# Solvable groups and affine actions on the line

Joaquín Brum and Nicolás Matte Bon and Cristóbal Rivas and Michele Triestino

###### Abstract.

We prove a structural result for orientation-preserving actions of finitely
generated solvable groups on real intervals, considered up to semi-conjugacy.
As applications we obtain new answers to a problem first considered by J. F.
Plante, which asks under which conditions an action of a solvable group on a
real interval is semi-conjugate to an action on the line by affine
transformations. We show that this is always the case for actions by $C^{1}$
diffeomorphisms on closed intervals. For arbitrary actions by homeomorphisms,
for which this result is no longer true (as shown by Plante), we show that a
semi-conjugacy to an affine action still exists in a local sense, at the level
of germs near the endpoints. Finally for a vast class of solvable groups,
including all solvable linear groups, we show that the family of affine
actions on the line is robust, in the sense that any action by homeomorphisms
on the line which is sufficiently close to an affine action must be semi-
conjugate to an affine action. This robustness fails for general solvable
groups, as illustrated by a counterexample.

MSC2020: Primary 37C85, 20F16, 20E08, 20F60, 57M60. Secondary 37E05, 37B05.

###### Key words and phrases:

Group actions on the real line, solvable groups, actions on real trees,
$C^{1}$ actions

##  1\. Introduction

###  1.1. Background

This work is about the study of group actions on one-manifold. This means,
given a group $G$, to try to understand and classify the possible
representations $\varphi\colon G\to\operatorname{\mathsf{Homeo}}_{0}(X)$ into
the groups of orientation-preserving homeomorphisms, or into the group
$\operatorname{\mathsf{Diff}}^{r}_{0}(X)$ of diffeomorphisms of regularity
$C^{r}$ where $X$ is a connected one-manifold (that is, the circle or a real
interval). With no real loss of generality we restrict to actions which have
no global fixed points in the interior of $X$, which will be termed
_irreducible_ (indeed, an action with global fixed points can be decomposed as
a union of irreducible actions on a countable family of invariant intervals).
As usual in dynamics, actions can be considered up to (topological) conjugacy;
however, in the one-dimensional setting it is always possible to obtain many
non-conjugate actions from a given one by blowing up orbits as in the
classical Denjoy’s example. This redundancy can be avoided by considering a
weaker equivalence relation, called _semi-conjugacy_ , whose definition can be
traced back to Ghys [ghysbounded], although it has been unanimously formalized
only recently (see for instance the monograph by Kim, Koberda, and Mj [KKMj]).
When $X,Y$ are non-empty open intervals, two irreducible actions
$\varphi\colon G\to\operatorname{\mathsf{Homeo}}_{0}(X)$ and $\psi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(Y)$ are semi-conjugate if there is a
monotone $G$-equivariant map $h\colon X\to Y$ . We warn the reader that this
semi-conjugacy notion is slightly different from the homonymous notion
appearing in classical dynamical systems: notably, this is an equivalence
relation. (We refer to §2.1 for more details.)

In this article, we are interested in the case where the group $G$ is a
finitely generated solvable group, a case which has attracted considerable
attention, starting with the work of Plante [PlanteMeasure, Plante], whose
motivation was from the theory of codimension-one foliations. A basic source
of examples of solvable groups acting on one-manifolds are subgroups of the
real affine group

$\operatorname{\mathsf{Aff}}(\mathbb{R})=\\{x\mapsto
ax+b:a>0,b\in\mathbb{R}\\}.$

In a seminal article [Plante], Plante considered the following converse
problem: under which conditions an action of a solvable group $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ must be semi-conjugate to
an action by affine transformations $\varphi\colon
G\to\operatorname{\mathsf{Aff}}(\mathbb{R})$?

Plante provided an array of sufficient conditions for an affermative answer.
He deduced in particular that when $G$ is a finitely generated polycyclic
group, every irreducible action $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is semi-conjugate to an
affine action. In contrast, he showed that the result fails for general
solvable groups: he constructed a faithful and minimal action of the group
$\mathbb{Z}\wr\mathbb{Z}$ on $\mathbb{R}$ which is not semi-conjugate to an
affine action (we recall this construction in Section 6). The idea behind the
construction of Plante’s action of $\mathbb{Z}\wr\mathbb{Z}$ is quite robust
and can be generalized in multiple ways (see for instance our previous work
[BMRT, Example 7.11]); as a consequence, it turns out the possible actions of
general solvable groups on the real line can be quite wilder than expected.
Since Plante’s work, a number of papers have addressed the study of solvable
group actions on one-manifolds, and the special role played by affine actions
can be recognized as the main theme, see for instance works of Plante
[PlanteContinuous], Navas [NavasResolubles, NavasAmenable] and references
therein, Akhmedov [Akhmedov_Z, Akhmedov], Bonatti, Monteverde, Navas and the
third author [BMNR], Guelman and the third author [GuelmanRivas] (see also
work by Chiswell and Kropholler [ChiswellKropholler] and by the third author
and Tessera [RivasTessera] for tightly related results on left-orders on
solvable groups).

###  1.2. Results

In this paper we develop a new approach to study general actions of solvable
groups on the real line by homeomorphisms (we work under the slightly more
general assumption that $G$ is virtually solvable, i.e. contains a solvable
subgroup of finite index). As application, we obtain three results which
provide new answers to the problem originally considered by Plante, by
relating arbitrary actions to affine actions.

Our first result deals with actions on intervals by diffeomorphisms of class
$C^{1}$. Here and elsewhere, we say for short that an affine action
$\psi\colon G\to\operatorname{\mathsf{Aff}}(\mathbb{R})$ is _abelian_ if its
image is abelian.

######  Theorem A ($C^{1}$ actions of solvable groups).

Let $G$ be a finitely generated virtually solvable group and let
$\varphi\colon G\to\operatorname{\mathsf{Diff}}^{1}_{0}([0,1])$ be an
irreducible action. Then the restriction of $\varphi$ to $(0,1)$ is semi-
conjugate to an affine action $\psi\colon
G\to\operatorname{\mathsf{Aff}}(\mathbb{R})$. Moreover, the semi-conjugacy is
a topological conjugacy provided $\psi$ is non-abelian.

######  Remark 1.1.

The main contribution of the theorem is the existence of the semi-conjugacy.
When $\psi$ is a non-abelian affine action, any $C^{1}$ action semi-conjugate
to $\psi$ is automatically conjugate to $\psi$ by a result of Bonatti,
Monteverde, Navas, and the third author [BMNR, §4.2].

######  Remark 1.2.

Under the stronger assumption that actions be of class $C^{2}$, the result was
proved by Navas in [NavasResolubles], who also obtained an algebraic
description of solvable subgroups of
$\operatorname{\mathsf{Diff}}^{2}_{0}([0,1])$. Navas’s approach is based
essentially on the strong restriction on abelian groups of diffeomorphisms of
class $C^{2}$. Indeed classical results of Kopell [Kopell], Denjoy [Denjoy]
and Hölder [Holder] together imply that an abelian subgroup of
$\operatorname{\mathsf{Diff}}^{2}_{0}([0,1])$, whose action on $(0,1)$ is
irreducible, is conjugate to a group of translations of $\mathbb{R}$. Both
Kopell’s and Denjoy’s results fail in $C^{1}$ regularity, and this approach
does not suffice even to exclude that the Plante’s actions of
$\mathbb{Z}\wr\mathbb{Z}$ might be conjugate to a $C^{1}$ action. In fact our
proof of Theorem A proceeds by first reducing to this special case, and then
ruling out this possibility (see Proposition 7.1, which is inspired by
[BMNR]).

In the $C^{0}$ setting, our next result shows that the answer to Plante’s
problem is always affirmative in a _local_ sense, namely a semi-conjugacy to
an affine action can be found at the level of germs near fixed points.

######  Theorem B (Local semi-conjugacy near fixed points).

Let $G$ be a finitely generated virtually solvable group, and let
$\varphi\colon G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ be an action
without global fixed points on an interval $I$ of the form $(a,+\infty)$.

Then there exists an irreducible affine action $\psi\colon
G\to\operatorname{\mathsf{Aff}}(\mathbb{R})$ and a non-decreasing map $h\colon
I\to\mathbb{R}$, with $\lim_{x\to+\infty}h(x)=+\infty$ such that for every
$g\in G$ we have $\psi(g)(h(x))=h(\varphi(g)(x))$ for all $x\in\mathbb{R}$
sufficiently large. This action $\psi$ is unique up to affine conjugacy.

######  Remark 1.3.

Of course, equivalent results hold for irreducible actions on finite
intervals, considering neighborhoods of the endpoints.

Theorem B should be compared with a result of Chiswell and Kropholler
[ChiswellKropholler], stating that a solvable group is left-orderable if and
only if it is locally indicable (this is true more generally for amenable
groups, by a result of Witte Morris [WitteMorris]). In an equivalent
formulation, that result says that a finitely generated solvable group $G$ has
a non-trivial action $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ if and only if it admits a
non-trivial homomorphism to the group of translations $(\mathbb{R},+)$. The
reader may note that Theorem B easily recovers this result (using that every
non-trivial subgroup of $\operatorname{\mathsf{Aff}}(\mathbb{R})$ has a
homomorphism to $(\mathbb{R},+)$). An important difference is that Theorem B
uniquely associates to every action of $G$ on $\mathbb{R}$, a homomorphism to
$\operatorname{\mathsf{Aff}}(\mathbb{R})$ with an explicit dynamical meaning;
in contrast the proofs in [ChiswellKropholler] and [WitteMorris] proceed
through considerations on the set of all left-orders to deduce the existence
of a homomorphism to $\mathbb{R}$, but do not to establish an explicit
relation between a fixed action (or order) and such homomorphisms.

Our third main result addresses the question of rigidity of the family of
affine actions under small deformations. To this end, given a group $G$, we
denote by
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
the space of irreducible actions $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$. Recall that this space has
a natural topology, induced from the pointwise convergence topology of all
maps from $G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$, with respect to
the compact-open topology on $\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$.
In view of Plante’s problem, it is then natural to ask whether an action of a
solvable group which is sufficiently close to an affine action must be semi-
conjugate to an affine action. The following result provides an affirmative
answer for a vast class of solvable groups, namely those that are _virtually
metanilpotent_ , that is, admitting a nilpotent normal subgroup $N$ such that
$G/N$ is virtually nilpotent. This class includes in particular all virtually
solvable linear groups (that is, subgroups of $\mathsf{GL}(n,\mathbb{K})$ for
some field $\mathbb{K}$), thanks to a well-known result of Mal’cev [Malcev]
(see Remark 8.1).

######  Theorem C (Perturbations of affine actions).

Let $G$ be a finitely generated virtually metanilpotent group. Then, the
subset of
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
of all irreducible actions which are semi-conjugate to a non-abelian affine
action is open.

######  Remark 1.4.

Theorem C is false for general solvable group, already in the 3-step solvable
case: we construct in §8.2 a 3-step finitely generated solvable group $G$ and
a sequence of irreducible actions
$(\varphi_{n})\subset\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
which are not semi-conjugate to affine actions, yet converge to a non-abelian
affine action.

######  Remark 1.5.

Theorem C implies in particular that if $\psi\colon
G\to\operatorname{\mathsf{Aff}}(\mathbb{R})$ is a non-abelian affine action,
then every action $\varphi$ sufficiently close to $\psi$ remains semi-
conjugate to a non-abelian affine action. It is however not true in general
that ${\varphi}$ is semi-conjugate to $\psi$ itself: indeed even metabelian
groups often admit continuous paths of representations into
$\operatorname{\mathsf{Aff}}(\mathbb{R})$ (see e.g. Lemma 6.1).

###  1.3. The structure of general actions

Although affine actions are a central character in the results stated above,
the essence of this work is about the understanding of actions of solvable
groups on the line that are _not_ semi-conjugate to any affine action. Our
starting point is a result obtained in our previous work [BMRT, Theorem 7.18],
namely that every minimal action of a finitely generated solvable group $G$ on
the line is either conjugate to an affine action, or it is
$\mathbb{R}$-_focal_ in the terminology of [BMRT]. We review this notion and
its main properties in Section 3. An action $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is $\mathbb{R}$-focal if it
preserves a _lamination_ on the real line, that is, a collection $\mathcal{L}$
of bounded intervals without crossings, and the action of $G$ on $\mathcal{L}$
satisfies a hyperbolicity-like condition. As we recall in details in Section
3, from such an invariant lamination one may construct an action of $G$ on a
planar _directed (real) tree_ , which roughly speaking is a space $\mathbb{T}$
obtained by gluing together copies of the real line along intervals of the
form $[a,+\infty)$. The resulting $G$-action on $\mathbb{T}$ fixes an end
$\omega\in\partial\mathbb{T}$, and the original action of $G$ on the line can
be reconstructed from the action on the remaining ends
$\partial\mathbb{T}\smallsetminus\\{\omega\\}$, endowed with a natural linear
order. (One may intuitively think of $\mathbb{T}$ as hanging over the line in
the upper half plane, with one end in the vertical direction and the remaining
ends on the line.)

This leads us to study actions of solvable group $G$ on directed trees, which
is the purpose of Sections 4 and 5. The outcome is Theorem 5.2, which is a
central result of the article. That results says that if $G$ is a finitely
generated (virtually) solvable group, then every minimal $\mathbb{R}$-focal
action $\varphi\colon G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$
naturally comes with an associated action $\varphi^{\prime}\colon
G/N\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ of a proper quotient of
$G$, where $N\unlhd G$ is such that $G/N$ has solvable length strictly less
than $G$ (more precisely, $N$ will be the _Fitting subgroup_ of $G$). This new
action $\varphi^{\prime}$ retains information on the dynamics of $\varphi$ at
large scale. The precise relation is that $\varphi$ is _horograded_ by
$\varphi^{\prime}$, which means that the $G$-action on a directed tree
$\mathbb{T}$ associated to $\varphi$ is governed by $\varphi^{\prime}$ in the
vertical direction (see §3.3). A crucial feature of Theorem 5.2 is that it can
be applied inductively: if the horograding action $\varphi^{\prime}$ is not
semi-conjugate to an affine action, then it must be again $\mathbb{R}$-focal
and horograded by an action $\varphi^{\prime\prime}$ of a quotient of even
smaller solvable length. In finitely many steps, we must reach an affine
action $\psi\colon G\to\operatorname{\mathsf{Aff}}(\mathbb{R})$. This shows
that actions of a solvable groups on the line are naturally organized in a
tower of finitely many levels of complexity, where affine actions are the
simplest.

Theorem B is a direct consequence of Theorem 5.2, the action $\psi$ in its
statement being exactly the affine action obtained from $\varphi$ through the
inductive procedure above. In Section 6 we specialize the discussion to the
group $G=\mathbb{Z}\wr\mathbb{Z}$. We shall see that in this case Theorem 5.2
readily translates to a classification of actions of $G$ on the line up to
semi-conjugacy: these are either affine or one of the Plante’s action from
[Plante]. In addition, for every $\mathbb{R}$-focal action of a solvable group
on the line, the group $G$ must have a subgroup isomorphic to
$\mathbb{Z}\wr\mathbb{Z}$ whose action is semi-conjugate to a Plante action.
To prove Theorem A, we are reduced to show that this action cannot be semi-
conjugate to a $C^{1}$ action on the interval, which is done in Section 7.
Finally, in Section 8 we consider the class of metanilpotent groups. We shall
see that for these groups, Theorem 5.2 takes a much stronger form, and deduce
Theorem C.

In a companion paper in preparation [InPreparation], we will further use
Theorem 5.2 and the inductive approach developed here to provide more detail
results on the structure of semi-conjugacy classes in the space
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
for a finitely generated solvable group, through an analysis of the space
$\operatorname{\mathsf{Der}}_{\mu}(G)$ of normalized $\mu$-harmonic actions
(in the sense of [DKNP]). We shall see in particular that the quotient of
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
by the semi-conjugacy relation has a particularly nice structure (contrary to
arbitrary finitely generated groups, for which it might not even be a standard
Borel space).

### Note to the reader

While this paper builds on concepts introduced in our previous work [BMRT], we
made an effort to make the presentation as self-contained as possible. The
dependence on [BMRT] is limited to a few statements, all contained in Sections
7 and 8 there, which are largely independent from the rest of [BMRT]. Some
other statements from [BMRT, §§7-8] had to be strengthened and thus a complete
proof can be found here, or a sketch of proof is repeated for completeness.

##  2\. Notation and preliminaries

###  2.1. Actions on the line

Let us start with some notation and terminology. When a group $G$ acts on a
set $X$, we simply write $g.x$ for the action, unless there is risk of
confusion. If $\varphi:G\to\operatorname{\mathsf{Sym}}(X)$ is an action, we
write $\operatorname{\mathsf{Stab}}^{\varphi}(x)$ for the stabilizer of $x\in
X$ in $G$ under the action, and $\operatorname{\mathsf{Fix}}^{\varphi}(g)$ for
the subset of fixed points in $X$ of an element $g\in G$, and we write
$\operatorname{\mathsf{Supp}^{\varphi}}(g)=X\smallsetminus\operatorname{\mathsf{Fix}}^{\varphi}(g)$
for its support.

Let $X$ be a real interval with non-empty interior (that is, up to
diffeomorphism, one of the spaces $\mathbb{R},[0,1],[0,1)$). The set of
interior points of $X$ is denoted by $\operatorname{\mathsf{Int}}(X)$. The
group of orientation-preserving homeomorphisms of $X$ is denoted by
$\operatorname{\mathsf{Homeo}}_{0}(X)$, and similarly we denote the group of
orientation-preserving $C^{r}$ diffeomorphisms
$\operatorname{\mathsf{Diff}}_{0}^{r}(X)$, for $r\geq 1$. We will say that a
group action $\varphi\colon G\to\operatorname{\mathsf{Homeo}}_{0}(X)$ is
_irreducible_ if it has no fixed point in $\operatorname{\mathsf{Int}}(X)$. We
also recall the notion of semi-conjugacy for actions on intervals (see Kim,
Koberda, and Mj [KKMj]).

