# Locally moving groups acting on the line and $\mathbb{R}$-focal actions

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

###### Abstract.

We prove various results that, given a sufficiently rich subgroup $G$ of the
group of homeomorphisms on the real line, describe the structure of the other
possible actions of $G$ on the real line, and address under which conditions
such actions must be semi-conjugate to the natural defining action of $G$. The
main assumption is that $G$ should be locally moving, meaning that for every
open interval the subgroup of elements whose support is contained in such
interval acts on it without fixed points. One example (among many others) is
given by Thompson’s group $F$. A first rigidity result implies that if $G$ is
a locally moving group, every faithful minimal action of $G$ on the real line
by $C^{1}$ diffeomorphisms is conjugate to its standard action. It turns out
that the situation is much wilder when considering actions by homeomorphisms:
for a large class of groups, including Thompson’s group $F$, we describe
uncountably many conjugacy classes of minimal faithful actions by
homeomorphisms on the real line. To gain insight on such exotic actions, we
introduce and develop the notion of $\mathbb{R}$-focal action, a class of
actions on the real line that can be encoded by certain actions by
homeomorphisms on planar real trees fixing an end. Under a suitable finite
generation condition on a locally moving group $G$, we prove that every
minimal faithful action of $G$ on the line is either conjugate to the standard
action, or it is $\mathbb{R}$-focal and the action on the associated real tree
factors via a horofunction onto the standard action of $G$ on the line. This
establishes a tight relation between all minimal actions of $G$ on the line
and its standard action. Among the various applications of this result, we
show that for a large class of locally moving groups, the standard action is
locally rigid, in the sense that every sufficiently small perturbation in the
compact-open topology gives a semi-conjugate action. This is based on an
analysis of the space of harmonic actions on the line for such groups. Along
the way we introduce and study several concrete examples.

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

###### Key words and phrases:

Group actions on the real line, locally moving groups, actions on real trees,
local rigidity, left-orderable groups, groups of piecewise linear
homeomorphisms

###### Contents

  1. 1 Introduction
    1. 1.1 Background and overview
    2. 1.2 Actions by $C^{1}$ diffeomorphisms
    3. 1.3 On existence of exotic actions by homeomorphisms
    4. 1.4 Actions on planar real trees and $\mathbb{R}$-focal actions
    5. 1.5 A structure theorem for $C^{0}$ actions
    6. 1.6 Space of semi-conjugacy classes and local rigidity for groups in $\mathcal{F}_{0}$
    7. 1.7 A criterion for non-orientation-preserving groups
    8. 1.8 Some concrete groups in the class $\mathcal{F}_{0}$
    9. 1.9 Further consequences
      1. 1.9.1 Very large locally moving groups
      2. 1.9.2 Comparison with the case of groups acting on the circle
      3. 1.9.3 An application to non-smoothability
    10. 1.10 Outline of the paper
  2. 2 Notation and preliminaries
    1. 2.1 Actions on the line
      1. 2.1.1 Some notation on actions
      2. 2.1.2 Commuting actions
      3. 2.1.3 Semi-conjugacy
      4. 2.1.4 Canonical model
    2. 2.2 Harmonic actions and Deroin spaces
    3. 2.3 Preorders and group actions
      1. 2.3.1 Preordered sets
      2. 2.3.2 Preorders on groups
      3. 2.3.3 Dynamical realizations of actions on totally ordered sets
    4. 2.4 Bieri–Strebel groups
  3. 3 Deroin spaces and preorders
    1. 3.1 Equivalence of preorders
    2. 3.2 Relations between preorders and actions on the real line
    3. 3.3 Topology of the space of equivalence classes of preorders
    4. 3.4 Retraction to the Deroin space
    5. 3.5 Application to the Borel reducibility of the semi-conjugacy relation
  4. 4 Micro-supported and locally moving groups
    1. 4.1 Definitions
    2. 4.2 Structure of normal subgroups
    3. 4.3 Subgroups isomorphic to Thompson’s group
  5. 5 Dynamical trichotomy for actions of locally moving groups on the line
    1. 5.1 The trichotomy
    2. 5.2 First consequences of the trichotomy
    3. 5.3 First examples of exotic actions
      1. 5.3.1 Orders of germ type
      2. 5.3.2 Escaping sequences
      3. 5.3.3 An example where the rigid stabilizers have no fixed points
  6. 6 Differentiable actions of locally moving groups
    1. 6.1 Conradian actions and $C^{1}$ actions
    2. 6.2 Conradian $C^{1}$ actions of Thompson’s group $F$
    3. 6.3 $C^{1}$ actions of general locally moving groups
    4. 6.4 An application to non-smoothability
      1. 6.4.1 Thompson–Brown–Stein groups
      2. 6.4.2 An application to Bieri–Strebel groups on the line
  7. 7 $\mathbb{R}$-focal actions
    1. 7.1 Cross-free covers and $\mathbb{R}$-focal actions
    2. 7.2 A condition for $\mathbb{R}$-focality
    3. 7.3 $\mathbb{R}$-focal actions and alternatives
      1. 7.3.1 $\mathbb{R}$-focal actions and micro-supported groups
      2. 7.3.2 Actions of solvable groups
  8. 8 Planar trees and horograded $\mathbb{R}$-focal actions
    1. 8.1 Preliminaries on trees
      1. 8.1.1 Directed trees
      2. 8.1.2 End-completion and boundary
      3. 8.1.3 Horogradings
      4. 8.1.4 Metric and topologies on directed trees
      5. 8.1.5 Case of simplicial trees
      6. 8.1.6 Planar directed trees
      7. 8.1.7 Automorphisms and focal actions
    2. 8.2 Planar directed trees and $\mathbb{R}$-focal actions
    3. 8.3 Focal germ representation and dynamical classification of elements
    4. 8.4 Horograded $\mathbb{R}$-focal actions
    5. 8.5 $\mathbb{R}$-focal actions arising from simplicial trees
    6. 8.6 Constructing $\mathbb{R}$-focal actions from actions on ultrametric spaces
  9. 9 Some minimal $\mathbb{R}$-focal actions
    1. 9.1 Locally moving groups with cyclic group of germs
      1. 9.1.1 Actions from escaping orbits
      2. 9.1.2 A simplicial tree and $\mathbb{R}$-focality
    2. 9.2 Orders of germ type and semidirect products
    3. 9.3 A construction of $\mathbb{R}$-focal actions for groups of PL homeomorphisms
    4. 9.4 A construction of micro-supported $\mathbb{R}$-focal actions
      1. 9.4.1 Groups with many micro-supported actions
      2. 9.4.2 Groups with many differentiable micro-supported actions
  10. 10 A structure theorem for exotic actions of a class of locally moving groups
    1. 10.1 The class $\mathcal{F}$
    2. 10.2 The main theorem for the class $\mathcal{F}$
      1. 10.2.1 Sketch of the proof and preliminary observations
      2. 10.2.2 An invariant CF-cover
      3. 10.2.3 A planar directed tree
    3. 10.3 Non-orientation-preserving actions
    4. 10.4 An attractor in the Deroin space and local rigidity
      1. 10.4.1 Description of the Deroin space
      2. 10.4.2 Application to local rigidity
      3. 10.4.3 Smoothness of the semi-conjugacy relation
  11. 11 An illustrative example: Thompson’s group $F$
    1. 11.1 Main results in the case of $F$ and outline of the section
    2. 11.2 An analysis of the Deroin space of $F$
    3. 11.3 Simplicial $\mathbb{R}$-focal actions of $F$ and minimality of $[F,F]$
    4. 11.4 A plethora of $\mathbb{R}$-focal actions I: restriction preorders
      1. 11.4.1 A reinterpretation in terms of preorders
      2. 11.4.2 Restriction preorders on $F_{+}$
      3. 11.4.3 Some properties of the actions arising from restriction preorders
      4. 11.4.4 Some variations on the restriction preorder construction
    5. 11.5 A plethora of $\mathbb{R}$-focal actions II: ordering the orbit of a closed subset of $(0,1)$
      1. 11.5.1 A strategy to order $\mathcal{O}_{K}$
      2. 11.5.2 A concrete example
    6. 11.6 A plethora of $\mathbb{R}$-focal actions III: existence of non-simplicial $\mathbb{R}$-focal actions
  12. 12 Finitely generated locally moving groups with few exotic actions
    1. 12.1 Bieri–Strebel groups over the real line
    2. 12.2 A finitely generated group in the class $\mathcal{F}_{0}$ with no $\mathbb{R}$-focal actions
  13. 13 Uncountable groups
  14. 14 A results for actions on the circle

##  1\. Introduction

###  1.1. Background and overview

The study of group actions on one-manifolds is a classical topic at the
interface of dynamical systems, topology, and group theory, which is still
under intense development and has been the object of several monographs in the
recent years, such as Ghys [Ghys], Navas [Navas-book], Clay and Rolfsen
[ClayRolfsen], Deroin, Navas, and the third named author [GOD], Kim, Koberda,
and Mj [KKMj]; see also the surveys by Mann [MannHandbook] and Navas
[NavasICM]. The ultimate goal would be, given a group $G$, to be able to
describe all possible actions of $G$ on a one-manifold $M$ by homeomorphisms,
and as the topology of the manifold is rather elementary one expects that this
is more tractable than in higher dimension. In this framework, there is no
much loss of generality of considering the following simplifying assumptions:
the manifold $M$ is connected (that is, either the real line or the circle),
and the action of the group is _irreducible_ , in the sense that it preserves
the orientation and has no (global) fixed points. We then let
$\operatorname{\mathsf{Homeo}}_{0}(M)$ be the group of orientation-preserving
homeomorphisms of $M$, and write
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(M))$
for the space of irreducible actions of $G$ on $M$. Moreover, to avoid a
redundant description, we want to identify two actions if one is obtained from
the other with a change of variables (what is called a _conjugacy_); more
precisely, from a dynamical perspective, it is natural to study the space
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(M))$
up to _semi-conjugacy_. The definition of semi-conjugacy in this setting is
recalled in Section 2; here let us simply remind that every action
$\varphi\in\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(M))$
of a _finitely generated_ group is semi-conjugate either to a _minimal_ action
(i.e. an action all whose orbits are dense) or $\varphi(G)$ has a discrete
orbit, in which case $\varphi$ is semi-conjugate to _cyclic_ action (i.e. an
action by integer translations in the case of the real line or by rational
rotations in the case of the circle); moreover this minimal or cyclic model is
unique up to conjugacy (see e.g. Ghys [Ghys]). Thus studying actions of
finitely generated groups up to semi-conjugacy is essentially the same as to
study its minimal actions up to conjugacy.

The situation is particulary nice when $M$ is a circle, for which more tools
are available. In particular the _bounded Euler class_ is an complete
algebraic invariant for classifying irreducible actions up to semi-conjugacy
(see Ghys [Ghys-bounded, Ghys]), which generalizes to arbitrary groups the
rotation number for homeomorphisms. For instance, this has been successfully
used to understand the space
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{S}^{1}))$
for various discrete subgroups of Lie groups (see for instance Burger and
Monod [BurgerMonod], Matsumoto [MatsumotoSurface], Mann [MannThesis], or Mann
and Wolff [mann2018rigidity]), mapping class groups (Mann and Wolff
[MannWolff2018mcg]) or of Thompson’s group $T$ [GhysSergiescu, Ghys]. On the
other hand there is no known complete invariant to understand semi-conjugacy
classes of actions on $M=\mathbb{R}$ (see the discussion in §1.6), and here
there are fewer groups for which the space
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
is well understood (beyond deciding whether it is empty). Most results concern
rather small groups. For example, if $G$ does not contain a free semigroup on
two generators (e.g. if $G$ has subexponential growth), then every action in
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
is semi-conjugate to an action taking values in the group of translations
$(\mathbb{R},+)$ (see Navas [Navas2010]), and actions up to semi-conjugacy can
be completely classified for some classes of solvable groups such as
polycyclic groups (Plante [Plante]) or the solvable Baumslag–Soliltar groups
$\operatorname{\mathsf{BS}}(1,n)$ (see [RivasBS]). Some further classification
results for actions on the line can also be obtained by considering lifts of
actions on the circle, but the scope of this method is rather limited: for
instance one can show that the central lifts of the actions of the whole group
$\operatorname{\mathsf{Homeo}}_{0}(\mathbb{S}^{1})$ (Militon [Militon]), or of
Thompson’s group $T$ (see [MatteBonTriestino, Theorem 8.6]) are the unique
actions of such groups on the line up to semi-conjugacy.

