_{p}$-orbits. As a
strong limit, we can lift the point $p$ to $p^{\prime}\in T_{m}$ and the
directions $d_{j}~{}(1\leq j\leq s)$ to $d_{j}^{\prime}$ in $T_{m}$ so that
$p^{\prime}$ is fixed by all $g_{k}~{}(1\leq k\leq r)$ for large enough $m\geq
1$. By the equivariance of the morphism $T_{m}\to T$, the $T_{m}$-directions
$d_{j}^{\prime}$ have trivial stabilizers and $G_{p^{\prime}}\leq G_{p}$
contains the free factor generated by $g_{1},\ldots,g_{r}$. So $2r-2+s\leq
i[p^{\prime}]$. From Part (1), $i[p^{\prime}]$ is uniformly bounded by
$\iota(F\backslash T_{m})=c(F)-1$. Thus $G_{p}$ is finitely generated, there
are finitely many $G_{p}$-orbits of directions with trivial stabilizers at
$p$, and $i[p]\leq i[p^{\prime}]$.

Similarly, any finite set of branch points $p_{1},\ldots\in T$ in distinct
$F$-orbits can be lifted to branch points $p_{1}^{\prime},\ldots\in T_{m}$
with $i[p_{j}]\leq i[p_{j}^{\prime}]$ for large enough $m$. By equivariance,
the branch points $p_{1}^{\prime},\ldots$ are in distinct $F$-orbits. So we
get $i(F\backslash T)\leq c(F)-1$ from Part (1).

To show the inequality $i(F\backslash T)<c(F)-1$, we assume the equality
$i(F\backslash T)=c(F)-1$ and deduce a contradiction. Suppose each $T_{m}$ is
associated to finite real pretree $K_{m}\subset T$ that is the convex hull of
some partial $F$-orbit of a branch point $p_{0}\in T$.

For large $m\geq 1$, $K_{m}$ intersects each $F$-orbit $[p]$ of points in $T$
with $i[p]>0$ and $p^{\prime}\in\iota_{m}(K_{m}\cap[p])\subset T_{m}$ has
index $i[p^{\prime}]=i[p]$. Conversely, as $i(F\backslash T_{m})=i(F\backslash
T)$, every $F$-orbit $[p^{\prime}]$ of points in $T_{m}$ with
$i[p^{\prime}]>0$ intersects $\iota(K_{m}\cap[p])$ for some $F$-orbit of
points in $T$ with $i[p]=i[p^{\prime}]$. Fix a large enough $m\geq 1$.

We finally use the hypothesis that $T$ is a nontrivial strong limit of
$T_{1}\to T_{2}\to\cdots\to T$; i.e. the equivariant morphism $j_{m}\colon
T_{m}\to T$ is not injective. Thus, two distinct directions $d_{1},d_{2}$ at
some point $q^{\prime}\in T_{m}$ are mapped to the same direction at
$q\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=j_{m}(q^{\prime})\in
T$.

If $i[q^{\prime}]>0$, then $i[q]=i[q^{\prime}]$ implies $d_{1},d_{2}$ have
nontrivial stabilizers. Their image in $T$ also has a nontrivial cyclic
stabilizer and so the union of their stabilizers generate a cyclic subgroup;
therefore, some nondegenerate arc in $d_{1}\cup\\{q^{\prime}\\}\cup d_{2}$
intersecting both directions is fixed by a nontrivial element $y\in F$. But
Corollary A.3 states that $j_{m}$ is injective on the fixed-point set of a
nontrivial element — a contradiction.

If $q^{\prime}$ is a branch point with $i[q^{\prime}]=0$, then again
$d_{1},d_{2}$ have nontrivial stabilizers and the argument is the same. If
$q^{\prime}$ is not a branch point, then some nondegenerate arc in
$d_{1}\cup\\{q^{\prime}\\}\cup d_{2}$ intersecting both directions is
contained some translate $y\cdot\iota_{m}(K_{m})$ — this is where we use the
assumption $K_{m}$ was the convex hull of some branch points. We have another
contradiction as $j_{m}$ is injective on $\iota_{m}(K_{m})$ and we are done. ∎

##  Appendix B Solutions to some exercises

Preliminary to checking pretree axioms. Let $p,q,r$ be arbitrary points in a
pretree $(T,[\cdot,\cdot])$. We now state two facts (whose proofs are left to
the reader):

• (pseudocenters) $[p,q]\cap[q,r]\cap[p,r]$ is empty or a singleton.

• (subinterval) if $q\in[p,r]$, then $[p,q]\subset[p,r]$.

###### Checking pretree axioms are satisfied in Proposition IV.3.