######  Definition 2.1.

Let $X$ and $Y$ be real intervals with non-empty interior, and let
$\varphi\colon G\to\operatorname{\mathsf{Homeo}}_{0}(X)$ and $\psi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(Y)$ be two irreducible actions of a
group $G$. We say that $\varphi$ and $\psi$ are _semi-conjugate_ if there
exists a monotone map
$h\colon\operatorname{\mathsf{Int}}(X)\to\operatorname{\mathsf{Int}}(Y)$ such
that $h\circ\varphi(g)=\psi(g)\circ h$ for any $g\in G$. When $h$ is non-
decreasing, we say that that $\varphi$ and $\psi$ are _positively_ semi-
conjugate. When $h$ is an (orientation-preserving) homeomorphism, we say that
$\varphi$ and $\psi$ are _(positively) conjugate_.

A well-known argument (see [Navas-book, Section 2.1]) shows that if the group
$G$ is finitely generated, then every irreducible group action $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(X)$ is semi-conjugate to an action
$\psi\colon G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ which is either
_minimal_ (that is, every orbit is dense) or takes values in the group
$(\mathbb{Z},+)$ of integer translations, in which case we will say that
$\psi$ is _cyclic_. Moreover, such a minimal or cyclic representative of each
semi-conjugacy class is unique up to conjugacy.

We will often build actions of groups on real intervals starting from actions
on totally ordered sets. This strategy is quite general, but for the purpose
of this note we will restrict our attention to actions on ordered sets
$(\Omega,<)$ which are countable, have no minimal element neither maximal
element, and which are _densely ordered_ in the sense that for every
$\omega_{1}<\omega_{2}$ in $\Omega$ there is $\omega^{\prime}\in\Omega$ such
that $\omega_{1}<\omega^{\prime}<\omega_{2}$. The good thing about these sets
is that Cantor’s “back and forth argument” ensures that there is an order-
preserving bijection $t:(\Omega,<)\to(\mathbb{Q},<)$ where rationals are
equipped with is natural order (see for instance Clay and Rolfsen
[ClayRolfsen, Theorem 2.22]).

######  Definition 2.2.

Let $(\Omega,<)$ be a countable densely ordered set having no maximal neither
minimal element, $t:\Omega\to\mathbb{Q}$ an order-preserving bijection, and
let $\varphi:G\to\operatorname{\mathsf{Aut}}(\Omega,<)$ be a non-trivial
order-preserving action. A _dynamical realization_ of $\varphi$ is an
irreducible action
$\bar{\varphi}:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ with the
property that $\bar{\varphi}(g)(t(\omega))=t(\varphi(g)(\omega))$ for every
$g\in G$ and every $\omega\in\Omega$.

A dynamical realization always exists, see [ClayRolfsen, Section 2.4] for
details. Moreover, if $\bar{\varphi}_{1}$ and $\bar{\varphi}_{2}$ are two
dynamical realization of $\varphi:G\to\operatorname{\mathsf{Aut}}(\Omega,<)$,
then they are positively conjugate. Indeed, if $t_{1}$ and $t_{2}$ are the
corresponding bijections from $\Omega$ to $\mathbb{Q}$, then $h=t_{1}\circ
t_{2}^{-1}$ can be extended to the whole real line so that it conjugates
$\bar{\varphi}_{1}$ to $\bar{\varphi}_{2}$. This is why we usually refer to
_the_ dynamical realization of $\varphi$. The following result will be useful
later.

######  Lemma 2.3.

Let $(\Omega,<)$ be a countable, densely ordered set without maximal or
minimal element. Let $\varphi:G\to\operatorname{\mathsf{Aut}}(\Omega,<)$ be an
order-preserving action of a group $G$, whose dynamical realization
$\bar{\varphi}:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is minimal.
Let $\psi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ be an irreducible
action, and suppose that $\sigma\colon\Omega\to\mathbb{R}$ is a map which is
non-decreasing and $G$-equivariant:

$\sigma(\varphi(g)\omega)=\psi(g)\sigma(\omega)\quad\quad\text{for every }g\in
G,\omega\in\Omega.$

Then, $\sigma$ is injective, and $\psi$ and $\bar{\varphi}$ are positively
semi-conjugate. In particular if $\psi$ is minimal, then it is conjugate to
$\overline{\varphi}$.

###### Proof.

Let $t:\Omega\to\mathbb{Q}$ be the injection used to define the dynamical
realization $\bar{\varphi}$. We define a map
$\tilde{\sigma}\colon\mathbb{R}\to\mathbb{R}$ by setting
$\tilde{\sigma}(x)=\sup\\{\sigma(\omega)\colon t(\omega)\leq x\\}$. Then
$\tilde{\sigma}$ is non-decreasing and equivariant with respect to the
$\bar{\varphi}$-action on the source $\mathbb{R}$ and the $\psi$-action on the
target line, showing that $\psi$ is semi-conjugate to $\overline{\varphi}$. We
claim that $\tilde{\sigma}$ is injective. To see this, consider the subset

$S=\\{x\in\mathbb{R}:\tilde{\sigma}\text{ is constant on a neighborhood of
}x\\}.$

It is direct to see that $S$ is open and $\overline{\varphi}$-invariant, and
therefore it must be empty by minimality of $\overline{\varphi}$. To conclude
the proof of the claim notice that if $\tilde{\sigma}(x)\neq\tilde{\sigma}(y)$
for $x<y$ then, monotonicity of $\tilde{\sigma}$ would imply $(x,y)\subseteq
S$, which is a contradiction. In particular, since $t$ is injective, so is
$\sigma=\tilde{\sigma}\circ t$.

Assume now that $\Psi$ is minimal. In this case, the image of $\tilde{\sigma}$
must be dense, or otherwise its closure would be a proper closed
$\Psi$-invariant subset. Therefore, since $\tilde{\sigma}$ is monotone,
injective, and has dense image, it must be a homeomorphism. This shows that
$\overline{\varphi}$ and $\psi$ are conjugate. ∎

###  2.2. Virtually solvable groups and Fitting series

In this paper, we will mostly be interested in groups $G$ which are _virtually
solvable_ , that is, contain a (normal) solvable subgroup of finite index. We
refer the reader to the book by Lennox and Robinson [LDR] for a general
reference on the theory of solvable groups.

Recall that the _Fitting subgroup_ $\operatorname{\mathsf{Fit}}(G)$ of a group
$G$ is the subgroup generated by all nilpotent normal subgroups of $G$.
Fitting’s theorem states that the group generated by two nilpotent normal
subgroups of $G$ is again nilpotent. In particular,
$\operatorname{\mathsf{Fit}}(G)$ is in fact the union of all nilpotent normal
subgroups. The following basic lemma will be used without mention.

######  Lemma 2.4.

Let $G$ be a group, and $G_{0}\unlhd G$ be a normal solvable subgroup of
finite index. Then $\operatorname{\mathsf{Fit}}(G)$ contains every nilpotent
normal subgroup of $G_{0}$.

###### Proof.

Let $N\unlhd G_{0}$ be nilpotent. Then $N$ has finitely many conjugate
subgroups in $G$, all contained in $G_{0}$. By Fitting’s theorem the subgroup
$M$ generated by such conjugates is nilpotent, and it is normal in $G$. Hence
$N\leq M\leq\operatorname{\mathsf{Fit}}(G)$. ∎

Note that in the previous situation $\operatorname{\mathsf{Fit}}(G)$ contains
in particular the last non-trivial term of the derived series of $G_{0}$.

######  Definition 2.5.

Let $G$ be a virtually solvable group. The _upper Fitting series_ of $G$ is
defined by setting $F_{1}=\operatorname{\mathsf{Fit}}(G)$, and $F_{i+1}$ to be
the preimage of $\operatorname{\mathsf{Fit}}(G/F_{i})$ in $G$. Since $G$ is
virtually solvable, there must exist some $k\in\mathbb{N}$ such that $G/F_{k}$
is finite. We call the smallest such $k$ the _virtual Fitting length_ of $G$
and denote it by $\operatorname{\mathsf{vf}}(G)$.

##  3\. Generalities on $\mathbb{R}$-focal actions

The notion of $\mathbb{R}$-focal action was introduced in our previous work
[BMRT]. We recall and further study this notion and its consequences. We
follow [BMRT, Sections 7, 8] with some slight differences in the terminology
(which will be pointed out).

###  3.1. Minimal laminations

We identify the space of bounded open intervals of the real line with the set
$\mathbb{R}^{(2)}=\\{(a,b)\in\mathbb{R}^{2}:a<b\\}$, and endow it with the
topology induced from $\mathbb{R}^{2}$. We say that two intervals are _non-
crossed_ if they are either disjoint or nested.

######  Definition 3.1.

A _lamination_ of the real line is a non-empty closed subset
$\mathcal{L}\subset\mathbb{R}^{(2)}$ consisting of pairwise non-crossed
intervals.

######  Definition 3.2.

An action $\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is
_$\mathbb{R}$ -focal_ if it preserves a lamination $\mathcal{L}$ such that for
every $l\in\mathcal{L}$ there exists a sequence $(g_{n})\subset G$ such that
$(g_{n}.l)$ is an increasing exhaustion of the line. When we want to emphasize
the role of the lamination, we will say that $\varphi$ is $\mathbb{R}$-focal
_with respect to $\mathcal{L}$_.

Definition 3.2 is slightly different from the definition of $\mathbb{R}$-focal
action introduced in [BMRT], where we were not assuming that $\mathcal{L}$ be
closed (the terminology being _cross-free cover_ instead of lamination).
However, the closure of a $G$-invariant cross-free cover is a lamination, and
therefore the two definitions are equivalent.

######  Definition 3.3.

Let $\mathcal{L}\subset\mathbb{R}^{(2)}$ be a lamination which is invariant
under an action $\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$.
We say that $\mathcal{L}$ is a _minimal_ $\varphi$-invariant lamination if the
subset $\mathcal{L}$ is minimal with respect to inclusion, among the family of
$\varphi$-invariant laminations.

In other terms, $\mathcal{L}$ is a minimal $\varphi$-invariant lamination if
and only if the subset $\mathcal{L}\subset\mathbb{R}^{(2)}$ is a minimal set
(in the classical dynamical sense) for the diagonal action induced by
$\varphi$ on $\mathbb{R}^{(2)}$. When there is no risk of confusion, we will
simply adopt the terminology “minimal lamination” instead of “minimal
$\varphi$-invariant lamination”.

######  Remark 3.4.

The property of being an $\mathbb{R}$-focal action is invariant under semi-
conjugacy, and every $\mathbb{R}$-focal action is semi-conjugate to a minimal
$\mathbb{R}$-focal action. See [BMRT, Proposition 7.6].

######  Remark 3.5.

Assume that $\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is a
_minimal_ action. If $\mathcal{L}$ is any $\varphi$-invariant lamination, then
$\varphi$ is automatically $\mathbb{R}$-focal with respect to $\mathcal{L}$.
See [BMRT, Proposition 7.7].

Our first general result is that, in analogy to minimal invariant sets for
actions on the real line, minimal laminations always exist for
$\mathbb{R}$-focal actions of finitely generated groups.

######  Proposition 3.6.

Let $G$ be a finitely generated group, and let
$\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ be an
$\mathbb{R}$-focal with respect to a lamination $\mathcal{L}$. Then,
$\mathcal{L}$ contains a minimal $\varphi$-invariant lamination.

###### Proof.

Consider a finite symmetric generating system $S=\\{g_{1},\ldots,g_{n}\\}$ for
$G$, and let $\|\cdot\|_{S}$ be the associated word length. As $\mathcal{L}$
exhausts the real line, we can take an interval $l\in\mathcal{L}$ so that
$h.l\cap l\neq\varnothing$ for every $h\in S$. The non-crossing condition
gives that for every $h\in S$ either $h.l$ or $h^{-1}.l$ contains $l$. This
implies that the subset $L:=\bigcup_{h\in S}h.l$ is an interval which belongs
to $\mathcal{L}$. We can then consider the subset
$\mathcal{K}:=\\{k\in\mathcal{L}:l\subseteq k\subseteq L\\}$, which is a
compact subset of $\mathcal{L}$. We want to show that any orbit in
$\mathcal{L}$ intersects $\mathcal{K}$. Take $k_{0}\in\mathcal{L}$. If
$k_{0}\in\mathcal{K}$, there is nothing to prove. Otherwise, by definition of
$\mathbb{R}$-focal action we can find an element $h\in G$ such that
$L\subseteq h.k_{0}$, and such that $n:=\|h\|_{S}$ is minimal among lengths of
elements with this property. Write $h=f_{n}\cdots f_{1}$, with $f_{i}\in S$
for every $i\in\\{1,\ldots,n\\}$. By the minimality assumption on $h$, the
interval $k_{*}=f_{n}^{-1}h.k_{0}$ does not contain $L$. On the other hand, by
definition of $L$, we have $f_{n}.l\subseteq L$, and since $L\subseteq
h.k_{0}=f_{n}.k_{*}$, we must have $l\subseteq k_{*}$. As $l\subseteq L$, this
gives $k_{*}\cap L\neq\varnothing$, so that by the non-crossing condition we
must have $l\subseteq k_{*}\subseteq L$, or equivalently
$k_{*}\in\mathcal{K}$, as desired.

Consider now the families

$\mathcal{C}=\\{C\subset\mathcal{L}:C\text{ is }\varphi\text{-invariant and
closed}\\},\quad\mathcal{C}_{\mathcal{K}}=\\{C\cap\mathcal{K}:C\in\mathcal{C}\\}.$

Denote by $i:\mathcal{C}\to\mathcal{C}_{\mathcal{K}}$ the map given by
$C\mapsto C\cap\mathcal{K}$. Since every $\varphi$-orbit meets $\mathcal{K}$,
we have that $i$ is a partial-order-isomorphism with respect to inclusion. On
the other hand, since $\mathcal{K}$ is compact, every decreasing chain in
$\mathcal{C}_{\mathcal{K}}$ has an infimum, so that after Zorn’s lemma we can
find a $\varphi$-invariant lamination which is minimal with respect to
inclusion in $\mathcal{C}$. ∎

###  3.2. Minimal actions on planar trees

Let us recall some further terminology from [BMRT, Section 8]. To study
$\mathbb{R}$-focal actions it is helpful to study actions on a certain type of
(real) trees with a preferred direction to infinity. The precise setting is
the following.

######  Definition 3.7.

A _directed tree_ is a partially ordered set (_poset_ for short)
$(\mathbb{T},\triangleleft)$ with the following properties.

  1. (T1)

For every $v\in\mathbb{T}$ the subset $\\{u\in\mathbb{T}\colon
v\trianglelefteq u\\}$ is totally ordered and order-isomorphic to the half-
line $[0,+\infty)$.

  2. (T2)

Every pair of points $u,v\in\mathbb{T}$ has a smallest common upper bound,
denoted $u\curlywedgeuparrow v$.

  3. (T3)

There exists a countable subset $\Sigma\subset\mathbb{T}$ such that for every
distinct $u,v\in\mathbb{T}$ with $u\trianglelefteq v$ there exists
$z\in\Sigma$ such that $u\trianglelefteq z\trianglelefteq v$.

We say that a point $u\in\mathbb{T}$ is _below_ $v\in\mathbb{T}$ (or that $v$
is _above_ $u$) if $u\neq v$ and $u\trianglelefteq v$, and write
$u\triangleleft v$. When $u$ is below $v$, we write
$[u,v]=[v,u]=\\{w\in\mathbb{T}:u\trianglelefteq w\trianglelefteq v\\}$ for the
arc between $u$ and $v$, and for general $u$ and $v$ we set
$[u,v]=[u,u\curlywedgeuparrow v]\cup[v,u\curlywedgeuparrow v]$. We also write
$]u,v[=[u,v]\smallsetminus\\{u,v\\}$, and similarly we define $]u,v]$ and
$[u,v[$. Arcs allow to define path-components and introduce the notion of
_branching points_ , that is, points $v\in\mathbb{T}$ such that
$\mathbb{T}\smallsetminus\\{v\\}$ has at least three path-components. We write
$\operatorname{\mathsf{Br}}(\mathbb{T})$ for the collection of branching
points. We say that a directed tree $(\mathbb{T},\triangleleft)$ is
_simplicial_ if $\operatorname{\mathsf{Br}}(\mathbb{T})\cap[v,w]$ is finite
for every $v,w\in\mathbb{T}$. A point $v\in\mathbb{T}$ is called a _leaf_ if
it $\mathbb{T}\smallsetminus\\{v\\}$ has only one path-component, or
equivalently, if $v$ is minimal with respect to $\triangleleft$. We will only
be interested in trees which have no leaves (as justified by Remark 3.13
below).

The _end completion_ of a directed tree $(\mathbb{T},\triangleleft)$ is the
poset $(\bar{\mathbb{T}},\triangleleft)$ obtained by adding a unique
$\triangleleft$-maximal point $\omega$, called the _focus_ , and completing
all unbounded maximal totally $\triangleleft$-ordered subsets by adding a
least point. The set
$\partial\mathbb{T}:=\overline{\mathbb{T}}\smallsetminus\mathbb{T}$ is called
the _boundary_ of $\mathbb{T}$. We write
$\partial^{\ast}\mathbb{T}:=\partial\mathbb{T}\smallsetminus\\{\omega\\}$.
Given a point $v\in\mathbb{T}$, we denote by $\partial
U_{v}\subset\partial\mathbb{T}\smallsetminus\\{\omega\\}$ the collection of
boundary points corresponding to maximal totally $\triangleleft$-ordered
subsets containing the point $v$, and we call it the _shadow_ of $v$.