The main goal of this work is to prove structure theorems for actions on the
real line of various classes of groups which arise as sufficiently rich
subgroups of the group of homeomorphisms of an interval. Throughout the
introduction we fix an open interval $X=(a,b)\subseteq\mathbb{R}$ with
$a\in\mathbb{R}\cup\\{-\infty\\}$ and $b\in\mathbb{R}\cup\\{+\infty\\}$. We
are interested in studying actions on the real line and intervals of groups
$G\subset\operatorname{\mathsf{Homeo}}_{0}(X)$. The action of $G$ on $X$ will
be referred to as _the standard action_ of $G$. Of course there would be no
loss of generality in assuming $X=\mathbb{R}$, but since we will be dealing
with more general actions of $G$ on the real line it is convenient to keep a
separate notation for $X$. The precise assumptions we will make on $G$ will
depend on the situation, but the most important one is that $G$ is a _locally
moving_ group. To recall this notion, given an open interval $I\subset X$, we
denote by $G_{I}\subset G$ the subgroup consisting of elements that fix
$X\smallsetminus I$ pointwise (called the _rigid stabilizer_ of $I$).

######  Definition 1.1.

Let $X=(a,b)$ be an open interval. We say that a subgroup
$G\subset\operatorname{\mathsf{Homeo}}_{0}(X)$ is _locally moving_ if for
every $I\subset X$ the subgroup $G_{I}$ acts on $I$ without fixed points.

The most basic example of a locally moving group is
$\operatorname{\mathsf{Homeo}}_{0}(X)$ itself. However there are also many
finitely generated (and even finitely presented) groups that admit a locally
moving action on an interval: a relevant example is Thompson’s group $F$, as
well as many other related groups studied in the literature.

The main question addressed in this work is the following.

######  Main Question.

Let $G\subset\operatorname{\mathsf{Homeo}}_{0}(X)$ be a locally moving group
of homeomorphisms of an open interval $X$. What are the possible actions of
$G$ on the real line, and in particular under which conditions an action
$\varphi\in\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
must be semi-conjugate to its standard action on $X$?

Evidence towards rigidity of actions of locally moving groups on the real line
comes from a general reconstruction theorem of Rubin [Rubin, Rubin2] holding
for groups of homeomorphisms of locally compact spaces. In this (very special)
case, Rubin’s theorem implies that any group isomorphism between two locally
moving groups of homeomorphisms of intervals must be implemented by a
topological conjugacy; equivalently for a locally moving group
$G\subset\operatorname{\mathsf{Homeo}}_{0}(X)$, the standard action on $X$ is
the unique faithful _locally moving_ action of $G$. This result does not allow
to draw many conclusions on more general actions of $G$ which may fail to be
locally moving (see Lodha [Coherent] for some conditions ensuring this). Some
further evidence towards rigidity comes from the fact that the group
$\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is known to admit a unique
action on the real line up to conjugacy by a result of Militon [Militon] (see
also Mann [Mann]), and the same result was recently shown for the group of
compactly supported diffeomorphisms of $\mathbb{R}$ by Chen and Mann
[ChenMann]. However, for smaller (e.g. finitely generated) locally moving
groups very little appears to be known, even for well-studied cases such as
Thompson’s group $F$. In fact it turns out that smaller groups admit a much
rich variety of actions on the line than one might guess based on the previous
results, and satisfy a combination of rigidity and flexibility phenomena. In
particular many of them (but not all!) admit a a vast class of “exotic”
actions on the real line.

Before proceeding to discuss our main results, let us clarify what we mean (or
rather what we _don’t_ mean) by “exotic” action in this context. When
$G\subset\operatorname{\mathsf{Homeo}}_{0}(X)$ is a locally moving group
acting on $X=(a,b)$, it is often possible to obtain new actions of $G$ on the
real line by considering actions induced from _proper quotients_ of $G$. To
explain this, we denote by $G_{c}\subset G$ the normal subgroup of $G$
consisting of compactly supported elements (that is those elements that act
trivially on a neighborhood of $a$ and $b$), then a well-known simplicity
argument shows that the commutator subgroup $[G_{c},G_{c}]$ of $G_{c}$ is
simple and contained in every non-trivial normal subgroup of $G$ (see
Proposition 4.4). Thus, if $G=[G_{c},G_{c}]$ then $G$ is simple, and otherwise
the group $G/[G_{c},G_{c}]$ is the largest proper quotient of $G$. In
particular whenever $G$ is finitely generated the associated _groups of germs_
$\operatorname{\mathsf{Germ}}(G,a)$ and $\operatorname{\mathsf{Germ}}(G,b)$
are non-trivial quotients of $G$ which moreover can be faithfully represented
inside $\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ (see [Mann]). Thus
finitely generated locally moving groups admit non-trivial, yet non-faithful,
actions on the real line factoring throughout its germs111Even if one
restricts to study faithful actions, it should be kept in mind that a faithful
action can be _semi-conjugate_ to an action arising from a proper quotient.
Indeed, one can start from a non-faithful action of a countable group and
perform a “Denjoy’s blow up” by replacing each point of an orbit by an
interval, and then extend the action to these intervals in a faithful way by
identifying them equivariantly with $X$. This phenomenon can never happen for
actions which are _minimal and faithful_ , and we will often restrict to such.
. Non-faithful actions of $G$ correspond to actions of the largest quotient
$G/[G_{c},G_{c}]$ and can be studied separately. For example the largest
quotient of Thompson’s group $F$ coincides with its abelianization
$F=F/[F,F]\cong\mathbb{Z}^{2}$, whose actions on the real line are always
semi-conjugate to an action by translations arising from a homomorphism to
$(\mathbb{R},+)$. In view of this, when
$G\subset\operatorname{\mathsf{Homeo}}_{0}(X)$ is locally moving, we will
reserve the term _exotic action_ for actions
$\varphi\in\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
which are _not semi-conjugate to the standard action of $G$ on $X$, nor to any
action induced from the quotient $G/[G_{c},G_{c}]$_.

###  1.2. Actions by $C^{1}$ diffeomorphisms

We begin with a result for actions of locally moving groups on the line by
diffeomorphisms of class $C^{1}$, which states that such actions are never
exotic in the sense clarified above.

######  Theorem 1.2 ($C^{1}$ rigidity of locally moving groups).

Let $X$ be an open interval and let
$G\subset\operatorname{\mathsf{Homeo}}_{0}(X)$ be a locally moving group. Then
every irreducible action $\varphi\colon
G\to\operatorname{\mathsf{Diff}}^{1}_{0}(\mathbb{R})$ is semi-conjugate either
to the standard action of $G$ on $X$, or to a non-faithful action (induced
from an action of the largest quotient $G/[G_{c},G_{c}]$).

For actions on closed intervals one can rule out the non-faithful case under
some mild additional assumption on $G$. Given an interval $X=(a,b)$, we say
that a group $G\subset\operatorname{\mathsf{Homeo}}_{0}(X)$ has _independent
groups of germs at the endpoints_ if for every $g_{1},g_{2}\in G$, there is
$g\in G$ which coincides with $g_{1}$ on a neighborhood of $a$ and with
$g_{2}$ on a neighborhood of $b$.

######  Corollary 1.3.

Let $X$ be an open interval and let
$G\subset\operatorname{\mathsf{Homeo}}_{0}(X)$ be a locally moving group with
independent groups of germs at the endpoints. Let $\varphi\colon
G\to\operatorname{\mathsf{Diff}}^{1}([0,1])$ be a faithful action with no
fixed point in $(0,1)$. Then the $\varphi$-action of $G$ on $(0,1)$ is semi-
conjugate to its standard action on $X$.

The condition in the previous corollary is satisfied by many locally moving
groups. In particular Corollary 1.3 applies to Thompson’s group $F$. However
an interesting feature of the proof of Theorem 1.2 is that the group $F$ plays
a key role in an intermediate step. In fact this proof combines three
ingredients. The first is a general trichotomy for $C^{0}$ actions of locally
moving groups on the line (Theorem 5.3), which is the common building ground
for all the results in this paper. The second is the fact that every locally
moving group over an interval contains many copies of $F$ (Proposition 4.8),
as follows from an argument based on a presentation of $F$ going back to Brin
[Brin] and extensively exploited and popularized by Kim, Koberda, and Lodha
[KKL] under the name “2-chain lemma”. These two results together imply that if
a locally moving group $G$ admits an exotic $C^{1}$ action on the line, then
one can find an embedding of $F$ in the $C^{1}$ centralizer of a
diffeomorphism of a compact interval without fixed points in the interior. The
third and last step consists in showing that the group $F$ cannot admit such
an embedding (this uses the $C^{1}$ Sackseteder’s theorem from [DKN-acta] and
an elementary version of Bonatti’s approximate linearization [BonattiProches,
BMNR], together with algebraic properties of $F$). We note that, while the
abundance of copies of $F$ inside rich groups of homeomorphisms of the line is
a long-standing fact [Brin, KKL], this seems to have been rarely exploited to
prove general results about such groups.

###  1.3. On existence of exotic actions by homeomorphisms

The rigidity displayed in Theorem 1.2 fails in the $C^{0}$ setting. Perhaps
the simplest way to give counterexamples is to consider countable groups of
compactly supported homeomorphisms. For such groups, one can always obtain
exotic actions via two general constructions presented in §5.3.1 and §5.3.2,
which yield two proofs of the following fact.

######  Fact 1.4.

Let $X$ be an open interval, and let
$G\subset\operatorname{\mathsf{Homeo}}_{0}(X)$ be a countable group of
compactly supported homeomorphisms of $X$ acting minimally on $X$. Then $G$
admits irreducible actions on $\mathbb{R}$ which are not semi-conjugate to its
standard action on $X$, nor to any non-faithful action of $G$.

While this observation is formally sufficient to rule out the $C^{0}$ version
of Theorem 1.2, it is not fully satisfactory, for instance because a group $G$
as in Fact 1.4 cannot be finitely generated. In fact, the proofs from §5.3.1
and §5.3.2 yield actions which admit no non-empty closed invariant set on
which the group acts minimally (in particular, the action is _not_ semi-
conjugate to any minimal action nor to any cyclic action); this phenomenon is
somewhat degenerate, and cannot arise for a finitely generated group (see
[Navas-book, Proposition 2.1.12]). Much more interesting is the fact that many
(finitely generated) locally moving groups admit exotic actions which are
_minimal and faithful_. Various constructions of such actions of different
nature will be provided in Section 9, and more constructions can be found in
Section 11 in the special case of Thompson’s group $F$. Here we only mention
the following existence criteria, based on a construction in §9.2, which are
satisfied by some well-studied groups.

######  Proposition 1.5 (Criteria for existence of minimal exotic actions).

For $X=(a,b)$, let $G\subset\operatorname{\mathsf{Homeo}}_{0}(X)$ be a
finitely generated subgroup. Assume that $G$ acts minimally on $X$ and
contains non-trivial elements of relatively compact support in $X$, and that
at least one of the following holds.

  1. (a)

The group $G$ is a subgroup of the group of piecewise projective
homeomorphisms of $X$.

  2. (b)

The groups of germs $\operatorname{\mathsf{Germ}}(G,b)$ is abelian and its
non-trivial elements have no fixed points in a neighborhood of $b$.

Then there exists a faithful minimal action $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ which is not topologically
conjugate to the action of $G$ on $X$ (nor to any non-faithful action of $G$).

###  1.4. Actions on planar real trees and $\mathbb{R}$-focal actions

We now leave briefly aside locally moving groups to introduce a concept that
plays a central role in this paper: the notion of $\mathbb{R}$-_focal action_.
This will be the main tool to understand exotic actions on the line of a vast
class of locally moving groups (see Theorem 1.13 below).