Next we introduce the notion of _planar tree_. Roughly speaking, this means a
directed tree together with an order on branches which corresponds to a
preferred way to draw it in the plane. The definition we give is slightly
different from that appearing in [BMRT], but equivalent (see Remark 3.9
below). To state it, when $X$ is a set and $\triangleleft$, $\prec$ are two
partial orders on $X$, we say that $\triangleleft$ and $\prec$ are
_transverse_ if for every $u,v\in X$ it holds that $u$ and $v$ are
$\triangleleft$-related if and only if they are not $\prec$-related.

######  Definition 3.8.

Let $(\mathbb{T},\triangleleft)$ be a directed tree. A _planar structure_ on
$(\mathbb{T},\triangleleft)$ is a partial order $\prec$ on $\mathbb{T}$ which
is transverse to $\triangleleft$ and satisfies the _compatibility condition_ :

(CC) $\text{for every }u,u_{0},v,v_{0}\in\mathbb{T}\text{ with }u\prec
v,\,u_{0}\trianglelefteq u\text{ and }v_{0}\trianglelefteq v,\text{ it holds
that }u_{0}\prec v_{0}.$

We say that $(\mathbb{T},\triangleleft,\prec)$ is a _planar directed tree_ (or
simply _planar tree_) if $(\mathbb{T},\triangleleft)$ is a directed tree and
$\prec$ is a planar structure on $(\mathbb{T},\triangleleft)$.

######  Remark 3.9.

In [BMRT, Definition 8.2], the definition of planar tree was given instead by
fixing a linear order $\prec_{v}$ on the set of path-components of
$\mathbb{T}\smallsetminus\\{v\\}$ below any point
$v\in\operatorname{\mathsf{Br}}(\mathbb{T})$ (also called _directions_ below
$v$). The two definitions are equivalent, and it is immediate to pass from one
to the other. We prefer to adopt this alternative definition because in this
way we can avoid introducing the related terminology, which is unnecessary for
this work.

######  Remark 3.10.

Any lamination $\mathcal{L}$ admits two distinct partial orders, defined by
the inclusion relation $\subset$ and the order of the real line $<$ (for
$l,k\in\mathcal{L}$, we write $l<k$ when $\sup l\leq\inf k$). These two
partial orders are transverse and satisfy the compatibility condition (CC).

Note that whenever $(\mathbb{T},\triangleleft,\prec)$ is a planar tree, the
partial order $\prec$ naturally induces a linear order on
$\partial^{*}\mathbb{T}=\partial\mathbb{T}\smallsetminus\\{\omega\\}$, by
declaring $\xi\prec\eta$ if there exist $v\in\mathbb{T}$ and $w\in\mathbb{T}$
satisfying $\xi\triangleleft v,\eta\triangleleft w$, and $v\prec w$.

We next want to describe the dynamics of group actions on (planar) directed
trees.

######  Definition 3.11.

Let $G\subseteq\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft)$ be a
group of automorphisms of a directed tree.

  * •

$G$ is _focal_ if for any $u,v\in\mathbb{T}$ there exists $g\in G$ such that
$v\triangleleft g.u$.

  * •

The action of $G$ is _d-minimal_ if for every
$v_{1},v_{2},w_{-},w\in\mathbb{T}$ with $v_{1}\triangleleft v_{2}$ and
$w_{-}\triangleleft w$, there exist $g\in G$ and $w^{\prime}\in[w_{-},w[$ such
that $g.[w^{\prime},w]\subset]v_{1},v_{2}[$.

  * •

The action of $G$ is _simplicial_ if $(\mathbb{T},\triangleleft)$ is
simplicial and for every $v,w\in\operatorname{\mathsf{Br}}(\mathbb{T})$ the
stabilizer $\operatorname{\mathsf{Stab}}^{\Phi}(\\{v,w\\})$ acts trivially on
$[v,w]$.

When $\Phi:G\to\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft)$ is an
action on a directed tree, we say that $\Phi$ is _focal_ , _d-minimal_ , or
_simplicial_ , according to what $\Phi(G)$ is.

######  Remark 3.12.

Any d-minimal action is focal. The letter “d” stands for _directionally_.
Indeed the definition says that it is possible to send every point $w$ inside
any interval $[v_{1},v_{2}[$, and moreover this can be done so that any given
direction below $w$ (defined by $w_{-}$) is mapped to the direction defined by
the point $v_{1}$.

######  Remark 3.13.

If $(\mathbb{T},\triangleleft)$ admits a focal group of automorphisms, then it
has no leaves.

Any group action $\Phi\colon
G\to\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft)$ extends to an
action on $\overline{\mathbb{T}}=\mathbb{T}\cup\partial\mathbb{T}$, which
fixes the focus $\omega$. As a consequence, we obtain an action on the subset
$\partial^{*}\mathbb{T}=\partial\mathbb{T}\smallsetminus\\{\omega\\}$. In the
case the focal action preserves a planar structure $\prec$ on $\mathbb{T}$, we
get that the linear order on $\partial^{*}\mathbb{T}$ is preserved by the
extension of the focal action. Fixing an end $\xi\in\partial^{*}\mathbb{T}$ we
get an order-preserving action on the totally ordered set $(G.\xi,\prec)$
where $\prec$ is the order induced by the linear order on
$\partial^{\ast}\mathbb{T}$. We proved in [BMRT, Proposition 8.19] that, for a
given focal action $\Phi$, the dynamical realization of the action on
$(G.\xi,\prec)$ gives a minimal $\mathbb{R}$-focal action that does not depend
on the choice of $\xi\in\partial^{*}\mathbb{T}$ (up to positive conjugacy). We
call this action the _dynamical realization_ of $\Phi$. Conversely, by [BMRT,
Proposition 8.20], any minimal $\mathbb{R}$-focal action of a countable group
$G$ can be reconstructed (up to positive semi-conjugacy) as the dynamical
realization of a focal action $\Phi$ of $G$ on a planar directed tree. In this
case we say that $\Phi$ _represents_ the $\mathbb{R}$-focal action.

The following lemma gives an alternative characterization of this notion.

######  Lemma 3.14.

Let $G$ be a countable group and $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ a minimal
$\mathbb{R}$-focal action. Let $\Phi\colon
G\to\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft,\prec)$ be a focal
action on a planar tree. Then $\Phi$ represents $\varphi$ if and only if there
exists a non-decreasing $G$-equivariant map
$\sigma\colon(\partial^{*}\mathbb{T},\prec)\to(\mathbb{R},<)$. Moreover such a
map $\sigma$ is necessarily injective on each $G$-orbit.

###### Proof.

Assume first that $\Phi$ represents $\varphi$. This means that there exists
$\xi\in\partial^{*}\mathbb{T}$ and an equivariant order preserving bijection
$t\colon G.\xi\to\mathbb{Q}$, which can be extended extend to a non-decreasing
$G$-equivariant map $\sigma:\partial^{\ast}\mathbb{T}\to\mathbb{R}$ given by
$\sigma(\eta)=\sup\\{t(\xi^{\prime})\colon\xi^{\prime}\in
G.\xi,\xi^{\prime}\preceq\eta\\}$. Conversely, assume that there is such a map
$\sigma$. Then for every $\xi\in\partial^{*}\mathbb{T}$, the restriction of
$\sigma$ to $G\cdot\xi$ satisfies the assumption of Lemma 2.3. We deduce both
that $\sigma$ is injective on $G\cdot\xi$ and that $\varphi$ is the dynamical
realization of $\Phi$. ∎

The next result is a more precise version of [BMRT, Proposition 8.20].

######  Proposition 3.15.

Let $G$ be a finitely generated group and let
$\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ be a minimal
action which is $\mathbb{R}$-focal. Then, $\varphi$ can be represented by a
focal action
$\Phi:G\to\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft,\prec)$ which
is either d-minimal, or simplicial and transitive on branching points.

###### Proof.

As we are assuming that $G$ is finitely generated, after Proposition 3.6, we
can take a minimal $\varphi$-invariant lamination $\mathcal{L}$. Since the
subset of isolated leaves is both $\varphi$-invariant and open in
$\mathcal{L}$, minimality of $\mathcal{L}$ implies that either $\mathcal{L}$
is discrete or $\mathcal{L}$ is perfect.

We first treat the discrete case. In this case, each $\varphi$-orbit in
$\mathcal{L}$ is open, so that by minimality $\mathcal{L}$ consists of a
single $\varphi$-orbit. We consider the simplicial tree $\mathbb{T}$
associated with the partial order on $\mathcal{L}$ defined by the inclusion
relation $\subset$. It is straightforward to check that $(\mathbb{T},\subset)$
is a simplicial directed tree, where $\operatorname{\mathsf{Br}}(\mathbb{T})$
is naturally identified with $\mathcal{L}$. Moreover, the partial order
relation $<$ induced by the order on the real line makes
$(\mathbb{T},\subset,<)$ a planar directed tree (see Remark 3.10). The action
of $\varphi$ on $\mathcal{L}$ naturally extends to a simplicial action
$\Phi:G\to\operatorname{\mathsf{Aut}}(\mathbb{T},\subset,<)$. Finally,
consider the map $\iota:\partial^{\ast}\mathbb{T}\to\mathbb{R}$ that sends the
boundary point $\xi$, represented by a $\subset$-monotone ray
$r_{\xi}:[0,+\infty)\to\mathbb{T}$, to the intersection of nested closed
intervals

$\iota(\xi):=\bigcap_{l\in\mathsf{Im}(r_{\xi})\cap\mathcal{L}}\overline{l}.$

It is clear that $\iota(\xi)$ does not depend on the choice of $r_{\xi}$. To
see that $\iota$ is well defined (that is, it takes value in $\mathbb{R}$) we
need to check that $\iota(\xi)$ has empty interior. To see this, first notice
that
$\mathsf{Int}(\iota(\xi_{1}))\cap\mathsf{Int}(\iota(\xi_{2}))=\varnothing$ for
any $\xi_{1}\neq\xi_{2}$. Thus, if some $\xi\in\partial^{\ast}\mathbb{T}$ had
non-empty interior we would obtain a proper $\varphi$-invariant open subset,
contradicting minimality of the action $\varphi$. It is direct to check that
$\iota$ is $<$-monotone and equivariant with respect to $\Phi$ and $\varphi$.
We conclude that $\varphi$ is represented by $\Phi$.

Assume now that $\mathcal{L}$ is perfect. Say that a leaf $l\in\mathcal{L}$ is
_accumulated from above_ if the subset $\\{k\in\mathcal{L}:k\supsetneq l\\}$
accumulates on $l$. Analogously we define when a leaf is _accumulated from
below_. Given $l\in\mathcal{L}$ write

$l^{\ast}=\mathsf{Int}\left(\bigcap_{k\in\mathcal{L},\,k\supsetneq
l}k\right),$

and note that $l^{*}\in\mathcal{L}$ as $\mathcal{L}$ is closed in
$\mathbb{R}^{(2)}$.

######  Claim.

The subset $\mathcal{L}^{\ast}:=\\{l^{\ast}:l\in\mathcal{L}\\}$ coincides with
the subset of leaves in $\mathcal{L}$ accumulated from above, and the equality
$(l^{\ast})^{\ast}=l^{\ast}$ holds for every $l\in\mathcal{L}$.

###### Proof of claim.

First notice that, by definition, if $l\in\mathcal{L}$ is accumulated from
above, then $l=l^{\ast}$ and therefore $l\in\mathcal{L}^{\ast}$. Take now
$k=l^{\ast}$ in $\mathcal{L}^{\ast}$. If $l=k$ then, by definition again, $k$
is accumulated from above. On the other hand, when $l\neq k$, the leaf $k$ is
not accumulated from below. Thus, since $\mathcal{L}$ is perfect, $k$ must be
accumulated from above. Also notice that $k^{\ast}=k$ holds in both cases. ∎

We want to show that the action induced by $\varphi$ on
$(\mathcal{L}^{*},\subset,<)$ is the desired focal action. We first need to
verify that $(\mathcal{L}^{*},\subset)$ is a directed tree. We start with
condition (T1): for fixed $l\in\mathcal{L}^{*}$, we need to show that the
subset $R=\\{k\in\mathcal{L}^{*}:l\subseteq k\\}$ is order-isomorphic to
$[0,+\infty)$. Consider the function $j:(R,\subset)\to[0,+\infty)$, defined by
$j(k)=|k|-|l|$, where $|\cdot|$ denotes the size of an interval. After the
claim, the function $j$ is order preserving and unbounded, and every point in
its image is accumulated from the right. We can then find monotone surjective
map $h:[0,+\infty)\to[0,+\infty)$, which is bijective on $j(R)$. The
composition $h\circ j$ gives the desired order-isomorphism. We next focus on
condition (T2), and for this fix $l_{1},l_{2}\in\mathcal{L}^{*}$. Consider the
subset $S=\\{k\in\mathcal{L}^{*}:(l_{1}\cup l_{2})\subset k\\}$, and define
$L:=\operatorname{\mathsf{Int}}\left(\bigcap_{k\in S}k\right)$. We need to
show that $L\in\mathcal{L}^{*}$, or equivalently, by the claim, that $L$ is
accumulated from above. If not, since $\mathcal{L}$ is perfect, $L$ must be
accumulated from below by leaves in $\mathcal{L}^{*}$. Taking a leaf
$L^{\prime}\in\mathcal{L}^{*}$ strictly contained in $L$ and sufficiently
close to $L$, we must have $L^{\prime}\cap l_{i}\neq\varnothing$ for
$i\in\\{1,2\\}$, and by the non-crossing condition, this implies
$L^{\prime}\supset(l_{1}\cup l_{2})$, contradicting the choice of $L$. For
condition (T3), take any $l_{0}\in\mathcal{L}^{\ast}$ and consider the
countable subset $\Sigma:=\\{g.l_{0}:g\in G\\}$. Using the claim and the fact
that $\mathcal{L}$ is a minimal lamination , for every distinct
$l,k\in\mathcal{L}^{*}$ we can find an element $g\in G$ such that $l\subseteq
g.l_{0}\subseteq k$, as desired.

As for the discrete case, the linear order $<$ on the real line makes
$(\mathcal{L}^{*},\subset,<)$ a planar directed tree, so that the action
induced by $\varphi$ on $\mathcal{L}^{*}$ preserves this structure, and
represents the $\mathbb{R}$-focal action $\varphi$. It remains to prove that
this action, call it
$\Phi:G\to\operatorname{\mathsf{Aut}}(\mathcal{L}^{*},\subset,<)$, is
d-minimal (so that it is automatically focal, see Remark 3.12). For this, take
leaves $l_{1}\subset l_{2}$ and $k_{1}\subset k_{2}$ in $\mathcal{L}^{\ast}$,
and we want to find an element $g\in G$ and a leaf $k\in[k_{1},k_{2}]$ such
that $g.[k,k_{2}]\subset]l_{1},l_{2}[$. Taking a smaller $l_{2}$ if necessary,
we can assume that $l_{2}$ is also accumulated from below. Consider the leaf
$\overline{k}$ in $\mathcal{L}$ given by

$\overline{k}:=\bigcup_{k\in\mathcal{L};k_{1}\subset k\subsetneq k_{2}}k,$

and note that $\overline{k}$ is accumulated from below. Again, by the claim
and minimality of $\mathcal{L}$, there exists an element $g\in G$ such that
$l_{1}\subsetneq g.\overline{k}\subsetneq l_{2}$. As $l_{2}$ is accumulated
from below, and $k_{2}$ is either $\overline{k}$ or its successor, we must
have $l_{1}\subsetneq g.k_{2}\subsetneq l_{2}$. As $\overline{k}$ is
accumulated from below, we can also find a leaf $k\in\mathcal{L}^{*}$ such
that $l_{1}\subsetneq g.k\subsetneq g.\overline{k}$. This gives the desired
conclusion. ∎

###  3.3. Horograding actions

One of the advantages when passing from an $\mathbb{R}$-focal action to a
focal representation on a directed tree, is that some new information about
the dynamics might become visible at the level of the tree. This happens when
there is a consistent way of measuring how the distance from the focus changes
under the action. Let us recall from [BMRT, §8.1.3] that an _(increasing)
horograding_ of a directed tree $(\mathbb{T},\triangleleft)$ is an increasing
map $\pi:\mathbb{T}\to\mathbb{R}$ such that for every $u,v\in\mathbb{T}$
verifying $u\triangleleft v$, the restriction of $\pi$ to the arc
$[u,v]\subset\mathbb{T}$ is an order-preserving bijection onto the interval
$[\pi(u),\pi(v)]\subset\mathbb{R}$. Every directed tree admits horogradings,
but they are not always adapted to study focal actions. For this, given an
action $\Phi:G\to\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft)$, we
say that $\Phi$ is _horograded_ by an action
$j:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ if there exists a
$G$-equivariant surjective horograding $\pi:\mathbb{T}\to\mathbb{R}$. We also
say that an $\mathbb{R}$-focal action
$\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is horograded by
an action $j:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$, if $j$ is a
horograding for a focal tree representation of $\varphi$.

Horogradings are good to detect the dynamical behavior of single elements, and
for this we recall the following terminology.

######  Definition 3.16.

A homeomorphism $h\in\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is an
_expanding pseudohomothety_ if there exists a compact subset
$K\subset\mathbb{R}$ such that $h(U)\supset\overline{U}$ for every open subset
$U\supset K$. When, $h^{-1}$ is an expanding pseudohomothety we say that $h$
is a _contracting pseudohomothety_. In the case where $K$ can be taken to be a
point, we simply say $h$ is an (expanding or contracting) _homothety_. In
contrast, we say that $h$ is _totally bounded_ if
$\operatorname{\mathsf{Fix}}(h)$ accumulates on both $+\infty$ and $-\infty$.

It was shown in [BMRT, Corollary 8.28] that, in an $\mathbb{R}$-focal action
$\varphi\colon G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$, every
element is either a pseudohomothety or totally bounded. In the case of a
horograded $\mathbb{R}$-focal action, this can be determined by the
horograding action:

######  Proposition 3.17 (Proposition 8.33 in [BMRT]).

Let $\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ be a minimal
$\mathbb{R}$-focal action positively horograded by an action
$j:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$. For any element $g\in
G$, we have the following.

  1. (1)

If $\operatorname{\mathsf{Fix}}(j(g))$ does not accumulate on $+\infty$, then
$\varphi(g)$ is a pseudohomothety, which is expanding if $j(g)(x)>x$ for every
large enough $x$, and contracting otherwise.

  2. (2)

Otherwise, $\varphi(g)$ is totally bounded.

Moreover, in the former situation, if
$\operatorname{\mathsf{Fix}}(j(g))=\varnothing$ then $\varphi(g)$ is a
homothety.

The following result establishes a more explicit relation between an
$\mathbb{R}$-focal action and an action that horogrades it. Roughly speaking
it says that if $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is horograded by $j\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$, then there are locally
defined semi-conjugacies at the level of germs near $\infty$.

######  Proposition 3.18.

Let $\varphi\colon G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ be a
minimal $\mathbb{R}$-focal action, and assume that $\varphi$ can be horograded
by $j\colon G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$. Then, there
exist two maps $h_{+},h_{-}\colon\mathbb{R}\to\mathbb{R}$ satisfying the
following conditions.

  1. (1)

$h_{+}$ is monotone non-decreasing,

  2. (2)

$h_{-}$ is monotone non-increasing,

  3. (3)

we have $\lim_{x\to+\infty}h_{+}(x)=\lim_{x\to-\infty}h_{-}(x)=+\infty$,

  4. (4)

for every $g\in G$ there exists $M>0$ such that
$h_{+}(\varphi(g)(x))=j(g)(h_{+}(x))$ for every $x>M$, and
$h_{-}(\varphi(g)(x))=j(g)(h_{-}(x))$ for every $x<-M$.

###### Proof.

Let $\Phi\colon
G\to\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft,\prec)$ be a focal
action representing $\varphi$ together with a horograding
$\pi\colon\mathbb{T}\to\mathbb{R}$ that is $G$-equivariant with respect to
$\Phi$ and $j$. Given a boundary point $\xi\in\partial^{\ast}\mathbb{T}$, let
$\iota:G.\xi\to\mathbb{R}$ be an order-preserving map which is $G$-equivariant
with respect to $\Phi$ and $\varphi$. For $v\in\mathbb{T}$, recall that we
denote $\partial U_{v}\subset\partial^{\ast}\mathbb{T}$ the shadow of $v$,
consisting of points $\xi\in\partial^{\ast}\mathbb{T}$ such that
$\xi\triangleleft v$. Set $I_{v}=\overline{\iota\Big{(}G.\xi\cap\partial
U_{v}\Big{)}}$, which is a compact interval. Choose a point
$v_{0}\in\mathbb{T}$. The collection $\\{I_{v}\colon v_{0}\trianglelefteq
v\\}$ is a totally ordered family of intervals which exhausts $\mathbb{R}$.
For $x\in\mathbb{R}$, set $v_{x}=\inf_{\triangleleft}\\{v\triangleright
v_{0}\colon x\in I_{v}\\}$. Let $I_{v_{0}}=[a,b]$ and note that if $x\leq y$
both lie in $[b,+\infty)$ then we have $v_{x}\trianglelefteq v_{y}$, and if
they lie in $(-\infty,a]$ we have $v_{y}\trianglelefteq v_{y}$. Now define
$h_{+}$ and $h_{-}$ by letting $h_{+}(x)=\pi(v_{x})$ for $x\in[b,+\infty)$ and
$h_{-}(x)=\pi(v_{x})=(-\infty,a]$ (extend them arbitrarily to the rest of
$\mathbb{R}$ by ensuring that $h_{+}$ is non-decreasing and $h_{-}$ is non-
increasing). Then $h_{+}$ and $h_{-}$ satisfy the desired properties. ∎

###  3.4. Abelian horogradings and invariant $\mathbb{R}$-tree metrics

Recall that a _metric_ $\mathbb{R}$_-tree_ is a metric space $(X,d)$ where
every pair of points $v,w\in X$ can be joined by a unique arc, and this arc
can be chosen to be a geodesic. We say that a directed tree
$(\mathbb{T},\triangleleft)$ has a _compatible_ metric if $\mathbb{T}$ is
endowed with an $\mathbb{R}$-tree metric where the subsets of the form $[v,w]$
associated to $\triangleleft$, coincide with the geodesic segments associated
to the $\mathbb{R}$-tree metric. In [BMRT, Section 8.1.4] it is shown that
every directed tree can be endowed with such a compatible $\mathbb{R}$-tree
metric. However, in the presence of an action
$\Phi:G\to\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft)$ it is not
always possible to find $\Phi$-invariant compatible metrics. The following
proposition describes when it is the case in terms of the existence of abelian
horogradings. Its proof is roughly contained in [BMRT, Proposition 8.34] and
[BMRT, Remark 8.30] up to minor terminological differences. This point of view
will be helpful in Section 8.

######  Proposition 3.19.

Let $\Phi:G\to\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft)$ be a
focal action on a directed tree. Then, $\Phi$ preserves a compatible metric on
$(\mathbb{T},\triangleleft)$ if and only if $\Phi$ can be horograded by an
action by translations $j\colon G\to(\mathbb{R},+)$. Moreover, if $\Phi$ is
simplicial then $\Phi$ is horograded by a cyclic action.

###### Sketch of proof.

First suppose that $\Phi$ is horograded by an action
$j:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ whose image consists on
translations, and denote $\pi:\mathbb{T}\to\mathbb{R}$ its associated
horograding. Then, define the distance $d$ on $\mathbb{T}$ so that

$d(w_{1},w_{2})=\Big{(}\pi(w_{1}\curlywedgeuparrow
w_{2})-\pi(w_{1})\Big{)}+\Big{(}\pi(w_{1}\curlywedgeuparrow
w_{2})-\pi(w_{2})\Big{)}.$

It is direct to check that $d$ is a compatible metric on $\mathbb{T}$. Also
notice that, since $j$ is an action by translations, $d$ is $\Phi$-invariant.

For the converse, suppose $\Phi$ preserves a compatible metric $d$ on
$(\mathbb{T},\triangleleft)$. Denote by $l_{d}(w_{1},w_{2})$ the length
induced by $d$ on the geodesic $[w_{1},w_{2}]$. Fix $v\in\mathbb{T}$,
$x\in\mathbb{R}$ and define $\pi:\mathbb{T}\to\mathbb{R}$ as

$\pi(w)=x+l_{d}([v,v\curlywedgeuparrow w])-l_{d}(w,v\curlywedgeuparrow w).$

Notice that $\pi$ is the unique horograding satisfying:

  1. (1)

$\pi(v)=x$

  2. (2)

the restriction of $\pi$ to any ray of the form $[w,\omega[$ is an order-
preserving isometry.

By considering all possible values of $x\in\mathbb{R}$ we deduce that there
exists exactly a one-parameter family of horogradings that satisfy condition
(2), and that all these horogradings can be obtained as $T_{y}\circ\pi$ where
$T_{y}:\mathbb{R}\to\mathbb{R}$ is the translation by $y\in\mathbb{R}$. Since
$\Phi$ preserves $d$, $\pi\circ\Phi(g)$ satisfies condition (2) for every
$g\in G$. Putting all together we deduce that, for every $g\in G$, there exist
a unique $t_{g}\in\mathbb{R}$ such that $\pi\circ\Phi(g)=T_{t_{g}}\circ\pi$.
Thus, the map $g\mapsto t_{g}$ is a morphism and $\Phi$ is horograded by the
action $j:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ defined as
$j(g)=T_{t_{g}}$.

In the particular case where $\Phi$ is simplicial, one can consider the
compatible metric such that $d(v,w)=1$ for every pair of adjacent branching
points, that is for every $v,w\in\operatorname{\mathsf{Br}}(\mathbb{T})$ so
that $]v,w[\cap\operatorname{\mathsf{Br}}(\mathbb{T})=\varnothing$. Since
$\Phi$ is simplicial this metric is $\Phi$ invariant. Finally, it is direct to
check that

$t_{g}=l_{d}([v,v\curlywedgeuparrow g.v])-l_{d}([g.v,v\curlywedgeuparrow
g.v]),$

which belongs to $\mathbb{Z}$. This implies that $\Phi$ is horograded by a
cyclic action as desired. ∎

##  4\. Elliptic subgroups in focal actions

In this section we explain a technical mechanism that allows, given a
d-minimal action $\Phi\colon
G\to\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft)$ on a directed tree
and a suitable normal subgroup $N$ of $G$, to “mod out $N$” and construct an
action of $G/N$ on a smaller directed tree. This mechanism will be an
important tool to study $\mathbb{R}$-focal actions of solvable groups, by
successively simplifying them.

###  4.1. Elliptic subgroups

######  Definition 4.1.

An automorphism $g\in\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft)$ of
a directed tree is _elliptic_ if for every $v\in\mathbb{T}$ verifying
$g.v\trianglelefteq v$, one has $g.v=v$. A subgroup
$G\subseteq\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft)$ of
automorphisms is elliptic, if it contains only elliptic elements. Similarly,
when $\Phi:G\to\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft)$ is an
action such that $\Phi(G)$ is elliptic, we say that $\Phi$ is elliptic.

######  Remark 4.2.

When the directed tree $(\mathbb{T},\triangleleft)$ is simplicial (or more
generally, when an action preserves an $\mathbb{R}$-tree metric), the
definition above corresponds to the classical notion of elliptic isometry.
Indeed, for isometric actions on simplicial trees, every automorphism is
either elliptic (which is equivalent to having a fixed point in $\mathbb{T}$)
or _hyperbolic_. Recall that hyperbolic automorphisms admit a unique
translation axis (an embedded line on which the automorphism acts by
translation), and in the case of directed trees, the axis contains the focus
in its boundary.

######  Proposition 4.3.

Let $G\subseteq\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft)$ be a
focal subgroup of automorphisms of a directed tree, whose action is either
simplicial or d-minimal. Then any normal subgroup of $G$ is either focal or
elliptic.

###### Proof.

Let $N\lhd G$ be a normal subgroup. Assume first that the action of $G$ is
simplicial. We distinguish two cases according to whether the image of $N$
contains or not hyperbolic automorphisms of $\mathbb{T}$ (see Remark 4.2). If
$N$ does not contain any hyperbolic element, then it is elliptic. If $N$ does
contain a hyperbolic element, let $\mathbb{S}$ be the union of the axes of its
hyperbolic elements. Note that $\mathbb{S}$ is a directed subtree, and the
action of $N$ on $\mathbb{S}$ is focal. As $N$ is normal, $\mathbb{S}$ is also
$G$-invariant. Finally, take a point $v\in\mathbb{S}$ and note that as $G$ is
focal, for any $u\in\mathbb{T}$ there exists $g\in G$ such that
$g.u\in[v,\omega[\subset\mathbb{S}$. This shows that $\mathbb{T}=\mathbb{S}$,
and we have already observed that the action of $N$ on $\mathbb{T}=\mathbb{S}$
is focal, as desired.

Consider next the case where the action of $G$ is d-minimal and assume by
contradiction that $N$ is neither elliptic nor focal.

######  Claim.

For any $v\in\mathbb{T}$ and $v^{\prime}\triangleleft v$, there exist $k\in N$
and $w\in[v^{\prime},v[$ such that $v\triangleleft k.w$.

###### Proof.

Since $N$ is not elliptic, we can choose a point $v_{0}\in\mathbb{T}$ and
$h\in N$ such that $v_{0}\triangleleft h.v_{0}$. By d-minimality, for given
$v^{\prime}\triangleleft v$, we can find $w\in[v^{\prime},v[$ and $g\in G$
such that $g.[w,v]\subset\,]v_{0},h.v_{0}[$. By the choice of $h$, we must
then have $g.v\triangleleft h.v_{0}\triangleleft hg.w$. Applying $g^{-1}$, we
get $v\triangleleft g^{-1}hg.w$. As $N$ is normal and $h\in N$, the element
$k:=g^{-1}hg\in N$ is the element we were looking for. ∎

Now since $N$ is not focal, there exists $u\in\mathbb{T}$ such that
$N.u\cap[u,\omega[$ is upper bounded, and set
$v=\sup_{\triangleleft}(N.u\cap[u,\omega[)$. After the claim, we can take
$k\in N$ and $w\in[u,v[$ such that $v\triangleleft k.w$. Note that the above
relation holds for any $w^{\prime}\in[w,v]$. In particular, for any
$w^{\prime}\in N.u\cap[w,\omega[$, we will have $v\triangleleft k.w^{\prime}$,
and as $k.w^{\prime}\in N.u\cap[u,\omega[$, this contradicts the choice of
$v$. ∎

From the previous proposition, we obtain the following.

######  Corollary 4.4.

Let $G\subseteq\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft)$ be a
focal subgroup of automorphisms of a directed tree, and let $N$ be a normal
subgroup of $G$ whose centralizer in $G$ is non-trivial. Assume further that
the action of $G$ is either simplicial or d-minimal, and that
$\mathbb{T}\neq\mathbb{R}$. Then, $N$ is elliptic.

The proof of the corollary is also based on the next general result for focal
subgroups.

######  Lemma 4.5.

Let $G\subseteq\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft)$ be a
focal subgroup of automorphisms of a directed tree $\mathbb{T}\neq\mathbb{R}$.
Assume that the action of $G$ is either $d$-minimal or simplicial. Let $H<G$
be a focal subgroup of $G$. Then the centralizer of $H$ in $G$ is trivial.

###### Proof.

By contradiction, we assume there exists a non-trivial automorphism $k\in G$
centralizing $G$. We first claim that there exists a point $v\in\mathbb{T}$
with $v$ and $k.v$ that are not $\triangleleft$-related. To see this, assume
first that the action of $G$ is $d$-minimal. This assumption, together with
the fact that $\mathbb{T}\neq\mathbb{R}$, implies that every non-empty open
path $]v,w[\subset\mathbb{T}$ contains branching points. Choose
$w\in\mathbb{T}$ such that $k.w\neq w$. If $w,h.w$ are not
$\triangleleft$-related, we choose $v=w$. Else assume, say, that
$w\triangleleft k.w$, and let $v_{0}\in]w,k.w[$ be a branching point. Let
$v\triangleleft v_{0}$ lie in a different path-component of
$\mathbb{T}\smallsetminus\\{v_{0}\\}$ as $w$. Then $v,k.v$ are not
$\triangleleft$-related. If the action is simplicial, choose
$w\in\operatorname{Br}(\mathbb{T})$ such that $k.w\neq w$, and let
$v_{1},v_{2}$ be two distinct branching points below $w$. Then at least one
between $v=v_{1}$ or $v=v_{2}$ must be such that $k.v$ and $v$ are not
$\triangleleft$-related.

Let now $v$ be as in the claim. Write $w:=v\curlywedgeuparrow k.v$. Since $H$
is focal we can take $g\in H$ so that $g.v\triangleright w$. On the one hand,
since $v$ and $k.v$ are not $\triangleleft$-related, neither are $g.v$ and
$gk.v$. Thus, since $k.v\triangleleft w\triangleleft g.v$, we conclude that
$k.v$ and $gk.v$ are non $\triangleleft$-related. On the other hand, since
$v\triangleleft w\triangleleft g.v$ we have that $k.v\triangleleft kg.v=gk.v$
which contradicts the previous conclusion. ∎

###### Proof of Corollary 4.4.

Combine Lemma 4.5 with Proposition 4.3. ∎

###  4.2. Modding out elliptic normal subgroups

The aim of this section is to explain that if $\Phi\colon
G\to\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft)$ is an action and
$N$ is a normal elliptic subgroup, then $N$ can be mod out to obtain a new
action of $G/N$ on a directed tree (Proposition 4.10 below). This is quite
intuitive: the idea is to consider the action on the quotient $\mathbb{T}/N$.
The issue however, is that the space $\mathbb{T}/N$ might not be a directed
tree. Fortunately, this can be readily solved by passing to a slightly smaller
quotient. This the goal of the discussion below.

######  Definition 4.6.

We say that a poset $(\mathbb{T},\triangleleft)$ is a _pre-directed tree_ if
it satisfies conditions (T1) and (T3) in the definition of directed tree and

  1. (T2’)

for every pair of points $u,v\in\mathbb{T}$ there is $w\in\mathbb{T}$ such
that $u\triangleleft w$ and $v\triangleleft w$.

We define d-minimality for actions on pre-directed tree in the same way as in
the case of directed trees.

######  Remark 4.7.

As an example, think of the space obtained by gluing two copies of
$\mathbb{R}$ along the open ray $(0,+\infty)$. Although we did not consider
any topology on a directed tree, one may think of pre-directed tree as a
directed tree which is possibly not Hausdorff.

######  Definition 4.8.

Let $(\mathbb{T}_{1},\triangleleft)$ and $(\mathbb{T}_{2},\triangleleft)$ be
pre-directed trees. We say that a map $\pi:\mathbb{T}_{1}\to\mathbb{T}_{2}$ is
a _grading_ if it is surjective and for every $v,w\in\mathbb{T}_{1}$ such that
$v\triangleleft w$, we have that $\pi(v)\triangleleft\pi(w)$ and the
restriction of $\pi$ to $[v,w]$ is an order-preserving isomorphism onto
$[\pi(v),\pi(w)]$. If in addition, for $i\in\\{1,2\\}$, one has that
$\Phi_{i}\colon G\to\operatorname{\mathsf{Aut}}(\mathbb{T}_{i},\triangleleft)$
are order-preserving actions admitting a $G$-equivariant grading from
$\mathbb{T}_{1}$ onto $\mathbb{T}_{2}$, we say that $\Phi_{1}$ can be _graded_
by $\Phi_{2}$.

######  Lemma 4.9.

Let $\Phi:G\to\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft)$ be an
action on a pre-directed tree. Then, there exists an action
$\widehat{\Phi}:G\to\operatorname{\mathsf{Aut}}(\widehat{\mathbb{T}},\triangleleft)$
on a directed tree, grading $\Phi$. Moreover, if $\Phi$ is d-minimal so is
$\widehat{\Phi}$.