In order to define this notion, we say that a collection $\mathcal{S}$ of open
bounded real intervals is a _cross-free cover_ if it covers $\mathbb{R}$ and
every two intervals in $\mathcal{S}$ are either disjoint or one is contained
in the other.

######  Definition 1.6.

Let $G$ be a group. An action $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is $\mathbb{R}$-focal if
there exists a bounded open interval $I\subset\mathbb{R}$ whose $G$-orbit is a
cross-free cover.

The terminology comes from group actions on trees (and Gromov hyperbolic
spaces). In this classical setting, an isometric group action on a tree
$\mathbb{T}$ is called _focal_ if it fixes a unique end
$\omega\in\partial\mathbb{T}$ and contains hyperbolic elements (which
necessarily admit $\omega$ as an attracting or repelling fixed point). The
dynamics of an $\mathbb{R}$-focal action on the line closely resembles the
dynamics of the action on the boundary of a focal action on a tree. In fact
this is more than an analogy: every $\mathbb{R}$-focal action on the line can
be encoded by an action on a tree, except that we need to consider group
actions on _real trees_ (or $\mathbb{R}$-trees) by homeomorphisms (not
necessarily isometric). Let us give an overview of this connection (for more
precise definitions and details we refer to Section 8).

Recall that a real tree is a metrizable space $\mathbb{T}$ where any two
points can be joined by a unique path, and which admits a compatible metric
which makes it geodesic. By a _directed tree_ we mean a (separable) real tree
$\mathbb{T}$ together with a preferred end $\omega\in\partial\mathbb{T}$,
called the _focus_ 222While this is not exactly the definition of directed
tree which we will work with (see Definition 8.1), it can be taken as
equivalent definition for the purpose of this introduction (the connection is
explained in §8.1.4).. If $\mathbb{T}$ is a directed tree with focus $\omega$,
we write
$\partial^{*}\mathbb{T}:=\partial\mathbb{T}\smallsetminus\\{\omega\\}$. An
action of a group $G$ on a directed tree $\mathbb{T}$ (by homeomorphisms) is
always required to fix the focus. In this topological setting we will say that
such an action is _focal_ if for every $v\in\mathbb{T}$ there exists a
sequence $(g_{n})\subset G$ such that $(g_{n}.v)$ approaches $\omega$ along
the ray $[v,\omega[$.

By a _planar directed tree_ we mean a directed tree $\mathbb{T}$ endowed with
a _planar order_ , which is the choice of a linear order on the set of
directions below every branching point of $\mathbb{T}$ (one can think of
$\mathbb{T}$ as embedded in the plane; see §8.2 for a formal definition). Note
that in this case the set $\partial^{*}\mathbb{T}$ inherits a linear order
$\prec$ in a natural way. Assume that $\mathbb{T}$ is a planar directed tree,
and that $\Phi\colon G\to\operatorname{\mathsf{Homeo}}(\mathbb{T})$ is a focal
action of a countable group which preserves the planar order. Then $\Phi$
induces an order-preserving action of $G$ on the ordered space
$(\partial^{*}\mathbb{T},\prec)$. From this action one can obtain an action
$\varphi\in\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
on the real line (for instance by considering the Dedekind completion of any
orbit in $\partial^{*}\mathbb{T}$, see §2.3.3 and §8.2 for details), which we
call the _dynamical realization_ of the action of $\Phi$. It turns out that
such an action is always minimal and $\mathbb{R}$-focal. In fact, we have the
following equivalence, which can be taken as an alternative definition of
$\mathbb{R}$-focal actions.

######  Proposition 1.7.

Let $G$ be a countable group. An action $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is minimal and
$\mathbb{R}$-focal if and only if it is conjugate to the dynamical realization
of a focal action by homeomorphisms of $G$ on some planar directed tree.

We will say that an $\mathbb{R}$-focal action $\varphi$ can be represented by
an action $\Phi\colon G\to\operatorname{\mathsf{Homeo}}(\mathbb{T})$ on a
planar directed tree if it is conjugate to the dynamical realization of
$\Phi$. Note that in general such an action $\Phi$ representing $\varphi$ is
not unique.

Examples of $\mathbb{R}$-focal actions appear naturally in the context of
solvable groups. In fact, the notion of $\mathbb{R}$-focal action was largely
inspired by an action on the line of the group $\mathbb{Z}\wr\mathbb{Z}$
constructed by Plante [Plante] to give an example of action of a solvable
group on the line which is not semi-conjugate to any action by affine
transformations (i.e. transformations of the form $x\mapsto ax+b$). See
Example 7.10 for a generalization of Plante’s construction to arbitrary wreath
products. For finitely generated solvable groups we obtain the following
dichotomy.

######  Theorem 1.8.

Let $G$ be a finitely generated solvable group. Then every irreducible action
$\varphi\colon G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is either
semi-conjugate to an action by affine transformations, or to a minimal
$\mathbb{R}$-focal action.

######  Remark 1.9.

This should be compared with the result by Plante, which motivated the
construction of the action of $\mathbb{Z}\wr\mathbb{Z}$ mentioned above, that
every irreducible action of a solvable group of _finite (Prüfer) rank_ is
semi-conjugate to an affine action (a weaker condition on the group is
actually sufficient, see [Plante, Theorem 4.4] for details).

A distinctive feature of $\mathbb{R}$-focal actions is that the action of
individual elements of the group satisfy a dynamical classification which
resembles the classification of isometries of trees into elliptic and
hyperbolic elements. Namely if
$G\subset\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is a subgroup whose
action is $\mathbb{R}$-focal, then every element $g\in G$ satisfies one of the
following (see Corollary 8.27).

  * •

Either $g$ is _totally bounded_ : its set of fixed points accumulates on both
$\pm\infty$.

  * •

Or $g$ is a _pseudohomothety_ : it has a non-empty compact set of fixed points
$K\subset\mathbb{R}$ and either every $x\notin[\min K,\max K]$ satisfies
$|g^{n}(x)|\to\infty$ as $n\to+\infty$ (in which case we say that $g$ is an
_expanding_ pseudohomothety), or the same holds as $n\to-\infty$ (in which
case we say that $g$ is _contracting_). If $K$ is reduced to a single point,
we further say that $g$ is a _homothety_.

Moreover the dynamical type of each element can be explicitly determined from
the $G$-action on a planar directed tree $\mathbb{T}$ representing the
$\mathbb{R}$-focal action by looking at the local dynamics near the focus
$\omega\in\partial\mathbb{T}$.

We finally discuss another concept: the notion of _horograding_ of an
$\mathbb{R}$-focal action of a group $G$ by another action of $G$. This will
be crucial in the sequel, as it will allow us to establish a relation between
exotic actions of various locally moving groups and their standard actions.
Assume that $\mathbb{T}$ is a directed tree with focus $\omega$. An
_increasing horograding_ of $\mathbb{T}$ by a real interval $X=(a,b)$ is a map
$\pi\colon\mathbb{T}\to X$ such that for every $v\in\mathbb{T}$ the ray
$[v,\omega[$ is mapped homeomorphically onto the interval $[\pi(v),b)$. This
is a non-metric analogue of the classical metric notion of horofunction
associated with $\omega$. A _decreasing horograding_ is defined analogously
but maps $[v,\omega[$ to $(a,\pi(v)]$, and a _horograding_ is an increasing or
decreasing horograding. If $G$ is a group acting both on $\mathbb{T}$ and $X$
we say that $\pi$ is a $G$-_horograding_ if its is $G$-equivariant. This leads
to the following definition.

######  Definition 1.10.

Assume that $\varphi\colon G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$
is a minimal $\mathbb{R}$-focal action, and that $j\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(X)$ is another action of $G$ on some
open interval $X$. We say that $\varphi$ can be (increasingly or decreasingly)
horograded by $j$ if $\varphi$ can be represented by an action on a planar
directed tree $\Phi\colon G\to\operatorname{\mathsf{Homeo}}(\mathbb{T})$ which
admits an (increasing or decreasing) $G$-horograding $\pi\colon\mathbb{T}\to
X$.

The existence of such a horograding is a tight relation between $\varphi$ and
$j$, which is nevertheless quite different from the notion of semi-conjugacy:
here the action of $G$ on $X$ plays the role of a hidden “extra-dimensional
direction” with respect to the real line on which $\varphi$ is defined. For
instance, in the presence of an increasing $G$-horograding, the type of each
element in $\varphi$ can be determined from its germ on the rightmost point of
$X$ as follows.

######  Proposition 1.11.

Let $\varphi\colon G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ be a
minimal $\mathbb{R}$-focal action, and assume that $\varphi$ can be
increasingly horograded by an action $j\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(X)$ on an interval $X=(a,b)$. Then we
have the following alternative.

  1. (i)

If the fixed points of $j(g)$ accumulate on $b$ then $\varphi(g)$ is totally
bounded.

  2. (ii)

Else $\varphi(g)$ is a pseudohomothety, which is expanding if $j(g)(x)>x$ for
$x$ in a neighborhood of $b$, and contracting otherwise. Moreover if $j(g)$
has no fixed points in $X$, then $\varphi(g)$ is a homothety.

###  1.5. A structure theorem for $C^{0}$ actions

With the notion of $\mathbb{R}$-focal action in hand, we now go back to our
original problem and state a structure theorem for actions on the line by
homeomorphisms of a vast class of locally moving group.

######  Definition 1.12 (The classes $\mathcal{F}$ and $\mathcal{F}_{0}$).

For $X=(a,b)$, let $G\subset\operatorname{\mathsf{Homeo}}_{0}(X)$ be a
subgroup, and recall that for an interval $I=(x,y)\subseteq X$ we denote by
$G_{(x,y)}$ the rigid stabilizer of $I$. Write $G_{+}:=\bigcup_{x<b}G_{(a,x)}$
and $G_{-}:=\bigcup_{x>a}G_{(x,b)}$ for the subgroups of elements with trivial
germ at $a$ and $b$ respectively. Consider the following conditions.

  1. (i)

$G$ is locally moving.

  2. (ii)

There exist two finitely generated subgroups $\Gamma_{\pm}\subset G_{\pm}$ and
$x,y\in X$ such that $G_{(a,x)}\subset\Gamma_{+}$ and
$G_{(y,b)}\subset\Gamma_{-}$.

  3. (iii)

There exists an element of $G$ without fixed points in $X$.

We say that $G$ belongs to the class $\mathcal{F}$ if it satisfies (i) and
(ii), and that it belongs to the class $\mathcal{F}_{0}$ if it satisfies (i),
(ii), and (iii).

Note that condition (ii) trivially holds true provided there exist $x,y\in X$
such that $G_{(a,x)}$ and $G_{(y,b)}$ are finitely generated. In practice this
weaker condition will be satisfied in many examples, but (ii) is more flexible
and more convenient to handle.

The class $\mathcal{F}_{0}$ contains many well-studied examples of finitely
generated locally moving groups, including Thompson’s group $F$, and all
Thompson–Brown–Stein groups $F_{n_{1},\ldots,n_{k}}$ [Stein], the groups of
piecewise projective homeomorphisms of Lodha–Moore [LodhaMoore], and several
other Bieri–Strebel groups [BieriStrebel]. It also contains various groups
which are far from the setting of groups of piecewise linear or projective
homeomorphisms: for instance every countable subgroup of
$\operatorname{\mathsf{Homeo}}_{0}(X)$ is contained in a finitely generated
group belonging to $\mathcal{F}_{0}$ (see Proposition 10.2).

Our main result is a qualitative description for the actions on the line of
groups in the class $\mathcal{F}$, stating that such actions can be classified
into three types of behavior.

######  Theorem 1.13 (Main structure theorem for actions of groups in
$\mathcal{F}$).

Let $X$ be an open interval and let
$G\subset\operatorname{\mathsf{Homeo}}_{0}(X)$ be a group in the class
$\mathcal{F}$. Then every irreducible action $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is semi-conjugate to an
action in one of the following families.

  1. (1)

_(Non-faithful)_ An action which factors through the largest quotient
$G/[G_{c},G_{c}]$.

  2. (2)

_(Standard)_ An action which is conjugate to the standard action of $G$ on
$X$.

  3. (3)

_(Exotic)_ A minimal faithful $\mathbb{R}$-focal action which can be
horograded by the standard action of $G$ on $X$.