###### Proof.

Given $v\in\mathbb{T}$, write $\widehat{v}:=\\{w\in\mathbb{T}:v\triangleleft
w\\}$ and $\widehat{\mathbb{T}}:=\\{\widehat{v}:v\in\mathbb{T}\\}$. We set
$\widehat{v}\triangleleft\widehat{w}$ if and only if
$\widehat{v}\supsetneq\widehat{w}$. Equivalently,
$\widehat{w}\triangleleft\widehat{v}$ if and only if $w\in\widehat{v}$. It is
direct to check that if $\Sigma\subset\mathbb{T}$ is a countable subset given
by condition (T3) for $(\mathbb{T},\triangleleft)$, then the subset
$\widehat{\Sigma}:=\left\\{\widehat{v}\in\widehat{\mathbb{T}}:v\in\Sigma\right\\}$
also satisfies condition (T3). In order to check condition (T1), for any
$v\in\mathbb{T}$ write

(4.1)
$L_{v}=\left\\{\widehat{w}\in\widehat{\mathbb{T}}:\widehat{v}\triangleleft\widehat{w}\right\\}=\left\\{\widehat{w}\in\widehat{\mathbb{T}}:w\in\widehat{v}\right\\}.$

From this, it follows that the poset $(\widehat{\mathbb{T}},\triangleleft)$
satisfies condition (T1) in the definition of directed tree and that the map
$\pi:\mathbb{T}\to\widehat{\mathbb{T}}$ defined by $\pi(v)=\widehat{v}$ is a
grading. Notice that the action $\Phi$ projects to an action
$\widehat{\Phi}:G\to\operatorname{\mathsf{Aut}}(\widehat{\mathbb{T}},\triangleleft)$
that grades $\Phi$ through the projection $\pi$. It also follows from (4.1)
that whenever $\Phi$ is d-minimal, so is $\widehat{\Phi}$. It remains to show
that $(\widehat{\mathbb{T}},\triangleleft)$ satisfies condition (T2) in the
definition of directed tree. For this, note that by condition (T2’), given
$v_{1},v_{2}\in\mathbb{T}$, there exists $w\in\mathbb{T}$ such that either
$\widehat{v_{1}}\cap\widehat{v_{2}}=\widehat{w}$ or
$\widehat{v_{1}}\cap\widehat{v_{2}}=\widehat{w}\cup\\{w\\}$. In both cases,
the set of common upper bounds of $\widehat{v_{1}}$ and $\widehat{v_{2}}$
equals $L_{{v_{1}}}\cap L_{{v_{2}}}=L_{w}\cup\\{\widehat{w}\\}$ and therefore
$\widehat{w}$ is the least common upper bound of $\widehat{v_{1}}$ and
$\widehat{v_{2}}$, as wanted. ∎

######  Proposition 4.10.

Let $\Phi\colon G\to\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft)$ be
an action on a directed tree. Assume that $N\unlhd G$ is a normal subgroup
whose action is elliptic. Then, $\Phi$ can be graded by an action
$\widehat{\Phi}\colon
G\to\operatorname{\mathsf{Aut}}(\widehat{\mathbb{T}},\triangleleft)$ on a
directed tree where $N$ acts trivially. Moreover, if $\Phi$ is d-minimal so is
$\widehat{\Phi}$.

###### Proof.

We will first show that $\Phi$ is graded by an action on a pre-directed tree
where $N$ acts trivially. For this, consider
$\widehat{\mathbb{T}}_{0}=\mathbb{T}/\Phi(N)$, namely the quotient space of
$\mathbb{T}$ under the action of $N$, and denote by
$\pi\colon\mathbb{T}\to\widehat{\mathbb{T}}_{0}$ the quotient projection. We
define a partial order on $\widehat{\mathbb{T}}_{0}$ by declaring
$x_{1}\trianglelefteq x_{2}$ if there exists $v_{i}\in\pi^{-1}(x_{i})$ for
$i\in\\{1,2\\}$, with $v_{1}\trianglelefteq v_{2}$. In order to check that the
relation is transitive, take $x_{1},x_{2},x_{3}\in\widehat{\mathbb{T}}_{0}$
satisfying $x_{1}\trianglelefteq x_{2}\trianglelefteq x_{3}$. By definition of
the partial order there are points $v_{i}\in\pi^{-1}(x_{i})$ for
$i\in\\{1,2\\}$ and $w_{i}\in\pi^{-1}(x_{i})$ for $i\in\\{2,3\\}$ such that
$v_{1}\trianglelefteq v_{2}$ and $w_{2}\trianglelefteq w_{3}$. On the other
hand, since the points $v_{2}$ and $w_{2}$ are in the same $N$-orbit, there
exists $h\in N$ such that $h.w_{2}=v_{2}$. Thus, we have $v_{1}\trianglelefteq
h.w_{2}\trianglelefteq h.w_{3}$ and, since $h.w_{3}$ projects to $x_{3}$, we
get $x_{1}\trianglelefteq x_{3}$, as desired. To check antisymmetry of
$\triangleleft$, suppose that for $x_{1},x_{2}\in\widehat{\mathbb{T}}_{0}$ it
holds $x_{1}\trianglelefteq x_{2}$ and $x_{2}\trianglelefteq x_{1}$. In this
case, again by the definition of the partial order, there exist
$v_{i},w_{i}\in\pi^{-1}(x_{i})$ for $i\in\\{1,2\\}$, such that
$v_{1}\trianglelefteq v_{2}$ and $w_{2}\trianglelefteq w_{1}$. Take
$h_{1},h_{2}\in N$ so that $w_{i}=h_{i}.v_{i}$ for $i\in\\{1,2\\}$. Then,
applying $h_{2}^{-1}$, we get $v_{1}\trianglelefteq v_{2}\trianglelefteq
h_{2}^{-1}h_{1}.v_{1}$. Since $h_{2}^{-1}h_{1}$ is elliptic, these must be
equalities, implying that $v_{1}=v_{2}$ and therefore $x_{1}=x_{2}$. This
finishes the proof that $(\widehat{\mathbb{T}}_{0},\triangleleft)$ is a poset.

We proceed to check that $(\widehat{\mathbb{T}}_{0},\triangleleft)$ satisfies
condition (T1) in the definition of pre-directed tree. Recall that for given
$x\in\widehat{\mathbb{T}}_{0}$, we write
$[x,\widehat{\omega}[\,=\\{y\in\widehat{\mathbb{T}}_{0}\colon x\trianglelefteq
y\\}$. We want to show that $[x,\widehat{\omega}[$ is order-isomorphic to the
half-line $[0,+\infty[$. To see this, take $v\in\pi^{-1}(x)$ and denote by
$\pi_{0}:[v,\omega[\to\widehat{\mathbb{T}}$ the restriction of the projection
$\pi$ to the line $[v,\omega[$. In order to prove the claim we will show that
$\pi_{0}$ is an order-preserving bijection onto $[x,\widehat{\omega}[$. Note
that, by construction, $\pi_{0}$ is order-preserving and takes values in
$[x,\widehat{\omega}[$. From the assumption that the action of $N$ is
elliptic, we get that $\pi_{0}$ is injective. Let us prove that $\pi_{0}$ is
surjective. For this, take $x^{\prime}\in[x,\widehat{\omega}[$. Then, there
exist $w^{\prime}\in\pi^{-1}(x^{\prime})$, $w\in\pi^{-1}(x)$ with
$w\trianglelefteq w^{\prime}$, and $h\in N$ with $h.w=v$. Since
$v=h.w\trianglelefteq h.w^{\prime}$ we deduce that
$h.w^{\prime}\in[v,\omega[$, and
$\pi_{0}(h.w^{\prime})=\pi_{0}(w^{\prime})=x^{\prime}$, as desired. This
proves condition (T1). Condition (T3) in the definition of pre-directed tree
follows taking a countable subset $\Sigma_{0}\subset\mathbb{T}$ and
considering the subset $\widehat{\Sigma}_{0}:=\pi(\Sigma_{0})$. Finally, for
condition (T2’), take $x_{1},x_{2}\in\widehat{T}_{0}$, and for $i\in\\{1,2\\}$
take $v_{i}\in\pi^{-1}(x_{i})$. As $\mathbb{T}$ is a directed tree, the least
common upper bound $v_{1}\curlywedgeuparrow v_{2}\in\mathbb{T}$ exists, and
set $y=\pi(v_{1}\curlywedgeuparrow v_{2})\in\mathbb{T}_{0}$. By the definition
of the partial order on $\mathbb{T}_{0}$, we see that $y$ is a common upper
bound for $x_{1}$ and $x_{2}$. Thus, we have proved that
$(\widehat{\mathbb{T}}_{0},\triangleleft)$ is a pre-directed tree. Notice that
the action $\Phi$ projects to an action on $\widehat{\mathbb{T}}_{0}$ which
clearly preserves $\triangleleft$. Denote by
$\widehat{\Phi}_{0}:G\to\operatorname{\mathsf{Aut}}(\widehat{\mathbb{T}}_{0},\triangleleft)$
this action and notice that $N$ is contained in its kernel. It follows from
the proof that $(\widehat{\mathbb{T}}_{0},\triangleleft)$ is a pre-directed
tree, that the projection map $\pi:\mathbb{T}\to\widehat{\mathbb{T}}_{0}$ is a
$G$-equivariant and surjective grading. Moreover, it follows directly from the
definitions that in the case where $\Phi$ is d-minimal, so is
$\widehat{\Phi}_{0}$.

Finally, by applying Lemma 4.9 we get a directed tree
$(\widehat{\mathbb{T}},\triangleleft)$ together with an action
$\widehat{\Phi}:G\to\operatorname{\mathsf{Aut}}(\widehat{\mathbb{T}},\triangleleft)$
which grades $\widehat{\Phi}_{0}$. Moreover, Lemma 4.9 also implies that if
$\widehat{\Phi}_{0}$ is d-minimal so is $\widehat{\Phi}$. Notice that, since
$N$ is contained in the kernel of $\widehat{\Phi}_{0}$ and gradings are
surjective, $N$ is also contained in the kernel of $\widehat{\Phi}$. Finally,
since the composition of gradings is a grading, the result follows. ∎

##  5\. $\mathbb{R}$-focal actions of solvable groups

###  5.1. Actions of solvable groups on directed trees

The results in the previous sections imply the following for actions of
solvable groups on directed trees. Recall that given a virtually solvable
group, we denote by $\operatorname{\mathsf{Fit}}(G)$ its Fitting subgroup and
by $\operatorname{\mathsf{vf}}(G)$ its virtual Fitting length (see §2.2).

######  Theorem 5.1.

Let $G$ be a virtually solvable group, and let $\Phi\colon
G\to\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft)$ be a d-minimal
action on a directed tree $\mathbb{T}\neq\mathbb{R}$. Then $\Phi$ can be
horograded by a minimal action $j\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ such that
$\operatorname{\mathsf{Fit}}(G)\subset\operatorname{\mathsf{ker}}j$.

###### Proof.

Since $\operatorname{\mathsf{Fit}}(G)$ is the union of nilpotent normal
subgroups, and the action of nilpotent normal subgroups is elliptic by
Corollary 4.4, we deduce that that the action of
$\operatorname{\mathsf{Fit}}(G)$ must be elliptic. Thus, applying Proposition
4.10 to $N=\operatorname{\mathsf{Fit}}(G)$ we obtain that $\Phi$ is graded by
a d-minimal action
$\Phi_{0}:G\to\operatorname{\mathsf{Aut}}(\mathbb{T}_{0},\triangleleft)$ where
$\operatorname{\mathsf{Fit}}(G)$ acts trivially. Since $\Phi_{0}$ is
d-minimal, $G/\operatorname{\mathsf{Fit}}(G)$ must be infinite and therefore
$\operatorname{\mathsf{vf}}(G/\operatorname{\mathsf{Fit}}(G))=\operatorname{\mathsf{vf}}(G)-1\geq
1$. In particular $\operatorname{\mathsf{vf}}(G)\geq 2$. We proceed with the
proof by induction on $\operatorname{\mathsf{vf}}(G)$. Assume that
$\operatorname{\mathsf{vf}}(G)=2$ and consider the d-minimal action
$\Phi_{0}:G\to\operatorname{\mathsf{Aut}}(\mathbb{T}_{0},\triangleleft)$ with
$\operatorname{\mathsf{Fit}}(G)\subset\operatorname{\mathsf{Ker}}(\Phi_{0})$
grading $\Phi$. Since
$\operatorname{\mathsf{vf}}(G/\operatorname{\mathsf{Fit}}(G))=1$ it must hold
that $\mathbb{T}_{0}=\mathbb{R}$ and therefore the conclusion follows by
setting $j=\Phi_{0}$. For the inductive step, consider a d-minimal action
$\Phi:G\to\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft)$ and its
corresponding grading action
$\Phi_{0}:G\to\operatorname{\mathsf{Aut}}(\mathbb{T}_{0},\triangleleft)$. We
distinguish two cases according to whether $\mathbb{T}_{0}=\mathbb{R}$ or not.
In the former case the conclusion follows by setting $j=\Phi_{0}$. In the
latter case, we can apply the induction hypothesis to $\Phi_{0}$ since it
factors through $G/\operatorname{\mathsf{Fit}}(G)$ which has strictly smaller
virtual Fitting length. Therefore $\Phi_{0}$ is horograded by an action
$j\colon G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ such that
$\operatorname{\mathsf{Fit}}(G)\subset\operatorname{\mathsf{ker}}j$ and the
same holds for $\Phi$. ∎

###  5.2. Actions of virtually solvable groups on the line

In [BMRT, Proposition 7.18] we observed that if $G$ is a finitely generated
solvable group, then its minimal actions on the line satisfy a dichotomy: they
are either conjugate to an affine action, or $\mathbb{R}$-focal. Combining
this result with Theorem 5.1, we obtain the following, which is the pivotal
result of this paper.

######  Theorem 5.2.

Let $G$ be a finitely generated virtually solvable group, and let
$\varphi\colon G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ be a minimal
action. Then either

  1. (1)

$\varphi$ is conjugate to an affine action, or

  2. (2)

$\varphi$ is $\mathbb{R}$-focal and can be horograded by an action $j\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ which factors through
$G/\operatorname{\mathsf{Fit}}(G)$. Moreover this action $j$ is either minimal
or cyclic.

###### Proof.

Let $\varphi$ be a minimal action. It was already shown in [BMRT, Proposition
7.18] that $\varphi$ conjugate to an affine action or $\mathbb{R}$-focal ($G$
is supposed there to be solvable but the proof works with minimal changes if
$G$ is virtually solvable). Let us repeat a sketch for completeness, and refer
to [BMRT, Proposition 7.18] for more details. Upon replacing $G$ by a quotient
(which remains virtually solvable), we can suppose that $\varphi$ is faithful.
Let $A$ be a an abelian normal subgroup of $G$. Then
$\operatorname{\mathsf{Fix}}^{\varphi}(A)=\varnothing$, by minimality. We
distinguish two cases according to whether $A$ acts freely or not. If $A$ acts
freely, then by Hölder’s theorem its action is semi-conjugate to an action by
translations associated to an embedding $\iota\colon A\to(\mathbb{R},+)$ (see
[GOD, Example 3.1.6]). If $\iota(A)$ is dense in $(\mathbb{R},+)$, then
$\varphi(A)$ has a unique non-empty closed minimal invariant subset
$\Lambda\subseteq\mathbb{R}$. As $A$ is normal, $\Lambda$ is preserved by
$\varphi(G)$, and thus by minimality $\Lambda=\mathbb{R}$ and $\varphi(A)$ is
actually conjugate to $\iota(A)$, and after conjugating we can assume
$\varphi(A)=\iota(A)$. Then the Lebesgue measure is the unique $A$-invariant
Radon measure up to a positive constant. It follows that every element of
$\varphi(G)$ sends the Lebesgue measure to a multiple of itself, and thus is
an affine map. In the case where $\iota(A)$ is cyclic, we have that
$\varphi(A)$ is conjugate to a cyclic group of translations centralized by
$\varphi(G)$, and $\varphi$ descends to a minimal action on the topological
circle $\mathbb{R}/\varphi(A)$. Since $G$ is amenable, this action preserves a
probability measure, and thus $\varphi$ preserves a non-zero Radon measure on
$\mathbb{R}$. In this case it follows that $\varphi$ is conjugate to an action
by translations, thus also affine. Assume now that $A$ acts non-freely; then
there exists $a\in A\smallsetminus\\{\mathsf{id}\\}$ such that
$\operatorname{\mathsf{Fix}}^{\varphi}(a)\neq\varnothing$. But
$\operatorname{\mathsf{Fix}}^{\varphi}(a)$ is an $A$-invariant closed subset
(as all elements of $A$ commute with $a$), and thus it must accumulate to both
$\pm\infty$. Let $I$ be a bounded connected component of
$\operatorname{\mathsf{Supp}^{\varphi}}(a)$. Then for every $g\in G$, $g.I$ is
a connected component of $\operatorname{\mathsf{Supp}^{\varphi}}(gag^{-1})$,
and since $gag^{-1}\in A$ commutes with $a$, we have that $g.I$ and $I$ cannot
cross. Thus the closure of the orbit of $I$ in $\mathbb{R}^{(2)}$ is a
$G$-invariant lamination $\mathcal{L}$, and since $\varphi$ is minimal it is
automatically $\mathbb{R}$-focal with respect to $\mathcal{L}$ [BMRT,
Proposition 7.7]. This shows that $\varphi$ is either conjugate to an affine
action, or it is $\mathbb{R}$-focal.

Now in the $\mathbb{R}$-focal case, by Proposition 3.15 we have that $\varphi$
can be represented by a focal action $\Phi\colon
G\to\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft,\prec)$ on a planar
directed tree, which is either d-minimal or simplicial and transitive on
branching points. If $\Phi$ is d-minimal, then by Theorem 5.1, $\Phi$ can be
horograded by an action $j\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ which satisfies the desired
conclusion. If it is simplicial, we have from Proposition 3.19 that $\Phi$ can
be horograded by a cyclic action $j\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$. Then every element $g\in
G$ with $g\notin\operatorname{\mathsf{ker}}j$ is a hyperbolic isometry. As
argued at the beginning of the proof of Theorem 5.1, Corollary 4.4 gives that
$\Phi(\operatorname{\mathsf{Fit}}(G))$ is elliptic, so that
$\operatorname{\mathsf{Fit}}(G)\subset\operatorname{\mathsf{ker}}j$. ∎

A crucial aspect of Theorem 5.2 is that if $\varphi$ falls in case (2), then
we may apply again the theorem to the action $j$ which horogrades $\varphi$.
Since this action factors through a quotient of smaller virtual Fitting
length, in finitely many steps we must reach an affine action, which remembers
some of the dynamics of the original action. This shows that actions of a
virtually solvable group on the line are naturally organized into finitely
many layers of complexity, the affine actions being the simplest type, and one
might study them inductively.

As a first simple illustration of this philosophy we prove Theorem B, which is
its most direct consequence. Indeed the combination of Theorem 5.2 and
Proposition 3.18 gives the following.

######  Proposition 5.3.

Let $G$ be a finitely generated virtually solvable group and
$\varphi\colon\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ a minimal
$\mathbb{R}$-focal action. Then, there exists an irreducible affine action
$\psi\colon G\to\operatorname{\mathsf{Aff}}(\mathbb{R})$ and two monotone maps
$h_{+},h_{-}\colon\mathbb{R}\to\mathbb{R}$ satisfying the following
conditions.

  1. (1)

$h_{+}$ is monotone non-decreasing,

  2. (2)

$h_{-}$ is monotone non-increasing,

  3. (3)

we have $\lim_{x\to+\infty}h_{+}(x)=\lim_{x\to-\infty}h_{-}(x)=+\infty$,

  4. (4)

for every $g\in G$ there exists $M>0$ such that
$h_{+}(\varphi(g)(x))=\psi(g)(h_{+}(x))$ for every $x>M$, and
$h_{-}(\varphi(g)(x))=\psi(g)(h_{-}(x))$ for every $x<-M$.

###### Proof.

We only discuss the existence of $h_{+}$, the other case being analogous. We
argue by induction on the virtual Fitting length
$\operatorname{\mathsf{vf}}(G)$ of $G$. Let $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ be a minimal
$\mathbb{R}$-focal action. Apply Theorem 5.2 to obtain an action $j\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ as in the statement that
horogrades $\varphi$. We apply Proposition 3.18 to obtain a map
$h^{\prime}_{+}$ as in its statement. If $j$ is an affine action (which
includes the possibility that it is cyclic), then we have already reached the
conclusion. Else, $j$ is $\mathbb{R}$-focal, and by the inductive hypothesis
for $G/\operatorname{\mathsf{Fit}}(G)$, there exists an irreducible affine
action $\psi$ and a non-decreasing map
$h^{\prime\prime}_{+}\colon\mathbb{R}\to\mathbb{R}$ satisfying
$\lim_{x\to+\infty}h^{\prime\prime}_{+}(x)=+\infty$ and
$h^{\prime\prime}_{+}(j(g)(x))=\psi(g)(h^{\prime\prime}_{+}(x))$ for any
sufficiently large $x\in\mathbb{R}$. We then set
$h_{+}=h^{\prime\prime}_{+}\circ h^{\prime}_{+}$. ∎

###### Proof of Theorem B.

If the action $\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is
semi-conjugate to an affine action, we simply have to take this action as the
desired $\psi$. Otherwise, by Theorem 5.2, we have that $\varphi$ is semi-
conjugate to a minimal $\mathbb{R}$-focal action, so that we can apply
Proposition 5.3, which gives the desired irreducible affine action $\psi$.

We next prove that $\psi$ is unique up to affine conjugacy. For this, assume
there exist two such irreducible affine actions $\psi_{1}$ and $\psi_{2}$,
with corresponding maps $h_{1}:I_{1}\to\mathbb{R}$ and
$h_{2}:I_{2}\to\mathbb{R}$. Then, for any $g\in G$ and sufficiently large
$x\in\mathbb{R}$, we have that

$\psi_{2}(g)h_{2}h_{1}^{-1}(x)=h_{2}\varphi(g)h_{1}^{-1}(x)=h_{2}h_{1}^{-1}\psi_{1}(g)(x),$

so that the actions $\psi_{1}$ and $\psi_{2}$ are “semi-conjugate near
$+\infty$” by the map $h:=h_{2}\circ h_{1}^{-1}$, defined on some interval of
the form $(c,+\infty)$. For $i\in\\{1,2\\}$ we write $A_{i}=\psi_{i}(G)$.
Observe first that if $\psi_{1}(g)$ is trivial for some $g$, then
$\psi_{2}(g)$ is an affine map fixing the image of $h$, and thus is trivial as
well. Thus
$\operatorname{\mathsf{ker}}\psi_{1}=\operatorname{\mathsf{ker}}\psi_{2}$. It
follows that if one of the groups $A_{i}$ is a cyclic group (necessarily
generated by a translation), then the other is as well, and this case is
readily solved as any two positive translations are conjugate by an affine
map. We can thus assume that $A_{i}$ is not a cyclic group of translations
(and thus acts minimally on $\mathbb{R}$).

Given a finite subset $S$ of homeomorphisms of the line and a point
$x\in\mathbb{R}$, denote by $\mathcal{R}(S,x)$ the equivalence relation on the
interval $(x,+\infty)$ defined by $(y,z)\in\mathcal{R}(S,x)$ if there exist
$s_{1},\ldots,s_{n}\in S$ such that $s_{n}\cdots s_{1}(y)=z$ and $s_{i}\cdots
s_{1}(y)>x$ for every $i\in\\{1,\ldots,n\\}$.

######  Claim.

For $i\in\\{1,2\\}$, there exists $S_{i}\subset A_{i}$ such that for every
sufficiently large $x\in\mathbb{R}$, the relation $\mathcal{R}(S_{i},x)$ is
minimal (i.e. its equivalence classes are dense in $(x,+\infty)$).

###### Proof of claim.

Assume first that $A_{i}$ is contained in the group of translations. Since it
is finitely generated and not cyclic, it contains two rationally independent
translations $a,b$, and one readily checks that $\mathcal{R}(\\{a^{\pm
1},b^{\pm 1}\\},x)$ is minimal for every $x\in\mathbb{R}$. Otherwise $A_{i}$
contains a contracting homothety $a$ and a positive translation $b$. Up to
conjugating by an affine map, assume that $a(x)=\lambda x$ with $\lambda<1$
and $b(x)=x+1$. For $x>0$ and $n,m>0$ we have
$a^{m}b^{n}a^{-m}(x)=x+n\lambda^{m}$ and one readily checks that the
successive applications of $a^{\pm 1},b$ in the word $a^{m}b^{n}a^{-m}$ never
move $x$ to the left of itself. Since $\lambda^{m}$ can be made arbitrarily
small, we deduce that $\mathcal{R}(\\{a^{\pm 1},b\\},x)$ is minimal for every
$x>0$. ∎

Let now $\tilde{S}_{i}\subset G$ be finite subsets such that
$\psi_{i}(\tilde{S}_{i})=S_{i}$. Choose $x$ large enough so that
$\mathcal{R}(S_{1},x)$ and $\mathcal{R}(S_{2},h(x))$ are both minimal, and
such that the equivariance condition $h\circ\psi_{1}(s)(y)=\psi_{2}(s)\circ
h(y)$ holds for all $y\geq x$ and all $s\in\tilde{S}_{1}\cup\tilde{S_{2}}$.
Then the image of $h|_{(x,+\infty)}$ is invariant under the relation
$\mathcal{R}(S_{2},h(x))$. Thus, arguments similar to those in Lemma 2.3 show
that this image must be dense, implying that $h$ is continuous on
$(x,+\infty)$. Again as in the proof of Lemma 2.3, the minimality of
$\mathcal{R}(S_{1},x)$ shows that $h|_{(x,+\infty)}$ must be injective and
thus a homeomorphism onto its image.

Now note that since
$K:=\operatorname{\mathsf{ker}}\psi_{1}=\operatorname{\mathsf{ker}}\psi_{2}$
we have that $A_{1}$ is abelian (and hence a group of translations) if and
only if $A_{2}$ is. Otherwise, both groups are non-abelian and their derived
subgroup consists of translations. In the first case, set $H=G$ and in the
second case set $H=[G,G]$. In either case $\psi_{i}(H)$ is a minimal group of
translations for $i\in\\{1,2\\}$. Let $g\in H$ be such that $\psi_{1}(g)$ is a
non-trivial positive translation. Then so is $\psi_{2}(g)$, by the
equivariance near $+\infty$ of the map $h$. Up to conjugating $\psi_{2}$ by an
affine map we can assume $\psi_{1}(g)=\psi_{2}(g)$. Let
$J_{1}\subset(x,+\infty)$ be a fundamental interval for the translation
$\psi_{1}(g)$ and set $J_{2}=h(J_{1})$, which is also a fundamental interval
for the translation $\psi_{1}(g)=\psi_{2}(g)$. Therefore, up to conjugating
the action $\psi_{2}$ by a translation, we can assume that $J_{1}=J_{2}$. Then
for any $i\in\\{1,2\\}$, $\psi_{i}|_{H}$ descends to a minimal actions by
rotations on the circle $J_{i}/\psi_{i}(g)$. Moreover, the homeomorphism
$h:(c,+\infty)\to(h(c),+\infty)$ induces a homeomorphism $\overline{h}$ of
$J_{1}/\psi_{1}(g)=J_{2}/\psi_{2}(g)$ which conjugates the two actions. This
implies that $\overline{h}$ is a rotation, and thus $h$ is a translation. As
$h(J_{1})=J_{1}$, we deduce that $h$ is trivial. It follows that for every
$g\in G$, the affine maps $\psi_{1}(g)$ and $\psi_{2}(g)$ coincide on a
neighborhood of $+\infty$, and thus are equal. ∎

For later use, let us record the following consequence of Proposition 5.3.

######  Lemma 5.4.

Let $G$ be a finitely generated virtually solvable group and let
$\varphi\colon G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ be a minimal
$\mathbb{R}$-focal action. Then there exists an element $h\in G$ whose image
is a pseudohomothety.

###### Proof.

Let $\psi\colon G\to\operatorname{\mathsf{Aff}}(\mathbb{R})$ be the affine
action given by Proposition 5.3. Choose $g\in G$ with
$\psi(g)\neq\mathsf{id}$. Assume, say, that $\psi(g)(x)>x$ for every
sufficiently large $x$. Then $\varphi(g)(x)>x$ for all $x$ close enough to
$+\infty$ and similarly $\varphi(g)(x)<x$ for all $x$ close enough to
$-\infty$. Thus $\varphi(g)$ is a pseudohomothety. ∎

##  6\. The case of $\mathbb{Z}\wr\mathbb{Z}$

The goal of this section is to apply the previous discussion to the
lamplighter group $\mathbb{Z}\wr\mathbb{Z}$. This gives an illustration of how
Theorem 5.2 can be applied in the simplest special case, and at the same time
the results of this section will be invoked in the proof of Theorem A.

We set $G:=\mathbb{Z}\wr\mathbb{Z}=L\rtimes\mathbb{Z}$, where
$L=\bigoplus_{\mathbb{Z}}\mathbb{Z}$ is the group of finitely supported
configurations $f\colon\mathbb{Z}\to\mathbb{Z}$, and $\mathbb{Z}$ acts on $L$
by shifting the indices. We consider the generating set
$\mathbb{Z}\wr\mathbb{Z}=\langle g,h_{0}\rangle$, where $g$ generates the
factor $\mathbb{Z}$, and $h_{0}\in L$ is the configuration given by
$h_{0}(0)=1$ and $h_{0}(n)=0$ for $n\neq 0$. We further denote by
$h_{n}=g^{n}h_{0}g^{-n}$ ($n\in\mathbb{Z}$) the elements of the canonical
basis of $L$. The group $G$ admits the infinite presentation:

(6.1) $G=\langle g,h_{0}\mid
h_{n}=g^{n}h_{0}g^{-n},\,[h_{n},h_{m}]=\mathsf{id}\quad(n,m\in\mathbb{Z})\rangle.$

The group $G$ is solvable (more precisely, metabelian) and it is clear that
$L=\operatorname{\mathsf{Fit}}(G)$.

###  6.1. Affine actions

For completeness, let us first recall that the group $G$ admits a continuous
family of affine actions on $\mathbb{R}$, which are readily classified:

######  Lemma 6.1.

Let $G=\mathbb{Z}\wr\mathbb{Z}$ with the standard generating pair $h_{0},g$
defined above. Then every irreducible affine action $\varphi\colon
G\to\operatorname{\mathsf{Aff}}(\mathbb{R})$ is positively conjugate by an
affine map to an action in one of the following families:

  * •

actions by translations obtained by setting

$\varphi(h_{0})\colon x\mapsto x+\alpha,\quad\varphi(g)\colon
x+\beta\quad\quad(\alpha^{2}+\beta^{2}=1);$

  * •

non-abelian affine actions obtained by setting

$\varphi(h_{0})\colon x\mapsto x\pm 1,\quad\varphi(g)\colon x\mapsto\lambda
x\quad\quad(\lambda>0,\lambda\neq 1).$

###### Proof.

First observe that the images of the generators under the actions above
satisfy the relations in (6.1), so that these actions are well-defined.
Conversely, assume $\varphi$ is an affine action of $G$. Assume by
contradiction that $\varphi(h_{0})$ is a homothety, with fixed point $p$. Then
$p$ cannot be fixed by $\varphi(g)$ or it would be a global fixed point. Thus
$\varphi(h_{1})$ is a homothety with fixed point $g.p\neq p$. This contradicts
that $\varphi(h_{0})$ and $\varphi(h_{1})$ commute. Thus $\varphi(h_{0})$ is a
translation (possibly trivial). If $\varphi(g)$ is also a translation, up to
conjugation by an affine map we are in the first case. If $\varphi(g)$ is a
homothety, then $\varphi(h_{0})$ cannot be trivial, and we can conjugate by an
affine map to suppose that $\varphi(h)$ is a translation by $\pm 1$ and that
$\varphi(g)$ fixes 0. ∎

######  Remark 6.2.

In the previous classification, actions by translations are obviously never
faithful, while non-abelian affine actions are faithful if and only if
$\lambda$ is a trascendental number.

###  6.2. Plante’s actions and classification

We now recall Plante’s construction of actions on the line $G$ that are not
semi-conjugate to any affine action [Plante]. One can identify the subgroup
$L$ with the ring of Laurent polynomials $\mathbb{Z}[X,X^{-1}]$, in which case
$h_{0}$ represents the constant polynomial $P_{0}(X)\equiv 1$. With this point
of view, the group $G=L\rtimes\mathbb{Z}$ admits a natural action on
$\mathbb{Z}[X,X^{-1}]$, where $L\cong\mathbb{Z}[X,X^{-1}]$ acts additively,
and $g$ acts by multiplication by $X$. This action of $G$ on
$\mathbb{Z}[X,X^{-1}]$ preserve two natural orders $\prec_{+}$ and $\prec_{-}$
of lexicographic type, obtained by looking at the sign of the coefficient of
the monomial of maximal or minimal power, respectively. More precisely, we
declare $P\succ_{+}0$ if the coefficient of the highest power of $X$ in $P(X)$
is positive. Similarly, we declare $P\succ_{-}0$ if the coefficient of the
lowest power of $X$ in $P(X)$ is positive. It is routine to check that these
orders are invariant under the action of $G$ defined above. Taking the
dynamical realization of such order-preserving actions (see §2.1), one obtains
two non-conjugate minimal $\mathbb{R}$-focal actions of $G$ on the real line,
that we call the _Plante’s actions_ of $\mathbb{Z}\wr\mathbb{Z}$. Note that in
the dynamical realization of $\prec_{+}$, the generator $g$ acts as an
expanding homothety, while in case of $\prec_{-}$ it acts as a contracting
homothety. See [BMRT, Example 7.11] for more details, and for a generalization
of this construction to other wreath products. The following result (together
with Lemma 6.1) classifies actions of $\mathbb{Z}\wr\mathbb{Z}$ on the line up
to semi-conjugacy.

######  Proposition 6.3.

Every irreducible action
$\varphi\colon\mathbb{Z}\wr\mathbb{Z}\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$
is semi-conjugate either to an affine action or to a Plante’s action.