Note that many finitely generated groups which belong to the class
$\mathcal{F}$, or even to the more restricted class $\mathcal{F}_{0}$, do
indeed admit exotic actions falling in (3), for instance as a consequence of
Proposition 1.5 (but not all groups in $\mathcal{F}_{0}$ do, see Theorem 1.20
below). Moreover in some cases there are uncountably many non-semi-conjugate
such actions (see for instance Theorem 1.18 below for the case of Thompson’s
group $F$), and the variety and flexibility of constructions suggest that in
general it is too complicated to obtain a reasonably explicit description of
all semi-conjugacy classes of exotic actions (however the word _explicit_ is
crucial here, as we shall see in Theorem 1.14 that such a description exists
in principle). Nevertheless, the main content of Theorem 1.13 is that every
exotic action of a group in $\mathcal{F}$ must be tightly related to the
standard action of $G$, although not via a semi-conjugacy but at the level of
a planar directed tree encoding the exotic ($\mathbb{R}$-focal) action. This
relation can be effectively exploited to study such exotic actions (in
particular the way in which every individual element of $G$ acts oi an exotic
action is determined by its standard action, see Proposition 1.11). This leads
to various applications that we describe now.

###  1.6. Space of semi-conjugacy classes and local rigidity for groups in
$\mathcal{F}_{0}$

The main structure theorem (Theorem 1.13) can be used to get an insight on the
structure of semi-conjugacy classes inside the space of irreducible actions
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
whenever $G$ is a group in $\mathcal{F}_{0}$. Recall that the space
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
can be endowed with the natural _compact-open topology_ , which means that a
neighborhood basis of a given action
$\varphi\in\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
is defined by considering for every $\varepsilon>0$, finite subset $S\subset
G$, and compact subset $K\subset\mathbb{R}$, the subset of actions

$\left\\{\psi\in\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))\colon\max_{g\in
S}\max_{x\in K}|\varphi(g)(x)-\psi(g)(x)|<\varepsilon\right\\}.$

A natural way to understand actions of a group $G$ up to semi-conjugacy would
be to study the quotient of the space
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
by the semi-conjugacy equivalence relation. Unfortunately this quotient space
is rather bad, for instance it is in general not Hausdorff and may even fail
to have the structure of a standard measurable space (see Remark 3.23). In
fact the semi-conjugacy equivalence relation need not be _smooth_ , that is it
need not be possible to classify the semi-conjugacy classes of actions in
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
by a Borel complete invariant (see the discussion in §3.5).

However, for groups in the class $\mathcal{F}_{0}$, Theorem 1.13 can be used
to show that a large part of the set of semi-conjugacy classes in
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(G))$
can in fact be parametrized by a well behaved space and it is possible to
select a system of representatives of each semi-conjugacy class with nice
properties. Under this identification, it turns out that the semi-conjugacy
class of the standard action of $G$ on $X$ corresponds to an _isolated point_
in this space. This is the content of the following result.

######  Theorem 1.14 (The space of semi-conjugacy classes for groups in
$\mathcal{F}_{0}$).

Let $X$ be an open interval and let
$G\subset\operatorname{\mathsf{Homeo}}_{0}(X)$ be a finitely generated group
in the class $\mathcal{F}_{0}$. Denote by $\sim$ the equivalence relation on
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
given by positive semi-conjugacy, and let
$\mathcal{U}\subset\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
be the $\sim$-invariant set of irreducible actions that are not semi-conjugate
to any action induced from the largest quotient $G/[G_{c},G_{c}]$. Then
$\mathcal{U}$ is open and the following hold.

  1. (i)

The quotient space $\mathcal{U}/_{\sim}$, with the quotient topology, is
Polish and locally compact.

  2. (ii)

There exists a continuous section
$\sigma\colon\mathcal{U}/_{\sim}\rightarrow\mathcal{U}$ which is a
homeomorphism onto its image, whose image is closed in $\mathcal{U}$ and
consists of faithful minimal actions.

  3. (iii)

The semi-conjugacy class of the standard action of $G$ on $X$ is an isolated
point in $\mathcal{U}/_{\sim}$.

######  Remark 1.15.

In the previous theorem, ruling out the actions induced from the largest
quotient $G/[G_{c},G_{c}]$ is necessary, as in this generality it may happen
that the semi-conjugacy equivalence relation on irreducible actions of
$G/[G_{c},G_{c}]$ might not be smooth, and a fortiori this may not be true for
$G$. However the theorem implies that the semi-conjugacy equivalence relation
on the set
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
is smooth provided the same holds for irreducible actions of $G/[G_{c},G_{c}]$
(see Corollary 10.24, or Theorem 11.6 in the case of Thompson’s group $F$).

The main dynamical significance of the previous result is that it implies a
local rigidity result. We say that an irreducible action
$\varphi\in\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
of a group $G$ is _locally rigid_ if there exists a neighborhood
$\mathcal{U}\subset\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
of $\varphi$ such that every $\psi\in\mathcal{U}$ is semi-conjugate to
$\varphi$. Otherwise, we say that the action of $\varphi$ is _flexible_.
Theorem 1.14 has the following direct consequence.

######  Corollary 1.16 (Local rigidity of the standard action for groups in
$\mathcal{F}_{0}$).

Let $X$ be an open interval and let
$G\subset\operatorname{\mathsf{Homeo}}_{0}(X)$ be a finitely generated group
in the class $\mathcal{F}_{0}$. Then the natural action of $G$ on $X$ is
locally rigid.

Corollary 1.16 provides a vast class of locally rigid actions on the real
line. For instance, by Proposition 10.2, it implies that every action on the
real line of a countable group $G$ can be extended to a locally rigid action
of some finitely generated overgroup.

A well-developed approach to local rigidity of group actions on the line is
through the space $\operatorname{\mathsf{LO}}(G)$ of left-invariant orders on
$G$, which is a totally disconnected compact space. Every such order gives
rise to an action in
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$,
and isolated points in $\operatorname{\mathsf{LO}}(G)$ produce locally rigid
actions (see Mann and the third author [MannRivas, Theorem 3.11]). This can be
used to show that some groups, for instance braid groups (see Dubrovin and
Dubrovina [DubrovinDubrovina], or the monograph by Dehornoy, Dynnikov,
Rolfsen, and Wiest [OrderingBraids]), do have locally rigid actions. However
the converse to this criterion does not hold, and this approach has been more
fruitful in the opposite direction, namely for showing that a group has no
isolated order from flexibility of the dynamical realization (see for instance
the works by Navas [Navas2010], or by Alonso, and the first and third named
authors [ABR, AlonsoBrum], as well as by Malicet, Mann, and the last two
authors [MMRT]). One difficulty underlying this approach is that it is usually
not easy to determine when two orders in $\operatorname{\mathsf{LO}}(G)$ give
rise to semi-conjugate actions.

To prove Theorem 1.14, we take a different approach to local rigidity, based
on a compact space suggested by Deroin [Deroin] as a dynamical substitute of
the space of left-orders. One way to construct this space is based on work by
Deroin, Kleptsyn, Navas, and Parwani [DKNP] on symmetric random walks on
$\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$. Given a probability measure
$\mu$ on $G$ whose support is finite, symmetric, and generates $G$, one
defines the _Deroin space_ $\operatorname{\mathsf{Der}}_{\mu}(G)$ as the
subspace of
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
of (normalized) _harmonic actions_ , that is, actions of $G$ for which the
Lebesgue measure is $\mu$-stationary (see §2.2 for details). The space
$\operatorname{\mathsf{Der}}_{\mu}(G)$ is compact and Hausdorff, with a
natural topological flow
$\Phi:\mathbb{R}\times\operatorname{\mathsf{Der}}_{\mu}(G)\to\operatorname{\mathsf{Der}}_{\mu}(G)$
defined on it by the conjugation action of the group of translations
$(\mathbb{R},+)$, and has the property that two actions in
$\operatorname{\mathsf{Der}}_{\mu}(G)$ are (positively semi-)conjugate if and
only if the are on the same $\Phi$-orbit. It was shown in [DKNP] that every
action in
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
can be semi-conjugated to one inside $\operatorname{\mathsf{Der}}_{\mu}(G)$ by
a probabilistic argument. In fact, we shall show in Section 3 that this can be
done in such a way that the new action depends continuously on the original
one (see Theorem 3.2). This continuous dependence implies a criterion for
local rigidity of minimal actions within the space
$\operatorname{\mathsf{Der}}_{\mu}(G)$ (see Corollary 3.4). The proof is based
on an alternative description of $\operatorname{\mathsf{Der}}_{\mu}(G)$ a
quotient of the space of left-invariant _preorders_ on $G$ (Theorem 3.20),
which also implies that $\operatorname{\mathsf{Der}}_{\mu}(G)$ does not depend
on the choice of the probability measure $\mu$ up to homeomorphism.

With these tools in hand, to prove Theorem 1.14 we study the shape of orbits
of the translation flow $\Phi$ on $\operatorname{\mathsf{Der}}_{\mu}(G)$ when
$G$ is a group in $\mathcal{F}_{0}$. Faithful actions correspond to an open
$\Phi$-invariant subset of $\operatorname{\mathsf{Der}}_{\mu}(G)$. Using
Theorem 1.13 and the properties of $\mathbb{R}$-focal actions we show that the
restriction of the flow $\Phi$ to this open subset has the simplest possible
type of orbit structure: there is a closed transversal $\mathcal{S}$
intersecting every $\Phi$-orbit, and $\Phi$ is conjugate to the vertical
translation on $\mathcal{S}\times\mathbb{R}$, so that its space of orbits can
be identified with $\mathcal{S}$. This analysis yields Theorem 1.14. (See
Figure 1 for a schematic representation for Thompson’s group $F$.)

Figure 1. The Deroin space of Thompson’s group $F$. The outer red circle
parametrizes the non-faithful actions (of $F^{ab}\cong\mathbb{Z}^{2}$) and it
is pointwise fixed by the flow $\Phi$. The remaining $\Phi$-orbits are the
faithful actions. The purple and blue orbits correspond to the standard action
and to its conjugate by the reflection, and are transversely isolated (this
gives local rigidity). The pencils of green and yellow orbits are the
$\mathbb{R}$-focal actions: both pencils contain uncountably many orbits,
admit a compact transversal to the flow, and the shown convergence to limit
points is uniform. See Figure 10 and §11.2 for details.

###  1.7. A criterion for non-orientation-preserving groups

We also observe that the situation is quite different if we leave the setting
of orientation-preserving actions. Indeed, in this case we have the following
result which follows from Corollary 1.16 (or can be more directly obtained in
the course of its proof). The proof is given in §10.3. We denote by
$\operatorname{\mathsf{Homeo}}(X)$ the group of all (not necessarily
orientation-preserving) homeomorphisms of an interval $X$.

######  Corollary 1.17 (Global rigidity in the non-orientation-preserving
case).

Let $X=(a,b)$ and let $G\subset\operatorname{\mathsf{Homeo}}(X)$ be a subgroup
such that $G\subsetneq\operatorname{\mathsf{Homeo}}_{0}(X)$ and
$G\cap\operatorname{\mathsf{Homeo}}_{0}(X)$ is in the class $\mathcal{F}$.
Then every faithful minimal action $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}(\mathbb{R})$ is conjugate to the standard
action on $X$.

###  1.8. Some concrete groups in the class $\mathcal{F}_{0}$

We have seen how Theorem 1.14 yields a global picture of actions on the line
for groups in the class $\mathcal{F}_{0}$. Nevertheless when $G$ is a group in
the class $\mathcal{F}$ the rigidity or abundance of its exotic actions turns
out to depend subtly on $G$. This difference is visible already among groups
of piecewise linear homeomorphisms.

Given an open interval $X=(a,b)$ we denote by $\operatorname{\mathsf{PL}}(X)$
the group of orientation-preserving PL homeomorphisms of $X$, with finitely
many discontinuity points for the derivatives. It turns out that this class of
groups exhibits a surprising mixture of rigidity and flexibility properties
and many illustrative examples of applications of our results arise as
subgroups of $\operatorname{\mathsf{PL}}(X)$. Note that Proposition 1.5
implies that every finitely generated locally moving group
$G\subset\operatorname{\mathsf{PL}}(X)$ admits a minimal faithful exotic
action on the real line.

The most famous example of group of PL homeomorphisms is Thompson’s group $F$.
Recall that $F$ is defined as the subgroup of
$\operatorname{\mathsf{PL}}((0,1))$ whose slopes are powers of 2 and whose
constant terms and breakpoints are in the ring $\mathbb{Z}[1/2]$ of dyadic
rationals. The group $F$ belongs to the class $\mathcal{F}_{0}$ and satisfies
the assumptions of most of the results in this paper. The reader can find in
Section 11 a detailed account of our results in this case. In particular every
faithful action $\varphi\colon
F\to\operatorname{\mathsf{Diff}}^{1}_{0}([0,1])$ without fixed points in
$(0,1)$ is semi-conjugate to the standard action (Corollary 1.3), every
irreducible exotic action $\varphi\colon
F\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is $\mathbb{R}$-focal and
horograded by the standard action of $F$ on $(0,1)$ (Theorem 1.13), and the
standard action of $F$ on $(0,1)$ is locally rigid (Corollary 1.16).

Despite these rigidity results, it turns out that the group $F$ admits a rich
and complicated universe of minimal exotic actions, and our work leaves some
interesting questions open: in particular we do not know whether $F$ admits
exotic actions that are locally rigid. In Section 11 we will discuss several
different constructions of minimal $\mathbb{R}$-focal actions of $F$, which
for instance leads to the following.

######  Theorem 1.18.

Thompson’s group $F$ admits uncountably many actions on the line which are
faithful, minimal, $\mathbb{R}$-focal and pairwise non-semi-conjugate.
Moreover, there are uncountably many such actions whose restrictions to the
commutator subgroup $[F,F]$ remain minimal and are pairwise non-semi-
conjugate.

This abundance of $\mathbb{R}$-focal actions can be explained in part by the
fact that the group $F$ admits many focal actions on _simplicial_ trees by
simplicial automorphisms (the Bass–Serre tree associated with any non-trivial
splitting as an ascending HNN extension [Geoghegan, Proposition 9.2.5]). Some
of these actions preserve a planar order on the tree, and many
$\mathbb{R}$-focal actions of $F$ arise in this way (a characterization will
be given in §11.3). However, we show in §11.6 that the group $F$ also admits
minimal $\mathbb{R}$-focal actions for which an associated planar directed
tree cannot be chosen to be simplicial, or even to carry an invariant
$\mathbb{R}$-tree metric. In fact the second claim in Theorem 1.18 is directly
related to this phenomenon. We refer to Section 11 for further discussion on
$\mathbb{R}$-focal actions of $F$.

In contrast, this abundance of exotic actions of the group $F$ already fails
for some tightly related groups of PL homeomorphisms. Given a real number
$\lambda>1$, we denote by $G(X;\lambda)$ the group of all PL homeomorphisms of
$X$ where all derivatives are integers powers of $\lambda$ and all constant
terms and breakpoints belong to $\mathbb{Z}[\lambda,\lambda^{-1}]$. Note that
the group $F$ is equal to $G((0,1);2)$. When $X=\mathbb{R}$, the group
$G(\mathbb{R};\lambda)$ belongs to $\mathcal{F}_{0}$ provided $\lambda$ is
algebraic, thus satisfies Theorem 1.13. However, in striking difference with
the case of $F$, we have the following.

######  Theorem 1.19 (PL groups with finitely many exotic actions).

Let $\lambda>1$ be an algebraic real number. Then the group
$G=G(\mathbb{R};\lambda)$ admits exactly three minimal faithful actions
$\varphi\colon G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ on the real
line up to (semi-)conjugacy, namely its standard action and two minimal
$\mathbb{R}$-focal actions (which can be horograded by its standard action).

Note that this shows that the two minimal exotic actions of
$G(\mathbb{R};\lambda)$ are locally rigid as well; in particular the standard
action need not be the unique locally rigid action for a group in
$\mathcal{F}_{0}$.

Theorem 1.19 is in fact a special case of Theorem 12.3 which applies to a more
general class of _Bieri–Strebel groups_ of the form $G(\mathbb{R};A,\Lambda)$,
which are groups of PL homeomorphisms defined by restricting derivatives to
belong to a multiplicative subgroup $\Lambda\subset\mathbb{R}^{*}_{+}$ and
constant terms and breakpoints to belong to a $\mathbb{Z}[\Lambda]$-submodule
$A\subset\mathbb{R}$ (see §2.4). The two exotic actions of the group
$G(\mathbb{R};\lambda)$ arise as a special case of a construction of exotic
actions of Bieri–Strebel groups explained in §9.3.

Building on the proof of Theorem 1.19, in §12.2 we also construct a finitely
generated locally moving group $G$ having no exotic actions at all. This group
is obtained by perturbing the group $G(\mathbb{R};2)$ locally in a controlled
manner.

######  Theorem 1.20 (A finitely generated locally moving group with no exotic
actions).

There exists a finitely generated subgroup
$G\subset\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ in the class
$\mathcal{F}_{0}$, such that every faithful minimal action $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is conjugate to the
standard action.

###  1.9. Further consequences

We conclude this introduction by mentioning some additional results obtained
along the way, which allow to recover and strenghten some previously known
results in the unified setting developed here.

####  1.9.1. Very large locally moving groups

As mentioned at the beginning of the introduction, two rigidity results were
known for some very large locally moving groups: namely a result of Militon
shows that the group $\operatorname{\mathsf{Homeo}}_{c}(\mathbb{R})$ of
compactly supported homeomorphisms of $\mathbb{R}$ admits a unique irreducible
action on $\mathbb{R}$ up to conjugacy, and a recent result of Chen and Mann
implies the same property for the group
$\operatorname{\mathsf{Diff}}^{r}_{c}(\mathbb{R})$ of compactly supported
diffeomorphisms in regularity $r\neq 2$. While the main focus of this paper is
on countable groups, our method also allows to obtain a criterion for
uniqueness of the action of a family of uncountable locally moving groups,
which recovers and generalizes those results. In order to do this, we need the
following relative version of a group property first considered by Schreier
(see [MR3242261, Problem 111]).

######  Definition 1.21.

Let $G$ be a group and let $H$ be a subgroup of $G$. We say that the pair
$(G,H)$ has _relative Schreier property_ if every countable subset of $H$ is
contained in a finitely generated subgroup of $G$.333The choice of _relative
Schreier property_ is inspired by the work of Le Roux and Mann [LeRouxMann].
Apparently there is no standard name in the literature, and such choice may be
ambiguous, as in other works _Schreier property_ is used for different
properties.

The following result is a special case of what we show in Section 13.

######  Theorem 1.22 (Uniqueness of action for some uncountable locally moving
groups).

Let $G\subset\operatorname{\mathsf{Homeo}}_{c}(\mathbb{R})$ be a perfect
subgroup of compactly supported homeomorphisms. Suppose that for every bounded
open interval $I\subset\mathbb{R}$ the following hold:

  1. (1)

the $G_{I}$-orbit of every $x\in I$ is uncountable;

  2. (2)

the pair $(G,G_{I})$ has relative Schreier property.

Then every action $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ without fixed points is
conjugate to the natural action of $G$.

####  1.9.2. Comparison with the case of groups acting on the circle

One may wonder whether a similar theory can be developed for locally moving
groups acting on the _circle_ rather than the real line. However this problem
turns out to be considerably simpler and less rich, as in this setting locally
moving groups essentially do not admit exotic actions at all. This phenomenon
is already well-known in some cases. In particular a result of Matsumoto
states that the group $\operatorname{\mathsf{Homeo}}_{0}(\mathbb{S}^{1})$
admits only one action on $\mathbb{S}^{1}$ up to conjugacy, and an unpublished
result of Ghys (announced in [Ghys]) shows that every action of Thompson’s
group $T$ on $\mathbb{S}^{1}$ is semi-conjugate to its standard action. The
original proofs of these results are both based on the computation of the
second bounded cohomology of these groups, and on its relation with actions on
the circle established by Ghys [Ghys-bounded] (a tool that does not have an
analogue for actions on real line). For Thompson’s group $T$ another proof is
provided in [LBMB-sub], which does not use the knowledge of bounded cohomology
of $T$, but relies in part on the nature of $T$ as a group of piecewise linear
homeomorphisms.

In fact, in Section 14 we observe that elementary considerations based on
commutations suffice to obtain a more general criterion (Theorem 14.2),
stating that if $G$ is a group of homeomorphisms of an arbitrary compact space
$X$ satisfying suitable assumptions, then every minimal faithful action of $G$
on the circle must factor onto its action on $X$. A special case of Theorem
14.2 (with $X=\mathbb{S}^{1}$) yields the following.

######  Theorem 1.23.

Let $G\subset\operatorname{\mathsf{Homeo}}_{0}(\mathbb{S}^{1})$ be a locally
moving subgroup. Then every faithful minimal action $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{S}^{1})$ factors onto the
standard action of $G$ on $\mathbb{S}^{1}$.

In fact, under some mild additional assumption on $G$ one obtains that the
action $\varphi$ is conjugate to (possibly a lift of) the standard action on
$\mathbb{S}^{1}$ via a self-cover of $\mathbb{S}^{1}$. We refer to Section 14
for more precise results and further discussion.

####  1.9.3. An application to non-smoothability

One possible direction of application of Theorem 1.2 is to use it as a tool to
show that certain locally moving group acting on the line cannot be isomorphic
to groups of diffeomorphisms (of suitable regularity). Results in this spirit
were obtained by Bonatti, Lodha, and the fourth named author in [BLT] for
various groups of piecewise linear and projective homeomorphisms. Here we
consider the Thompson–Brown–Stein groups $F_{n_{1},\cdots,n_{k}}$, which are a
natural generalization of Thompson’s group $F$ associated with any choice of
integers $n_{1},\ldots,n_{k}$ whose logarithms are linearly independent over
$\mathbb{Q}$ (see Definition 2.45). It was shown in [BLT] that if $k\geq 2$,
then the standard action of $F_{n_{1},\ldots,n_{k}}$ on $(0,1)$ cannot be
conjugate to an action by $C^{2}$ diffeomorphisms. With our methods we obtain
the following strengthening of that result.

######  Theorem 1.24.

When $k\geq 2$, any Thompson–Brown–Stein group of the form
$F_{n_{1},\ldots,n_{k}}$ admits no faithful action on the real line by
diffeomorphisms of class $C^{1+\alpha}$ for any $\alpha>0$.

In this direction we also mention a recent result of Kim, Koberda, and third
named author [KKR], who used our Theorem 1.2 to show that the free product
$F\ast\mathbb{Z}$ of Thompson’s group and the integers admits no faithful
action on $[0,1]$ by $C^{1}$ diffeomorphisms.

###  1.10. Outline of the paper

This paper is structured as follows. Sections 2 and 3 contain the required
general theory on group actions on the line. After recalling some elementary
preliminaries and notations in Section 2, we focus in Section 3 on the Deroin
space $\operatorname{\mathsf{Der}}_{\mu}(G)$ of $\mu$-harmonic actions on the
line of a finitely generated group $G$. The main purpose of that section is to
prove that there exists a continuous retraction from
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
to the space $\operatorname{\mathsf{Der}}_{\mu}(G)$ and a related local
rigidity criterion (Corollary 3.4), which will be used in the proofs of
Theorem 1.14 and Corollary 1.16 (see the discussion in §1.6). These results
appear to be new, although the arguments in the section are mostly based on
elementary facts on left-invariant preorders.

The remaining sections can be roughly divided into four parts. Part I contains
the central skeleton of the study locally moving groups on which the remaining
parts are built, while Parts II–IV are essentially independent from each other
and can be read after reading Part I (modulo some minor dependence between
them, mostly limited to examples or secondary results).