###### Proof.

Set $G=\mathbb{Z}\wr\mathbb{Z}$ and $L=\bigoplus_{\mathbb{Z}}\mathbb{Z}$, as
before. By Theorem 5.2, it is enough to show that every minimal
$\mathbb{R}$-focal action $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is conjugate to one of the
two Plante’s actions. So assume that $\varphi$ is $\mathbb{R}$-focal and
minimal. By Theorem 5.2, $\varphi$ can be horograded by an action
$j:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ which is either cyclic
or minimal, and where $\operatorname{\mathsf{Fit}}(G)$ acts trivially. Since
$\operatorname{\mathsf{Fit}}(G)=L$ and $G/L\cong\mathbb{Z}$, we deduce that
$j$ is a cyclic action. Moreover, since $j(g)$ generates the image of $g$, it
has no fixed points, and thus $\varphi(g)$ is a homothety by Proposition 3.17.
Also notice that, again by Proposition 3.17, the elements $\varphi(h_{n})$
($n\in\mathbb{Z}$) are totally bounded. Assume first that $\varphi(g)$ is
expanding, and let $p$ be its unique fixed point. Note that no element
$\varphi(h_{n})$ ($n\in\mathbb{Z}$) can fix $p$, as otherwise $p$ would be a
global fixed point for the action. Thus, upon conjugating $\varphi$ by an
orientation-reversing homeomorphism, we can assume that $h_{0}.p>p$ and thus
$h_{n}.p>p$ for every $n\in\mathbb{Z}$. For fixed $n\in\mathbb{Z}$, let
$I_{n}$ be the connected component of
$\operatorname{\mathsf{Supp}^{\varphi}}(h_{n})$ containing $p$. Then, for
every $n\in\mathbb{Z}$, the homothety $\varphi(g)$ sends $I_{n}$ to $I_{n+1}$
and thus $\overline{I_{n}}\subset I_{n+1}$; moreover for each $l<n$ the
element $h_{n}$ must send $I_{l}$ entirely to the right of itself, as it must
permute the connected components of
$\operatorname{\mathsf{Supp}^{\varphi}}(h_{l})$. From this, it easily follows
that for every element $f=h_{n_{1}}^{k_{1}}\cdots h_{n_{r}}^{k_{r}}$ with
$n_{1}<\cdots<n_{r}$ and $k_{r}\neq 0$, we have $f.p>p$ if $k_{r}>0$, and
$f.p<p$ otherwise. Since $g.p=p$, it follows that the the map that to any
element $f\in L\cong\mathbb{Z}[X,X^{-1}]$ associates $f.p$ is a
$G$-equivariant order-preserving embedding of
$(\mathbb{Z}[X,X^{-1}],\prec_{+})$ into $\mathbb{R}$, and thus $\varphi$ must
be conjugate with the Plante’s action associated with $\prec_{+}$ by Lemma
2.3. The case where $\varphi(g)$ is a contracting homothety is treated
similarly and gives rise to the Plante’s action associated with $\prec_{-}$.
Summarizing we get that, up to conjugating by an order-reversing
homeomorphism, $\varphi$ is either semi-conjugate to the Plante’s action
associated with $\prec_{+}$ or to that associated with $\prec_{-}$. ∎

###  6.3. Plante subactions in $\mathbb{R}$-focal actions

Plante’s actions of $\mathbb{Z}\wr\mathbb{Z}$ are in a precise sense the
“smallest” $\mathbb{R}$-focal actions for solvable groups:

######  Proposition 6.4.

Let $G$ be a virtually solvable group and let $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ be a minimal and faithful
$\mathbb{R}$-focal action. Then $G$ contains a subgroup isomorphic to
$\mathbb{Z}\wr\mathbb{Z}$ whose action is irreducible and semi-conjugate to a
Plante’s action.

###### Proof.

By Lemma 5.4 there exists an element $s\in G$ which acts as an expanding
pseudohomothety. Let $[\xi_{1},\xi_{2}]$ be an interval containing
$\operatorname{\mathsf{Fix}}^{\varphi}(s)$. Let $A$ be a non-trivial abelian
normal subgroup of $G$. Since the action of $G$ is minimal and faithful, we
have $\operatorname{\mathsf{Fix}}^{\varphi}(A)=\varnothing$, and and hence
there exists $a\in A$ such that $a.\xi_{0}>\xi_{1}$. Then
$\operatorname{\mathsf{Fix}}^{\varphi}(a)\cap\operatorname{\mathsf{Fix}}^{\varphi}(s)=\varnothing$
and thus $H:=\langle a,s\rangle$ acts irreducibly. The conjugates of $a$ by
powers of $s^{n}$ all belong to $A$ and thus commute. It follows that there is
well-defined homomorphism $\rho\colon\mathbb{Z}\wr\mathbb{Z}\to H$ sending
$h_{0}$ to $a$ and $g$ to $s$. Now $\varphi\circ\rho$ cannot be semi-conjugate
to an affine action (for instance because $\varphi(G)$ does not contain any
element conjugate to a translation, by Proposition 3.17), and thus it is semi-
conjugate to a Plante’s action by Proposition 6.3; we have a fortiori that
$\rho$ is an isomorphism, as Plante’s actions are faithful. ∎

##  7\. $C^{1}$ actions on intervals

In this section we prove Theorem A. After Proposition 6.4, we are reduced to
consider actions which are semi-conjugate to Plante’s actions of
$\mathbb{Z}\wr\mathbb{Z}$. Let us outline the main idea, which comes from the
work of Bonatti, Monteverde, Navas, and the third author [BMNR]. It is proven
there that for any $C^{1}$ action of the Bausmlag–Solitar group
$\operatorname{\mathsf{BS}}(1,n)=\langle a,b\mid aba^{-1}=b^{n}\rangle$ on the
closed interval, which is topologically conjugate to its standard affine
action, the derivative of the image of $a$ at its unique fixed point must be
equal to $n$. Observe that the affine action of
$\operatorname{\mathsf{BS}}(1,n)$ can also be seen as an action of
$\mathbb{Z}\wr\mathbb{Z}$, by precomposing it with the epimorphism
$\mathbb{Z}\wr\mathbb{Z}\to\operatorname{\mathsf{BS}}(1,n)$ defined on the
generators as $h_{0}\mapsto b$ and $g\mapsto a$. Letting $n$ increase we
obtain a sequence $(\varphi_{n})$ of actions of $\mathbb{Z}\wr\mathbb{Z}$. One
can show that the sequence $(\varphi_{n})$ can be conjugated to make converge
to a Plante’s action of $\mathbb{Z}\wr\mathbb{Z}$. This strongly suggests that
if Plante’s action of $\mathbb{Z}\wr\mathbb{Z}$ were conjugate to a $C^{1}$
action, the derivative of $g$ at its unique fixed point should be infinite,
thus getting a contradiction. While this limiting argument is not an actual
proof (as the conjugacy and the convergence are only $C^{0}$), it turns out
that the argument of in the proof of [BMNR, Proposition 4.13] can be adapted
to Plante’s actions to obtain the following.

######  Proposition 7.1.

Let $\rho:H\to\mathbb{Z}\wr\mathbb{Z}$ be an epimorphism from a finitely
generated group $H$, and let $\psi\colon
H\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ be the action obtained by
postcomposing $\rho$ with a Plante’s action. Let $\varphi\colon
H\to\operatorname{\mathsf{Homeo}}_{0}([0,1])$ be an action whose restriction
to $(0,1)$ is semi-conjugate to $\psi$. Then, $\varphi$ cannot be of class
$C^{1}$ on $[0,1]$.

###### Proof of Proposition 7.1.

Assume by contradiction that $\varphi$ is of class $C^{1}$. As $H$ is finitely
generated, using a trick attributed to Muller [Muller] and Tsuboi [Tsuboi]
(see also [BMNR]), we can assume, up to considering a $C^{0}$ conjugate action
of class $C^{1}$, that $\varphi(k)^{\prime}(0)=\varphi(k)^{\prime}(1)=1$ for
every $k\in H$. Recall that we denote by $h_{0}$ and $g$ the standard
generators of $\mathbb{Z}\wr\mathbb{Z}$, and $h_{n}=g^{n}h_{0}g^{-n}$. For
$n\in\mathbb{Z}$ we consider the subgroups $L_{n}:=\langle h_{j},j\geq
n\rangle$ of $\mathbb{Z}\wr\mathbb{Z}$. Choose corresponding preimages
$\widetilde{h}_{0}\in\rho^{-1}(h_{0})$ and $\widetilde{g}\in\rho^{-1}(g)$ in
$H$, and set
$\widetilde{h}_{n}=\widetilde{g}^{n}\widetilde{h}_{0}\widetilde{g}^{-n}$.
Assume that $\psi$ is the Plante’s action from the lexicographic order
$\prec_{+}$, so that $\psi(\widetilde{g})$ is an expanding homothety and
$\psi(\widetilde{h}_{0})$ moves to the right (the other case can be treated
identically). Let $\tau\colon(0,1)\to\mathbb{R}$ be a non-decreasing map that
semi-conjugates $\varphi$ to $\psi$, in the sense that
$\tau\varphi(k)=\psi(k)\tau$ for every $k\in G$. As Plante’s actions are
minimal (see §6), the semi-conjugacy $\tau$ is continuous. To avoid confusion,
we will use Latin letters to denote points of $\mathbb{R}$ (where the Plante’s
action lives), and Greek letters for points of $[0,1]$ (where the action
$\varphi$ lives). Let $x_{0}\in\mathbb{R}$ be the unique fixed point of
$\psi(g)$. Its preimage $\tau^{-1}(x_{0})$ in $(0,1)$ is a compact interval
(possibly reduced to one point), and we let $\xi_{0}$ denote its rightmost
point. Note that $\xi_{0}$ is fixed by $\phi(\widetilde{g})$ and that
$\widetilde{g}.\xi>\xi$ for any $\xi>\xi_{0}$. Consider the interval
$I_{0}=(x_{0},h_{0}.x_{0})$, and for $n>0$ set
$I_{-n}=g^{-n}.I_{0}=(x_{0},h_{-n}.x_{0})$. Observe that if $f_{1},f_{2}\in
L_{-n}$ are distinct, then the intervals $f_{1}.I_{-n-1}$ and $f_{2}.I_{-n-1}$
are disjoint (this follows from the fact that the interval $I_{-n-1}$ is
contained in a connected component of
$\operatorname{\mathsf{Supp}}^{\psi}(h_{-n-1})$, and that the subgroup
$L_{-n}$ preserves the set of such connected components and acts freely on
it). Set $J_{0}$ as the minimal open interval such that $\tau(J_{0})=I_{0}$,
and define $J_{-n}=\widetilde{g}^{-n}.J_{0}$. Note that $J_{-n}$ has $\xi_{0}$
as leftmost point and $\tau.J_{-n}=I_{-n}$ for $n\geq 0$.

Choose $0<\lambda<|D\varphi(\widetilde{g})(\xi_{0})|^{-1}$. Then there exists
a constant $C>0$ such that

(7.1) $|J_{-n}|\geq C\lambda^{n}.$

Fix $N>\frac{1}{\lambda}$ and let $\varepsilon>0$ be such that

(7.2) $(1-\varepsilon)^{2N+2}N\lambda>1.$

Let $\eta\in(0,1)$ be such that for any $\xi\in(\eta,1]$ and
$s\in\left\\{\widetilde{h}_{0},\widetilde{g},\widetilde{g}^{-1}\right\\}$, one
has $\varphi(s)^{\prime}(\xi)>1-\varepsilon$. Since the subgroup $L$ acts
without fixed points in the Plante’s action, we can find an element $a\in L$
such that $a.x_{0}>\tau(\eta)$; choosing a preimage
$\widetilde{a}\in\rho^{-1}(a)$ we have $\widetilde{a}.\xi_{0}>\eta$. Given
$N\in\mathbb{N}$ and an $n$-tuple of integers
$\underline{i}=(i_{0},\ldots,i_{n-1})\in\\{1,\ldots,N\\}^{n}$, set
$\widetilde{f}_{\underline{i}}=\widetilde{h}_{-n+1}^{i_{n-1}}\cdots\widetilde{h}_{0}^{i_{0}}$
and ${f}_{\underline{i}}=\pi\left(\widetilde{f}_{\underline{i}}\right)$. Note
that the intervals of the form
$\widetilde{f}_{\underline{i}}\widetilde{a}.J_{-n}$ are pairwise disjoint when
$\underline{i}$ varies. Indeed, applying the semi-conjugacy $\tau$ maps each
such interval to the corresponding
$f_{\underline{i}}a.I_{-n}=af_{\underline{i}}.I_{-n}$, and these are pairwise
disjoint since the elements $f_{\underline{i}}$ all belong to the subgroup
$L_{-n-1}$. We shall estimate from below the size of
$\widetilde{f}_{\underline{i}}\widetilde{a}.J_{-n}$. For this, note that
$\widetilde{f}_{\underline{i}}$ may be rewritten as

$\displaystyle\widetilde{f}_{\underline{i}}$
$\displaystyle=\left(\widetilde{g}^{-n+1}\widetilde{h}_{0}^{i_{n-1}}\widetilde{g}^{n-1}\right)\cdots\left(\widetilde{g}^{-2}\widetilde{h}_{0}^{i_{2}}\widetilde{g}^{2}\right)\left(\widetilde{g}^{-1}\widetilde{h}_{0}^{i_{1}}\widetilde{g}\right)\widetilde{h}_{0}^{i_{0}}$
$\displaystyle=\widetilde{g}^{-n+1}\widetilde{h}_{0}^{i_{-n+1}}\widetilde{g}\widetilde{h}_{0}^{i_{-n+2}}\widetilde{g}\cdots\widetilde{g}\widetilde{h}_{0}^{i_{1}}\widetilde{g}\widetilde{h}_{0}^{i_{0}}.$

Consider the sequence of intervals obtained by applying successively the terms
in this expression to the interval $\widetilde{a}.J_{-n}$. Observe that any
such interval stays inside $(\eta,1]$: indeed, each application of the
generator $\widetilde{g}$ or $\widetilde{h}_{0}$ moves the interval to the
right, and the final term $\widetilde{g}^{-n+1}$ moves it to the left but is
compensated by the previous $n-1$ instances of $\widetilde{g}$ with a positive
power. By the mean value theorem and the choice of $\eta$, at each application
of $\widetilde{g},\widetilde{g}^{-1}$ or $\widetilde{h}_{0}$ the size of the
interval may decrease by at most a factor $1-\varepsilon$. Hence

$\left|\widetilde{f}_{\underline{i}}\widetilde{a}.J_{-n}\right|\geq(1-\varepsilon)^{nN+2n-2}\left|\widetilde{a}.J_{-n}\right|\geq
CC_{1}(1-\varepsilon)^{nN+2n-2}\lambda^{n},$

where we have used (7.1), and the constant $C_{1}$ is
$\min_{\xi\in[0,1]}\varphi(\widetilde{a})^{\prime}(\xi)$. Summing over
$\underline{i}$ and using that the intervals are disjoint, we have

$1\geq\sum_{\underline{i}\in\\{1,\ldots,N\\}^{n}}\left|\widetilde{f}_{\underline{i}}\widetilde{a}.J_{-n}\right|\geq
CC_{1}(1-\varepsilon)^{-2}\left((1-\varepsilon)^{2N+2}N\lambda\right)^{n},$

which is impossible, since by (7.2) the right-hand side tends to $+\infty$ as
$n\to\infty$. This is the desired contradiction. ∎

We will also use the following result, whose proof readily follows from the
discussion in [BMNR, §4.2].

######  Lemma 7.2.

Let $G$ be a group and $\varphi\colon
G\to\operatorname{\mathsf{Diff}}_{0}^{1}([0,1])$ be an irreducible action.
Suppose that the $\varphi$-action on $(0,1)$ is semi-conjugate to an action
$\psi\colon G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$, and that there
exist elements $a,b\in G$ such that $\psi(a)$ is a homothety $x\mapsto\lambda
x$ with $\lambda>1$ and $\psi(b)$ is a translation $x\mapsto x+\alpha$ with
$\alpha\neq 0$. Then $\varphi$ is minimal on $(0,1)$, and thus conjugate to
$\psi$.

###### Proof of Theorem A.

Let $\varphi\colon G\to\operatorname{\mathsf{Diff}}^{1}([0,1])$ be an
irreducible action. If by contradiction the restriction of $\varphi$ to
$(0,1)$ is not semi-conjugate to any affine action, by Theorem 5.2 it is semi-
conjugate to a minimal $\mathbb{R}$-focal action $\psi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$, whose kernel we denote by
$N$. By Proposition 6.4 we have that $G/N$ contains a subgroup
$\mathbb{Z}\wr\mathbb{Z}$ that acts via an action which is semi-conjugate to a
Plante’s action. If we choose two preimages
$\widetilde{h}_{0},\widetilde{g}\in G$ of the corresponding generators of
$\mathbb{Z}\wr\mathbb{Z}$, the restriction of $\varphi$ to the group
$H=\left\langle\widetilde{h}_{0},\widetilde{g}\right\rangle$ satisfies all
assumptions of Proposition 7.1, giving a contradiction. The last sentence of
the theorem follows from Lemma 7.2. ∎

##  8\. The metanilpotent case

In this section we consider the class of virtually metanilpotent groups.
Recall that $G$ is virtually metanilpotent if it admits a finite index
subgroup $G_{0}\leq G$ which is metanilpotent, meaning that there is a normal
subgroup $N\unlhd G_{0}$ such that $N$ and $G_{0}/N$ are both nilpotent.

######  Remark 8.1.

The class of virtually metanilpotent groups contains in particular metabelian
groups and all virtually solvable linear groups, i.e. virtually solvable
subgroups of $\operatorname{\mathsf{GL}}(n,\mathbb{K})$ where $\mathbb{K}$ is
a field. Indeed by a classical theorem of Mal’cev [Malcev], every solvable
subgroup of $\operatorname{\mathsf{GL}}(n,\mathbb{K})$ has a finite index
subgroup which is triangularizable over the algebraic closure of $\mathbb{K}$,
and thus is nilpotent-by-abelian.

For this class of groups, the results in Section 5 become much stronger.
Theorem 5.1 specifies as follows.

######  Corollary 8.2.

Let $G$ be a virtually metanilpotent group and $\Phi\colon
G\to\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft)$ a d-minimal action
on a directed tree. Then $\Phi$ can be horograded by an action by translations
$j\colon G\to(\mathbb{R},+)$. Equivalently, $\Phi$ preserves a compatible
$\mathbb{R}$-tree metric.

###### Proof.

Let $G_{0}$ be a normal metanilpotent subgroup of $G$. By Theorem 5.1, $\Phi$
can be horograded by a minimal action
$j:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ which factors through
$G/\operatorname{\mathsf{Fit}}(G)$. The fact that $G$ be virtually
metanilpotent implies that $G/\operatorname{\mathsf{Fit}}(G)$ is virtually
nilpotent (using Lemma 2.4). By a classical result of Plante [PlanteMeasure]
(see also Navas [Navas-book, Theorem 2.2.39]), we deduce that
$j:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is conjugate to an
action by translations. The equivalent formulation in terms of invariant
metrics is a consequence of Proposition 3.19. ∎

######  Remark 8.3.

The reader may notice that the proof above works more generally for groups
with virtual Fitting length $\operatorname{\mathsf{vf}}(G)=2$. However one can
check that for finitely generated groups, this condition is equivalent to
being virtually metanilpotent.