  1. Part I

(Sections 4–5). General theory on locally moving groups acting on the line.
After introducing notations and terminology on locally moving groups of
homeomorphisms of intervals and proving some of their basic properties in
Section 4, we begin in Section 5 the study of their actions on the real line.
The main result of this part is Theorem 5.3, which establishes a dynamical
trichotomy for actions on the line of a locally moving group $G$ of
homeomorphisms, a crucial common ingredient for all the other results in this
paper. This result imposes a restrictive technical condition on all exotic
actions of $G$, which roughly speaking says that in every exotic action of $G$
there is “domination” of certain subgroups of $G$ over other. After proving
this, we give in §5.2 some of its simpler applications, such as a rigidity
result for actions with nice combinatorial properties (including actions by
piecewise analytic transformations). We conclude the section by giving some
first examples of exotic actions of locally moving groups in §5.3 (which allow
to prove Fact 1.4).

  2. Part II

(Section 6). Actions by $C^{1}$ diffeomorphisms. This part consists of Section
6 alone, where we study $C^{1}$ actions of locally moving group and complete
the proofs of Theorem 1.2 and Corollary 1.3.

  3. Part III

(Sections 7–12). $\mathbb{R}$-focal actions and topological dynamics of exotic
actions. Here we develop the notion of $\mathbb{R}$-focal action and apply it
to prove our main results for actions by homeomorphisms of locally moving
groups in the class $\mathcal{F}$. In Section 7 we introduce
$\mathbb{R}$-focal actions of groups on the line and establish their main
properties. In Section 8 we illustrate the connection of $\mathbb{R}$-focal
action with focal actions on trees. (Note that Sections 7–8 are not directly
concerned with locally moving groups, and could also be read directly after
reading Section 2.) In Section 9 we give various constructions and existence
criteria for exotic minimal $\mathbb{R}$-focal actions of locally moving
groups. Section 10 contains our main results on groups in the classes
$\mathcal{F}_{0}$$\subset$$\mathcal{F}$. We begin by proving our main
structure theorem (Theorem 1.13) for actions of groups in $\mathcal{F}$, and
then use it to study the Deroin space $\operatorname{\mathsf{Der}}_{\mu}(G)$
of a group in $\mathcal{F}_{0}$. As a corollary of this analysis and of the
results in Section 3 we obtain Theorem 1.14 and the local rigidity of the
standard action for groups in $\mathcal{F}_{0}$ (Corollary 1.16). In Section
11 we focus on the example of Thompson’s group $F$: we illustrate our main
rigidity results in this special case, and use them to analyze its Deroin
space $\operatorname{\mathsf{Der}}_{\mu}(F)$ (see Figure 10); then we proceed
to give various constructions of minimal exotic actions of the group $F$ and
discuss some of their properties. In Section 12 we provide examples of groups
in the class $\mathcal{F}_{0}$ whose $\mathbb{R}$-focal actions are much more
rigid: we classify the $\mathbb{R}$-focal actions of a class of Bieri–Strebel
groups of PL homeomorphisms (in particular we prove Theorem 1.19) and
construct a finitely generated locally moving group without exotic actions
(Theorem 1.20).

  4. Part IV

(Sections 13–14) Additional results. In this part we prove some additional
results which allow to recover and generalize some previously known results
from the literature in the framework of this paper. In Section 13 we prove
results on actions of some very large (uncountable) locally moving groups and
prove Theorem 1.22. In Section 14 we prove a result for actions on the circle,
which implies Theorem 1.23: this section is essentially self-contained and
independent on the other results in this paper (it only relies on Proposition
5.4).

### Acknowledgements

We are grateful to Sebastián Hurtado, Bertrand Deroin and Maxime Wolff for
fruitful discussions about the space of harmonic actions and to Maxime Wolff
also for discussions regarding the Plante product. We also acknowledge Todor
Tsankov for useful clarifications about Borel reducibility of equivalence
relations which motivated the discussion in §3.5, and Yash Lodha for many
discussions related to various locally moving groups of homeomorphisms of the
line.

J.B was supported by CONICYT via FONDECYT post-doctorate fellowship 3190719.
Part of this work was done during a visit of N.M.B. to the University of
Santiago de Chile funded by FONDECYT 1181548. N.M.B. and M.T. were partially
supported by the project ANR Gromeov (ANR-19-CE40-0007). M.T. was also
partially supported by the project ANER Agroupes (AAP 2019 Région
Bourgogne–Franche–Comté), and his host department IMB receives support from
the EIPHI Graduate School (ANR-17-EURE-0002). C.R. was partially supported by
FONDECYT 1181548 and FONDECYT 1210155.

##  2\. Notation and preliminaries

###  2.1. Actions on the line

In this work we are mainly concerned with orientation-preserving actions on
the real line, that is, homomorphisms
$\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$. We will almost
always be interested in actions without (global) fixed points, which will
sometimes be called _irreducible_ for short. Note that every action
$\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ can be described
in terms of irreducible actions, just considering all restrictions

$\begin{array}[]{cccc}\varphi_{J}:&G&\to&\operatorname{\mathsf{Homeo}}_{0}(J)\cong\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})\\\
&g&\mapsto&\varphi(g)\restriction_{J}\end{array},$

of the action $\varphi$ to minimal $\varphi(G)$-invariant open intervals
$J\subset\mathbb{R}$. We write
$\operatorname{\mathsf{Hom}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
for the space of order-preserving actions of the group $G$, endowed with the
compact open topology, and
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))\subseteq\operatorname{\mathsf{Hom}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
for the subspace of actions without fixed points.

####  2.1.1. Some notation on actions

Given $f\in\operatorname{\mathsf{Homeo}}(\mathbb{R})$, we write
$\operatorname{\mathsf{Fix}}(f)=\\{x\in\mathbb{R}\colon f(x)=x\\}$ for the set
of fixed points and
$\operatorname{\mathsf{Supp}}(f)=\mathbb{R}\smallsetminus\operatorname{\mathsf{Fix}}(f)$
for its _support_. Note that by definition $\operatorname{\mathsf{Supp}}(f)$
is an open set (we do _not_ take its closure unless specified). For
$G\subset\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ and $x\in\mathbb{R}$
we write $\operatorname{\mathsf{Stab}}_{G}(x)$ for the stabilizer of $x$. We
denote by $\operatorname{\mathsf{Homeo}}_{c}(\mathbb{R})$ the group of
homeomorphisms whose support is relatively compact. Given an action
$\varphi\colon G\to\operatorname{\mathsf{Homeo}}(\mathbb{R})$ and $g\in G$ we
set
$\operatorname{\mathsf{Fix}}^{\varphi}(g)=\operatorname{\mathsf{Fix}}(\varphi(g))$
and
$\operatorname{\mathsf{Supp}^{\varphi}}(g)=\operatorname{\mathsf{Supp}}(\varphi(g))$,
and $\operatorname{\mathsf{Stab}}^{\varphi}_{G}(x)$ for the stabilizer. When
there is no confusion we write $g.x$ instead of $\varphi(g)(x)$. The notation
$g(x)$ will be reserved to the case when $G$ is naturally given as a subgroup
$G\subset\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ to refer to its
standard action (but never to another action $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$).

For $x\in\mathbb{R}\cup\\{\pm\infty\\}$ we denote by
$\operatorname{\mathsf{Germ}}(x)$ the group of germs of homeomorphisms that
fix $x$. Recall that this is defined as the group of equivalence classes of
homeomorphisms $f\in\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ that fix
$x$, where two such homeomorphisms are identified if they coincide on a
neighborhood of $x$. By considering only one-sided neighborhoods, one gets two
groups $\operatorname{\mathsf{Germ}}_{-}(x)$ and
$\operatorname{\mathsf{Germ}}_{+}(x)$, the groups of right germs and the group
of left germs respectively, such that
$\operatorname{\mathsf{Germ}}(x)=\operatorname{\mathsf{Germ}}_{-}(x)\times\operatorname{\mathsf{Germ}}_{+}(x)$.
If $G$ is a group of homeomorphisms that fixes $x$, we denote by
$\operatorname{\mathsf{Germ}}(G,x)$ the group of germs induced by elements of
$G$, and by $\mathcal{G}_{x}\colon G\to\operatorname{\mathsf{Germ}}(G,x)$ the
associated homomorphism. Similarly we have groups of one-sided germs
$\operatorname{\mathsf{Germ}}_{\pm}(G,x)$. Note however that when $G$ acts
non-trivially only on one side of $G$ (e.g. if $x$ is an endpoint of its
support, or if $x=\pm\infty$), we omit the sign $\pm$ as this is clear from
the context.

####  2.1.2. Commuting actions

A particular case of what said above is that every homeomorphism
$g\in\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is basically determined by
its set of fixed points $\operatorname{\mathsf{Fix}}(g)$ and by how it acts on
every connected component of its support
$\operatorname{\mathsf{Supp}}(g)=\mathbb{R}\smallsetminus\operatorname{\mathsf{Fix}}(g)$.
Therefore, it is fundamental to understand the set of fixed points of a given
element, or at least to be able to say whether it is empty or not. For this, a
very useful relation is that for an element
$g\in\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ and a subgroup
$H\subseteq\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$, one has
$g(\operatorname{\mathsf{Fix}}(H))=\operatorname{\mathsf{Fix}}(gHg^{-1})$. In
particular, when $G\subseteq\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ and
$H\lhd G$ is a normal subgroup, then for every $g\in G$ one has
$g(\operatorname{\mathsf{Fix}}(H))=\operatorname{\mathsf{Fix}}(H)$. This holds
in particular for commuting subgroups, for which we have the following
observation, which is easily obtained using that the set of fixed point is a
closed subset.

######  Lemma 2.1.

Consider two commuting subgroups $H_{1}$ and $H_{2}$ of
$G\subseteq\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ (that is,
$[h_{1},h_{2}]=\mathsf{id}$ for every $h_{1}\in H_{1}$ and $h_{2}\in H_{2}$).
Suppose that both $\operatorname{\mathsf{Fix}}(H_{1})$ and
$\operatorname{\mathsf{Fix}}(H_{2})$ are non-empty. Then $H_{1}$ and $H_{2}$
admit a common fixed point.

This lemma will be used in the text without explicit reference.

####  2.1.3. Semi-conjugacy

It is customary in the field to consider actions up to _semi-conjugacy_. This
means that not only we do not really take care of the choice of coordinate on
$\mathbb{R}$ (which corresponds to the classical notion of _conjugacy_), but
we want to consider only the interesting part of the dynamics of the action.
This was first formalized by Ghys in his work on bounded Euler class, but the
definition has been unanimously fixed only recently. We follow here [KKMj,
Definition 2.1], although we allow order-reversing semi-conjugacies. For the
statement we will say that a map $h:\mathbb{R}\to\mathbb{R}$ is _proper_ if
its image $h(\mathbb{R})$ is unbounded in both directions of the line.

######  Definition 2.2.

Let $\varphi,\psi:G\to\operatorname{\mathsf{Homeo}}(\mathbb{R})$ be two
actions of a group $G$ on the real line. We say that $\varphi$ and $\psi$ are
_semi-conjugate_ if there exists a proper monotone map
$h:\mathbb{R}\to\mathbb{R}$ such that

(2.1) $h\varphi(g)=\psi(g)h\quad\text{for every }g\in G.$

Such a map $h$ is called a _semi-conjugacy_ between $\varphi$ and $\psi$. Note
that if $\varphi$ and $\psi$ are irreducible, the requirement that the map $h$
be proper follows automatically from the equivariance (2.1) and thus can be
dropped. When $h\in\operatorname{\mathsf{Homeo}}(\mathbb{R})$, we say that $h$
is a _conjugacy_ between $\varphi$ and $\psi$, in which case we say that $h$
is a _conjugacy_. When $h$ is nondecreasing, we say that $\varphi$ and $\psi$
are _positively_ semi-conjugate.

######  Remark 2.3.

Both conjugacy and semi-conjugacy are equivalence relations (for conjugacies
this is obvious, for semi-conjugacies the reader can check [KKMj, Lemma 2.2]
or [GOD, Proposition 1.1.16]). Notice that in the semi-conjugacy case, we do
not require that $h$ be continuous; indeed, being continuously semi-conjugate
is not even a symmetric relation.

Rarely (essentially only in Section 14) we will need the analogous notion of
semi-conjugacy for actions on the circle, which is defined as follows.

######  Definition 2.4.

Let $\varphi,\psi\colon G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{S}^{1})$
be two actions on the circle. They are semi-conjugate if and only if there
exists a group $\tilde{G}$ which is a central extension of $G$ of the form
$1\to C\to\tilde{G}\to G\to 1$, with $C\cong\mathbb{Z}$, and two semi-
conjugate actions
$\tilde{\varphi},\tilde{\psi}\colon\tilde{G}\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$
which both map $C$ to the group $\mathbb{Z}$ of integer translations, and
which descend to the quotient respectively to the actions $\varphi$ and $\psi$
of $G=\tilde{G}/C$ on $\mathbb{S}^{1}=\mathbb{R}/\mathbb{Z}$.

Given an action $\varphi:G\to\operatorname{\mathsf{Homeo}}(\mathbb{R})$, one
can consider the _reversed action_
$\widehat{\varphi}:G\to\operatorname{\mathsf{Homeo}}(\mathbb{R})$, defined by
conjugating $\varphi$ by the order-reversing isometry $x\mapsto-x$. After our
definition, the actions $\varphi$ and $\widehat{\varphi}$ are conjugate.

Given a monotone map $h:\mathbb{R}\to\mathbb{R}$, we denote by
$\operatorname{\mathsf{Gap}}(h)\subset\mathbb{R}$ the open subset of points at
which $h$ is locally constant. We also write
$\operatorname{\mathsf{Core}}(h)=\mathbb{R}\smallsetminus\operatorname{\mathsf{Gap}}(h)$.
Note that when $h:\mathbb{R}\to\mathbb{R}$ is a semi-conjugacy between two
actions $\varphi,\psi:G\to\operatorname{\mathsf{Homeo}}(\mathbb{R})$ without
fixed points (in the sense that the equivariance (2.1) holds), then
$\operatorname{\mathsf{Core}}(h)$ is a closed $\varphi(G)$-invariant subset
and $h(\operatorname{\mathsf{Core}}(h))=h(\mathbb{R})$. The following folklore
result is a sort of converse to this observation, and it describes the inverse
operation of Denjoy’s blow up of orbits (compare for instance [KKMj, Theorem
2.2]).

######  Theorem 2.5.

Let $\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ be an action
without fixed points and let $F\subset\mathbb{R}$ be a non-empty closed
$\varphi(G)$-invariant subset. Then there exists an action
$\varphi_{F}:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ and a
continuous positive semi-conjugacy $h:\mathbb{R}\to\mathbb{R}$ between
$\varphi$ and $\varphi_{F}$ such that $\operatorname{\mathsf{Core}}(h)=F$.

We follow with an easy application of Theorem 2.5 that will be repeatedly used
in the article when discussing actions coming from quotients.

######  Corollary 2.6.

Let $\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ be an action
without fixed points, and let $N\lhd G$ be a normal subgroup. Then $\varphi$
is semi-conjugate to an action of the quotient $G/N$ if and only if
$\operatorname{\mathsf{Fix}}^{\varphi}(N)\neq\varnothing$.

###### Proof.

Notice that, since $N$ is normal, the subset
$F=\operatorname{\mathsf{Fix}}^{\varphi}(N)$ is closed and
$\varphi(G)$-invariant. Assume that $F\neq\varnothing$; we consider the action
$\varphi_{F}$ given by Theorem 2.5. Clearly we have
$N\subseteq\operatorname{\mathsf{ker}}\varphi_{F}$. The other implication is
trivial. ∎

The following lemma basically states that the semi-conjugacy class is
determined by the action on one orbit.

######  Lemma 2.7.

Consider two actions
$\varphi,\psi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ without fixed
points. Let $\Omega\subset\mathbb{R}$ be a non-empty $\varphi(G)$-invariant
subset and $j_{0}:\Omega\to\mathbb{R}$ a monotone map which is equivariant in
the sense that it satisfies

$\psi(g)j_{0}=j_{0}\varphi(g)\restriction_{\Omega}\quad\text{for every }g\in
G.$

Then, $j_{0}$ can be extended to a semi-conjugacy $j:\mathbb{R}\to\mathbb{R}$
between $\varphi$ and $\psi$.

###### Proof.

Consider the map $j:\mathbb{R}\to\mathbb{R}$ defined by

$j(x)=\sup\\{j_{0}(y):y\in\Omega\text{ and }y\leq x\\}.$

Since $j_{0}$ is monotone, we get that $j$ is an extension of $j_{0}$. Also
note, it is direct from the definition of $j$ that it is monotone and
equivariant and therefore it defines a semi-conjugacy as desired. ∎

####  2.1.4. Canonical model

We next introduce a family of canonical representatives for the semi-conjugacy
relation when $G$ is finitely generated. Such representatives are well defined
up to conjugacy. Later, in §2.2, we will define a (much less redundant) family
of representatives for the semi-conjugacy relation consisting on normalized
$\mu$-harmonic actions.

######  Definition 2.8.

Let $\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ be an action,
we say that a non-empty subset $\Lambda\subset\mathbb{R}$ is a _minimal
invariant set_ for $\varphi(G)$ if it is closed, $\varphi(G)$-invariant and
contains no proper, closed and $\varphi(G)$-invariant subsets. When
$\Lambda=\mathbb{R}$, we say that the action $\varphi$ is _minimal_. When
$\Lambda$ is a proper perfect subset we say that $\Lambda$ is an _exceptional
minimal invariant set_. On the other hand, when the image of an action
$\varphi(G)$ is generated by a homeomorphism without fixed points, we say that
the action $\varphi$ is _cyclic_.

######  Remark 2.9.

Note that minimal invariant sets may not exist for general group actions. An
archetypical example is given by an action of the group
$\mathbb{Z}^{\infty}=\bigoplus_{\mathbb{Z}}\mathbb{Z}$ in which each canonical
generator has fixed points but there is no fixed point for the action (this
happens, for instance, in the dynamical realization of the lexicographic
ordering of $\mathbb{Z}^{\infty}$). However, when the acting group is finitely
generated there is always a minimal invariant set for the action; see e.g.
[Navas-book, Proposition 2.1.12].

######  Remark 2.10.

Let $h:\mathbb{R}\to\mathbb{R}$ be a semi-conjugacy between two actions
$\varphi,\psi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$, in the sense
that (2.1) holds. It is immediate to show that when $\psi$ is minimal, the
semi-conjugacy $h$ is continuous, and that when $\varphi$ is also minimal,
then $h$ is a conjugacy. Indeed, the subsets $\operatorname{\mathsf{Core}}(h)$
and $\overline{h(\mathbb{R})}$ are closed subsets which are respectively
$\varphi(G)$\- and $\psi(G)$-invariant.

The following notion corresponds to the _minimalization_ considered in [KKMj,
Definition 2.3].

######  Definition 2.11.

An action $\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is a
_canonical model_ if either it is a minimal action or it is a cyclic action.

We have the following consequence of Theorem 2.5.

######  Corollary 2.12.

Let $\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ be an action
without fixed points that admits a non-empty minimal invariant set (this is
automatic if $G$ is finitely generated). Then, $\varphi$ is semi-conjugate to
a canonical model. Moreover, the canonical model is unique up to conjugacy.

A similar discussion holds for actions on the circle. In this case we say that
an action $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{S}^{1})$ is in canonical model
if it is either minimal or conjugate to an action whose image is a finite
cyclic group of rotations (in which case we say that it is _cyclic_). However
a crucial simplifying difference is that in this case, by compactness, _every_
group action on the circle admits a non-empty minimal invariant set
(regardless on whether $G$ is finitely generated or not). This yields the
following well-known fact (see e.g. [Ghys]).

######  Proposition 2.13.

Let $G$ be a group, then every action $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{S}^{1})$ without fixed points is
semi-conjugate to a canonical model.

After Corollary 2.12, semi-conjugacy classes of finitely generated group
actions can be divided into two very different families: cyclic actions and
minimal actions. However, for practical purposes, it is preferable to split
further the case of minimal action. In this work the following classical
notion will be important.

######  Definition 2.14.

For $M\in\\{\mathbb{R},\mathbb{S}^{1}\\}$, we say that a minimal action
$\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(M)$ is _proximal_ if for every
non-empty open intervals $I,J\subsetneq M$ with $I$ bounded, there exists an
element $g\in G$ such that $g.I\subset J$.

######  Remark 2.15.

Note that if a subgroup $G\subseteq\operatorname{\mathsf{Homeo}}_{0}(M)$
commutes with a non-trivial element
$h\in\operatorname{\mathsf{Homeo}}_{0}(M)$, then the $G$-action cannot be
proximal. In fact this is a well-known observation for group actions on
arbitrary compact spaces.

A crucial feature of minimal actions on the circle and the line is that a
converse also holds. In the case of the circle this goes back to Antonov
[Antonov] and was rediscovered by Margulis [Margulis], and an analogous result
for the real line is provided by Malyutin [Malyutin]. Given a subgroup
$G\subseteq\operatorname{\mathsf{Homeo}}_{0}(M)$ with
$M\in\\{\mathbb{R},\mathbb{S}^{1}\\}$, its _centralizer_ (in
$\operatorname{\mathsf{Homeo}}_{0}(M)$) is the subgroup

$C(G):=\left\\{h\in\operatorname{\mathsf{Homeo}}_{0}(M):gh=hg\text{ for every
}g\in G\right\\}.$

We will write $C^{\varphi}:=C({\varphi(G)})$ when
$\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(M)$ is an action. Then we have
the following two results.

######  Theorem 2.16 (Antonov [Antonov], see [Ghys]).

Let $\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{S}^{1})$ be a
minimal action. Then we have the following alternative:

  * •

either $C^{\varphi}$ is trivial, in which case $\varphi$ is proximal, or

  * •

$C^{\varphi}$ is a finite cyclic subgroup without fixed points, and the action
on the topological circle $\mathbb{S}^{1}/C^{\varphi}$ is proximal, or

  * •

$C^{\varphi}$ is conjugate to the group of rotations $\mathbb{R}/\mathbb{Z}$
and $G$ is conjugate to a subgroup of it.

######  Theorem 2.17 (Malyutin [Malyutin], see also [GOD]).

Let $\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ be a minimal
action. Then we have the following alternative:

  * •

either $C^{\varphi}$ is trivial, in which case $\varphi$ is proximal, or

  * •

$C^{\varphi}$ is a cyclic subgroup without fixed points, and the action on the
topological circle $\mathbb{R}/C^{\varphi}$ is proximal, or

  * •

$C^{\varphi}$ is conjugate to the group of translations $(\mathbb{R},+)$ and
$G$ is conjugate to a subgroup of it.

###  2.2. Harmonic actions and Deroin spaces

In this subsection we recall the definition of the Deroin space. For that, we
fix a finitely generated group $G$, a probability measure $\mu$ on $G$ whose
support is finite, generates $G$, and satisfies the symmetry property

$\mu(g)=\mu(g^{-1})\quad\text{for every }g\in G.$

We will denote by $S\subset G$ the support of $\mu$.

######  Definition 2.18.

Given a nontrivial action
$\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$, we say that a
Radon measure $\nu$ on $\mathbb{R}$ is _stationary_ for the action $\varphi$
if for every Borel subset $A\subset\mathbb{R}$ one has

$\nu(A)=\sum_{g\in S}\nu(g^{-1}.A)\,\mu(g).$

When the Lebesgue measure $\operatorname{\mathsf{Leb}}$ on $\mathbb{R}$ is
stationary, we say that the action $\varphi$ is _$\mu$ -harnomic_.

Properties of $\mu$-harmonic actions are discussed in [GOD, §4.4], and
strongly rely on results of Deroin, Kleptsyn, Navas, and Parwani [DKNP] on
symmetric random walks in $\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$. As
remarked by Deroin, the space of $\mu$-harmonic actions of $G$ has several
remarkable properties. To describe this, we need first to discuss two
fundamental results.

######  Proposition 2.19 ([DKNP]).

With notation as above, the following properties hold.

  1. (1)

Every non-trivial $\mu$-harmonic action
$\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is a canonical
model.

  2. (2)

Conversely, every canonical model
$\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is conjugate to a
$\mu$-harmonic action
$\psi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$.

  3. (3)

Moreover, any two $\mu$-harmonic actions $\psi_{1}$ and $\psi_{2}$ of a group
$G$ are conjugate by an affine homeomorphism.