As a consequence, Theorem 5.2 translates here as follows.

######  Corollary 8.4.

. Let $G$ be a finitely generated, virtually metanilpotent group. Then every
minimal action $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is either conjugate to an
affine action, or it is $\mathbb{R}$-focal and can be horograded by an action
by translations $j\colon G\to(\mathbb{R},+)$.

It is interesting to note that the previous results imply that actions on the
line of a virtually metanilpotent group always arise from an isometric action
on a planar hyperbolic metric space, namely the hyperbolic plane, in the case
of affine actions, or an $\mathbb{R}$-tree, in the case of $\mathbb{R}$-focal
actions.

###  8.1. Perturbations of affine actions

Here we prove Theorem C. It will follow from the results above and from some
general results on perturbations of affine actions of a finitely generated
group.

######  Proposition 8.5.

Let $G$ be a finitely generated group. Then for any irreducible action
$\varphi_{0}\in\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
which is semi-conjugate to a non-abelian affine action, there exists an open
neighborhood in
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
which contains no action which is semi-conjugate to an action by translations.

###### Proof.

Let $S$ be a finite symmetric generating set of $G$ containing the identity.
As $\varphi_{0}$ is semi-conjugate to a non-abelian affine action
$\overline{\varphi}_{0}\colon G\to\operatorname{\mathsf{Aff}}(\mathbb{R})$, we
can choose $s\in S$ such that $\overline{\varphi_{0}}(s)$ is an expanding
homothety. Thus $\varphi_{0}(s)$ is an expanding pseudohomothety, and let $K$
be its compact set of fixed points. Choose $\xi_{-}<\min K,\xi_{+}>\max K$.
Since $\varphi_{0}(s^{n})(\xi_{+})$ tends to $+\infty$, we can choose $n>0$
such that $\varphi_{0}(s^{n})(\xi_{+})>\max_{t\in S}\varphi_{0}(t)(\xi_{+})$.
Let $\mathcal{U}$ be the set of actions
$\psi\in\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
such that $\psi(s^{n})(\xi_{+})>\max_{t\in S}\psi(t)(\xi_{+})$ and
$\psi(s)(\xi_{-})<\xi_{-}$. Then $\mathcal{U}$ is an open neighborhood of
$\varphi_{0}$ and we claim that it satisfies the desired conclusion. Assume by
contradiction that $\psi\in\mathcal{U}$ is semi-conjugate to an action given
by a non-trivial homomorphism $\overline{\psi}\colon G\to(\mathbb{R},+)$ by
translations. Since $\psi(s)(\xi_{-})<\xi_{-}$ and $\psi(s)(\xi_{+})>\xi_{+}$,
we must have $\overline{\psi}(s)=0$. On the other hand since $S$ is symmetric,
there exists $t\in S$ such that $\overline{\psi}(t)>0$. For such a $t$ we must
have $\psi(t)(\xi_{+})<\psi(s^{n})(\xi_{+})$ for every $n\in\mathbb{Z}$, a
contradiction with the definition of $\mathcal{U}$. ∎

######  Lemma 8.6.

Let $G$ be a finitely generated group, and let $S\subset G$ be a finite
symmetric generating subset. Let
$\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ be a minimal
$\mathbb{R}$-focal action horograded by an action by translations
$j:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$. Then, for every element
$g\in\operatorname{\mathsf{ker}}j$ and $p\in\mathbb{R}$, there exists $h\in
S\cup\\{sgs^{-1}:s\in S\\}$ such that

$h^{-1}.p<g^{n}.p<h.p$

for every $n\in\mathbb{Z}$.

###### Proof.

Take an element $g\in\operatorname{\mathsf{ker}}j$ and note that by
Proposition 3.17, $\varphi(g)$ is totally bounded. Fix a point
$p\in\mathbb{R}$. If $g.p=p$ the lemma follows from the fact that $\varphi$
has no global fixed points. Suppose now that $g.p\neq p$ and let
$J\subset\operatorname{\mathsf{Supp}^{\varphi}}(g)$ be the connected component
containing $p$. By Proposition 3.19, we can find an action
$\Phi:G\to\operatorname{\mathsf{Aut}}(\mathbb{T},\triangleleft,\prec)$ of $G$
on a planar directed tree representing $\varphi$ and which is isometric for a
compatible $\mathbb{R}$-tree metric. Denote by
$\sigma\colon(\partial^{*}\mathbb{T},\prec)\to(\mathbb{R},<)$ the map given in
Lemma 3.14 and set $U_{z}:=\\{w\in\partial^{\ast}\mathbb{T}\colon
w\triangleleft z\\}$. Set also $\tilde{J}=\sigma^{-1}(J)$ and define
$v:=\min\\{w\in\mathbb{T}:\tilde{J}\subseteq U_{w}\\}.$ In order to show that
$v$ is well defined, notice that the interval $\tilde{J}$ is bounded above and
below in $\partial^{*}\mathbb{T}$, thus we can choose
$\xi_{0},\xi_{1}\in\partial^{*}\mathbb{T}$ such that
$\xi_{0}\prec\xi\prec\xi_{1}$ for every $\xi\in\tilde{J}$. Then the point
$w=\xi_{0}\curlywedgeuparrow\xi_{1}$ satisfies $\tilde{J}\subseteq U_{w}$, as
it follows from the compatibility condition (CC) for the order $\prec$.
Finally, the existence of a $\triangleleft$-smallest point follows easily from
the definition of directed tree. Also notice that, by the $\Phi(g)$-invariance
of $\tilde{J}$ and the construction of $v$, it holds that $g.v=v$.

######  Claim 1.

Suppose that $w\triangleleft v$ and $g.w=w$. Then, the intersection
$U_{w}\cap\tilde{J}$ is empty.

###### Proof of claim.

Suppose by contradiction that there exists $\xi_{0}\in U_{w}\cap\tilde{J}$.
Note that, after the choice of $v$, the subset $U_{w}$ cannot contain
$\tilde{J}$. Therefore, we can consider $\xi_{1}\in\tilde{J}\smallsetminus
U_{w}$. Assume without loss of generality that $\xi_{0}\prec\xi_{1}$. By
$\prec$-convexity of $U_{w}$, we have that $U_{w}\prec\xi_{1}$. Since $\sigma$
is non-decreasing (Lemma 3.14), it follows that
$\sigma(U_{w})\leq\sigma(\xi_{1})$. Therefore, we conclude that
$\sup\sigma(U_{w})\in J$. On the other hand, since $g.w=w$ we have that
$g.\sigma(U_{w})=\sigma(U_{w})$ and therefore $\sup\sigma(U_{w})$ is a fixed
point of $\varphi(g)$ inside $J$, which is a contradiction. ∎

######  Claim 2.

Suppose that for some $h\in G$ one of the following holds:

  1. (1)

either $v$ and $h.v$ are $\prec$-related, or

  2. (2)

$h.v\triangleleft v$ and $g.(h.v)=h.v$.

Then, up to changing $h$ with its inverse, it holds that
$h^{-1}.p<g^{n}.p<h.p$ for every $n\in\mathbb{Z}$.

###### Proof of claim.

Suppose first that (1) holds. Up to changing $h$ with its inverse, we can
assume that $h^{-1}.v\prec v\prec h.v$. Therefore, the compatibility condition
(CC) implies that $U_{h^{-1}.v}\prec U_{v}\prec U_{h.v}$, which in turn
implies that $h^{-1}.\tilde{J}\prec\tilde{J}\prec h.\tilde{J}$. Thus, since
$\sigma$ is non-decreasing we obtain that $h^{-1}.J<J<h.J$ which concludes the
proof, since $J$ is the component of the support of $\varphi(g)$ containing
$p$. Assume next that (2) holds. By Claim 1, we have
$U_{h.v}\cap\tilde{J}=\varnothing$ is empty and therefore, since
$\tilde{J}\subset U_{v}$, we get that $h.\tilde{J}\cap\tilde{J}=\varnothing$.
Since $\tilde{J}$ is $\prec$-convex, up to changing $h$ with its inverse, we
can assume that $h^{-1}.\tilde{J}\prec\tilde{J}\prec h.\tilde{J}$. Therefore,
applying $\sigma$ we get that $h^{-1}.J<J<h.J$. ∎

Let $s\in S$ be a generator such that $j(s)$ is a positive translation. If
$s.v$ and $v$ are $\prec$-related, then we can conclude the proof using Claim
2, as condition (1) is satisfied. In the remaining case, since $j(s)$ is a
positive translation, we must have $v\triangleleft s.v$. Assume first that
$sgs^{-1}.v\neq v$. In this case, since $\Phi$ preserves an $\mathbb{R}$-tree
metric and $s.v\in\mathsf{Fix}^{\Phi}(sgs^{-1})$, it must hold that
$sgs^{-1}.v$ and $v$ are $\prec$-related. Thus, we conclude again using Claim
2, as condition (1) is satisfied. For the remaining case, suppose that
$sgs^{-1}.v=v$. Rewrite the equality as $gs^{-1}.v=s^{-1}.v$ and notice that
$s^{-1}.v\triangleleft v$. In this case, we conclude using Claim 2, setting
$h=s^{-1}$ in condition (2). ∎

######  Proposition 8.7.

Let $G$ be a finitely generated group. Let
$\mathcal{X}\subset\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
be the set of minimal actions of $G$ which are $\mathbb{R}$-focal and can be
horograded by an action by translations. Let $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ be an action which is semi-
conjugate to a non-abelian affine action. Then $\varphi$ has a neighborhood
$\mathcal{U}$ such that no element of $\mathcal{U}$ is semi-conjugate to an
action in $\mathcal{X}$.

###### Proof.

Let $\varphi_{0}:G\to\mathbb{R}$ be an action which is semi-conjugate to a
non-abelian affine action $\bar{\varphi}_{0}$, fix a point $p\in\mathbb{R}$.
and a finite symmetric generating subset $S\subset G$. Since $G$ is non-
abelian, there exists $g\in[G,G]$ such that $\bar{\varphi}_{0}(g)$ acts as a
translation, and thus $\varphi_{0}(g)$ acts without fixed points. We can
assume without loss of generality that $g.p>p$. Thus, for any finite subset
$F\subset G$, there exists $K=K_{F}\in\mathbb{N}$ such that for any $f\in F$
we have

$\varphi_{0}(g)^{-K}(p)<\varphi_{0}(f)(p)<\varphi_{0}(g^{K})(p).$

This is an open condition in the space of actions
$\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ with respect to
the compact-open topology, so we can find a neighborhood $\mathcal{U}_{F}$ of
$\varphi_{0}$ in the space of irreducible actions of $G$ such that the
condition above is satisfied. Assume
$\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is an action which
is semi-conjugate to an action $\bar{\varphi}\in\mathcal{X}$. Using Lemma 8.6,
we find an element $h\in F:=S\cup\\{sgs^{-1}:s\in S\\}$ such that
$\bar{\varphi}(h)(\bar{p})<\bar{\varphi}(g)^{n}(\bar{p})$ for any
$n\in\mathbb{Z}$. Therefore the analogue condition is satisfied for the action
$\varphi$: $\varphi(h)(p)<\varphi(g)^{n}(p)$ for any $n\in\mathbb{Z}$. We
deduce that $\varphi\notin\mathcal{U}_{F}$. As the finite subset $F$ does not
depend on $\varphi$, this gives that $\mathcal{U}_{F}$ contains no action
which is semi-conjugate to an action in $\mathcal{X}$. ∎

###### Proof of Theorem C.

Since $G$ is finitely generated, every
$\varphi\in\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
is semi-conjugate to an action which is either minimal or cyclic. After
Corollary 8.4, and by splitting further the affine actions into the non-
abelian ones and actions by translations, we obtain that every
$\varphi\in\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
is semi-conjugate to either

  1. (1)

a non-abelian affine action, or

  2. (2)

an action by translations, or

  3. (3)

a minimal $\mathbb{R}$-focal action which can be horograded by an action by
translations.

By Propositions 8.5 and 8.7, every action satisfying (1) has a neighborhood
consisting of actions which do not satisfy neither (2) nor (3). Thus the set
of such actions is open. ∎

###  8.2. A non-metanilpotent example

Here we show that the conclusion of Theorem C fails for solvable groups which
are not virtually metanilpotent. More precisely, we will construct a finitely
generated 3-step solvable group $G$ and a sequence
$(\varphi_{n})\subset\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
of minimal actions which are not semi-conjugate to affine actions, but
converge to a limit
$\varphi\in\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
which is conjugate to a non-abelian affine action.

As a basis for our construction, let
$H\subseteq\operatorname{\mathsf{Aff}}(\mathbb{R})$, be a finitely generated,
non-abelian group of affine transformations acting irreducibly on
$\mathbb{R}$. A precise choice of such an $H$ will be irrelevant; for
definiteness the reader may take $H$ to be the group generated by the two
transformations

$x\mapsto ax,\quad x\mapsto x+1\quad(a>0,a\neq 1).$

We denote by $j\colon H\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ the
action given by the standard inclusion.

Let $G$ be the wreath product $G=\mathbb{Z}\wr
H:=\bigoplus_{H}\mathbb{Z}\rtimes H$. We will simply write $\mathbb{Z}^{(H)}$
for the direct sum $\bigoplus_{H}\mathbb{Z}$, and identify it with the group
of finitely supported configurations $f\colon H\to\mathbb{Z}$. The semi-direct
product is taken with respect to the left-regular action $\lambda\colon
H\to\operatorname{\mathsf{Aut}}(\mathbb{Z}^{(H)})$, given by
$\lambda_{h}(f)(k)=f(h^{-1}k)$. For later use, we also denote by $\rho\colon
H\to\operatorname{\mathsf{Aut}}(\mathbb{Z}^{(H)})$ the right-regular action,
given by $\rho_{h}(f)(k)=f(kh)$. Note that these two actions commute.

We denote by $\pi_{H}\colon G\to H$ the projection to the quotient and by
$(\delta_{h})_{h\in H}$ the standard basis of $\mathbb{Z}^{(H)}$, where
$\delta_{h}\in\mathbb{Z}^{(H)}$ is the configuration taking the value 1 on $h$
and $0$ elsewhere. The identically zero configuration of $\mathbb{Z}^{(H)}$ is
denoted $\bar{0}$. Finally for $f\in\mathbb{Z}^{(H)}$ we write
$\operatorname{\mathsf{Supp}}(f)=\\{h\in H\colon f(h)\neq 0\\}$.

Chose a point $\xi_{0}\in\mathbb{R}$ which has a free orbit for the natural
affine action $j$ of $H$, and let $<_{H}$ be the associated left-order on $H$,
given by $h_{1}<_{H}h_{2}$ if $j(h_{1})(\xi_{0})<j(h_{2})(\xi_{0})$. We can
use this order to construct an action of $G$ on $\mathbb{R}$ in a way which is
analogous to the Plante’s action (as in [BMRT, Example 7.11]). Namely,
consider the order $\prec$ on $\mathbb{Z}^{(H)}$ given by $f_{1}\prec f_{2}$
if $f_{1}(k)<f_{2}(k)$, where $k=\max_{<_{H}}\\{h\colon f_{1}(h)\neq
f_{2}(h)\\}$. Let $\alpha\colon
G\to\operatorname{\mathsf{Aut}}(\mathbb{Z}^{(H)})$ be the “affine” action
obtained by letting $H$ act via the left-regular action and $\mathbb{Z}^{(H)}$
act on itself by translations. Explicitly, for
$g=(r,h)\in\mathbb{Z}^{(H)}\rtimes H=G$ and $f\in\mathbb{Z}^{(H)}$, we have

$\alpha(g)(f)=r+\lambda_{h}(f).$

This action preserves the order $\prec$ constructed above. By taking the
dynamical realization of the action $\alpha\colon
G\to\operatorname{\mathsf{Aut}}(\mathbb{Z}^{(H)},\prec)$, we obtain an action
$\psi\colon G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$. This action is
minimal and $\mathbb{R}$-focal. A proof of this, and a more detailed
discussion, can be found in [BMRT, Example 7.11].

Let $\iota\colon\mathbb{Z}^{(H)}\to\mathbb{R}$ be the associated equivariant
order-preserving map from the dynamical realization $\psi$. We suppose that
$\iota(\overline{0})=0$. Then $0$ is a global fixed point for $\psi(H)$, and
it is moved by all elements of $\psi(\mathbb{Z}^{(H)})$. Denote by $I_{h}$ the
connected component of the support of $\psi(\delta_{h})$ containing 0. Then
each $I_{h}$ is a bounded open interval and we have $I_{h_{1}}\subset
I_{h_{2}}$ if $h_{1}<_{H}h_{2}$. Moreover since $0$ is fixed by $\psi(H)$ and
$h\delta_{k}h^{-1}=\delta_{hk}$, we have $\psi(h)(I_{k})=I_{hk}$. In
particular if we denote by $p_{h}$ the rightmost point of $I_{h}$, the map
$h\mapsto p_{h}$ is an order-preserving, $H$-equivariant embedding of
$(H,<_{H})$ into $\mathbb{R}$. Finally we observe that the point $p_{h}$ is
fixed by $\psi(\delta_{k})$ for every $k<_{H}h$. Thus if
$f\in\mathbb{Z}^{(H)}$ is such that
$\max_{<_{H}}\operatorname{\mathsf{Supp}}(f)<_{H}h$, then
$\psi(f)(p_{h})=(p_{h})$.

Let now $t\in H$ be any element such that $j(t)$ is a positive translation.
Let $\tau\colon G\to G$ be the automorphism

$\tau(f,h)=(\rho_{t}(f),h),\quad(f,h)\in\mathbb{Z}^{(H)}\rtimes H.$

The fact that $\tau$ is an automorphism follows from the fact that $\rho_{t}$