After a careful reading of the proof of [DKNP, Proposition 8.1], there is a
natural way to reduce the symmetries in part (3) of the statement above, to
the group of translations. For this, given a homeomorphism
$h\in\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ consider the following
area function:

$\mathsf{A}^{h}(\xi)=\left\\{\begin{array}[]{lr}\int_{h^{-1}(\xi)}^{\xi}(h(\eta)-\xi)\,d\eta&\text{if
}h(\xi)\geq\xi,\\\\[5.0pt]
\int_{h(\xi)}^{\xi}(h^{-1}(\eta)-\xi)\,d\eta&\text{if }h(\xi)\leq\xi.\\\
\end{array}\right.$

Note that $\mathsf{A}^{h}(\xi)\geq 0$ for every $\xi\in\mathbb{R}$, and
$\mathsf{A}^{h}(\xi)=0$ if and only if $\xi\in\operatorname{\mathsf{Fix}}(h)$.
In the case $h(\xi)\geq\xi$, the quantity $\mathsf{A}^{h}(\xi)$ represents
indeed the area of the bounded planar region delimited by the segments
$[h^{-1}(\xi),\xi]\times\\{\xi\\}$, $\\{\xi\\}\times[\xi,h(\xi)]$ and the
graph of $h$ (when $h(\xi)\leq\xi$, one has to switch the endpoints of the
segments, and the same interpretation is valid). Being a two-dimensional area,
it should be clear that when considering the map $\widetilde{h}=aha^{-1}$,
obtained by conjugating $h$ by an affine map $a(\xi)=\lambda x+b$, one has the
following relation:

(2.2)
$\lambda^{2}\mathsf{A}^{h}(\xi)=\mathsf{A}^{\widetilde{h}}(a(\xi))\quad\text{for
every }\xi\in\mathbb{R}.$

######  Lemma 2.20 ([DKNP]).

With notation as above, let
$\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ be a
$\mu$-harmonic action. Then the expected area function
$\mathsf{A}^{\varphi}:\mathbb{R}\to\mathbb{R}$ defined by

$\mathsf{A}^{\varphi}(\xi)=\sum_{g\in S}\mathsf{A}^{\varphi(g)}(\xi)\,\mu(g),$

is constant and positive.

######  Definition 2.21.

With notation as above, we introduce the _Deroin space_
$\operatorname{\mathsf{Der}}_{\mu}(G)$ of normalized $\mu$-harmonic actions as
the subset of $\mu$-harmonic actions
$\varphi\in\operatorname{\mathsf{Hom}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
such that $\mathsf{A}^{\varphi}=1$, endowed with the induced compact-open
topology.

Note that as the group $G$ is finitely generated by the subset $S$, the
compact-open topology of
$\operatorname{\mathsf{Hom}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
is simply given by the product topology of
$\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})^{S}$, where
$\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is considered with the
topology of uniform convergence on compact subsets.

######  Theorem 2.22 ([DKNP, GOD]).

With notation as above, the following properties hold.

  1. (1)

The Deroin space $\operatorname{\mathsf{Der}}_{\mu}(G)$ is compact.

  2. (2)

Every action $\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(G)$ without fixed
points is positively semi-conjugate to a normalized $\mu$-harmonic action
$\psi\in\operatorname{\mathsf{Der}}_{\mu}(G)$, which is unique up to conjugacy
by a translation.

  3. (3)

Conjugacy by translations defines a topological flow
$\Phi:\mathbb{R}\times\operatorname{\mathsf{Der}}_{\mu}(G)\to\operatorname{\mathsf{Der}}_{\mu}(G)$,
called the _translation flow_. Explicitly, this is given by

$\Phi^{t}(\varphi)(g)=T_{t}\varphi(g)T_{-t},$

where $T_{t}:\xi\to\xi+t$ denotes the translation by $t\in\mathbb{R}$.

###### Sketch of proof.

After the identity (2.2) and Lemma 2.20, if $\varphi$ is a normalized
$\mu$-harmonic action, then also $\Phi^{t}(\varphi)$ is normalized for every
$t\in\mathbb{R}$. So the translation flow is well-defined.

After Corollary 2.12 and Proposition 2.19, we know that every action
$\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ without fixed
points, is semi-conjugate to a $\mu$-harmonic action, which is unique up to
affine rescaling. Moreover, the identity (2.2) and Lemma 2.20 give that after
an affine rescaling, we can assume that the $\mu$-harmonic action is
normalized. This gives (2).

Details for the fact that $\operatorname{\mathsf{Der}}_{\mu}(G)$ is compact
can be found in [deroin2020non, Proof of Theorem 5.4]. ∎

Although this will not be used, note that the group $G$ acts on the Deroin
space $\operatorname{\mathsf{Der}}_{\mu}(G)$ by the formula
$g.\varphi=\Phi^{\varphi(g)(0)}(\varphi)$ (see [GOD, (4.5)]). In other words,
the action on the parameterized $\Phi$-orbit of $\varphi$ is basically the
action $\varphi$.

Let us point out the following consequences of Theorem 2.22.

######  Corollary 2.23.

With notation as above, let $\varphi_{1}$ and $\varphi_{2}$ be two normalized
$\mu$-harmonic actions in $\operatorname{\mathsf{Der}}_{\mu}(G)$, which are
positively conjugate by a homeomorphism $h:\mathbb{R}\to\mathbb{R}$ that fixes
$0$. Then $\varphi_{1}=\varphi_{2}$.

######  Corollary 2.24.

With notation as above, let $\varphi_{1}$ and $\varphi_{2}$ be two normalized
$\mu$-harmonic actions in $\operatorname{\mathsf{Der}}_{\mu}(G)$. Then
$\varphi_{1}$ and $\varphi_{2}$ are positively conjugate if and only if they
belong to the same $\Phi$-orbit.

###  2.3. Preorders and group actions

####  2.3.1. Preordered sets

In this work, by a _preorder_ on a set $X$ we mean a transitive binary
relation $\leq$ which is _total_ , in the sense that for every $x,y\in X$ we
have $x\leq y$ or $y\leq x$ (possibly both relations can hold).

We write $x\lneq y$ whenever $x\leq y$ but it does not hold that $y\leq x$. On
the other hand, when $x\leq y$ and $y\leq x$ we say that $x$ and $y$ are
_equivalent_ and denote by $[x]_{\leq}$ the equivalence class of $x$ (we will
simply write $[x]$ when there is no risk of confusion).

######  Remark 2.25.

A preorder $\leq$ on $X$ induces a _total_ order on the set of equivalence
classes $\\{[x]_{\leq}:x\in X\\}$, by declaring $[x]<[y]$ whenever $x\lneq y$.

######  Definition 2.26.

We say that a map between preordered sets
$f:(X_{1},\leq_{1})\to(X_{2},\leq_{2})$ is _(pre)order-preserving_ if
$x\leq_{1}y$ implies $f(x)\leq_{2}f(y)$.

On the other hand, given a map $f:X_{1}\to(X_{2},\leq)$ we define the _pull-
back_ of $\leq$ by $f$ as the preorder $f^{\ast}(\leq)$ on $X_{1}$, denoted by
$\preceq$ here, so that $x_{1}\preceq x_{2}$ if and only if $f(x_{1})\leq
f(x_{2})$.

######  Definition 2.27.

Let $(X,\leq)$ be a preordered set. An _automorphism_ of $(X,\leq)$ is an
order-preserving bijection $f:(X,\leq)\to(X,\leq)$, whose inverse is also
order-preserving. We denote by $\operatorname{\mathsf{Aut}}(X,\leq)$ the group
of all automorphisms of $(X,\leq)$.

An _order-preserving action_ of a group $G$ on a preordered set $(X,\leq)$ is
a homomorphism $\psi:G\to\operatorname{\mathsf{Aut}}(X,\leq)$.

######  Remark 2.28.

Let $G$ be a group with actions
$\psi_{1}:G\to\operatorname{\mathsf{Sym}}(X_{1})$ and
$\psi_{2}:G\to\operatorname{\mathsf{Aut}}(X_{2},\leq)$. Let
$f:X_{1}\to(X_{2},\leq)$ be an equivariant map. Then the preorder
$f^{*}(\leq)$ is preserved by $\psi_{1}(G)$.

######  Definition 2.29.

Let $(X,\leq)$ be a preordered set and let $A\subset X$ be a subset. We say
that $A$ is _$\leq$ -convex_ if whenever $x\leq y\leq z$ and $x,z\in A$ it
holds that $y\in A$.

######  Remark 2.30.

It is a direct consequence of the definitions that when
$f:(X_{1},\leq_{1})\to(X_{2},\leq_{2})$ is order-preserving and
$A\subset(X_{2},\leq_{2})$ is $\leq_{2}$-convex, then the preimage $f^{-1}(A)$
is $\leq_{1}$-convex. This fact will be used several times without direct
reference.

######  Definition 2.31.

Given a partition $\mathfrak{P}$ of $X$, denote by $\mathfrak{P}(x)$ the atom
of $\mathfrak{P}$ containing the point $x\in X$. We say that a partition
$\mathfrak{P}$ of a preordered set $(X,\leq)$ is _$\leq$ -convex_ is every
atom of $\mathfrak{P}$ is a $\leq$-convex subset of $(X,\leq)$. Compare this
with Remark 2.25.

######  Remark 2.32.

When $\mathfrak{P}$ is a $\leq$-convex partition of a preordered set
$(X,\leq)$, there exists a total order $<_{\mathfrak{P}}$ on $\mathfrak{P}$
defined by the condition that $\mathfrak{P}(x)<_{\mathfrak{P}}\mathfrak{P}(y)$
if and only if $\mathfrak{P}(x)\neq\mathfrak{P}(y)$ and $x\lneq y$ (it is
immediate to verify that this does not depend on the choice of points $x$ and
$y$).

####  2.3.2. Preorders on groups

A preorder on a group $G$ is _left-invariant_ if $h\leq k$ implies $gh\leq gk$
for all $g,h,k\in G$. In other words, the left-multiplication gives an action
by automorphisms $G\to\operatorname{\mathsf{Aut}}(G,\leq)$. Recall also that a
preorder on $G$ is _bi-invariant_ if it preserved by left and right
multiplications.

By invariance and Remark 2.25, given a left-invariant preorder $\leq$ on $G$,
the equivalence class $[1]_{\leq}$ is a subgroup of $G$ and that $\leq$ is a
left-invariant order on $G$ if and only if $[1]_{\leq}=\\{1\\}$. The subgroup
$[1]_{\leq}$ is called the _residue_. We say that a left-invariant preorder
$\leq$ on $G$ is _trivial_ whenever $[1]_{\leq}=G$ and _non-trivial_
otherwise. We denote by $\operatorname{\mathsf{LPO}}(G)$ the set of all non-
trivial left-invariant preorders on $G$.

From now on, by a preorder on a group, we _always_ mean a non-trivial left-
invariant preorder. We endow $\operatorname{\mathsf{LPO}}(G)$ with the product
topology induced by the realization of preorders as subsets of
$\\{\leq,\gneq\\}^{G\times G}$. It turns out that with this topology,
$\operatorname{\mathsf{LPO}}(G)$ is a metrizable and totally disconnected
topological space, which is moreover compact whenever $G$ is finitely
generated (see Antolín and the third author [AntolinRivas], where preorders
are called relative orders). For a modern treatment of left-orders and left-
preorders see [GOD] and [AntolinDicksSunic, decaup2019preordered].

######  Definition 2.33.

Let $G$ be a group and let $\leq\in\operatorname{\mathsf{LPO}}(G)$ be a
preorder. The _positive cone_ of $\leq$ is the subset $P_{\leq}=\\{g\in
G:g\gneq 1\\}$. Similarly, the subset $N_{\leq}=\\{g\in G:g\lneq 1\\}$ defines
the _negative cone_ of $\leq$.

######  Remark 2.34.

A preorder $\leq\in\operatorname{\mathsf{LPO}}(G)$ induces a partition
$G=P_{\leq}\sqcup[1]_{\leq}\sqcup N_{\leq}$. Also note that $P_{\leq}$ and
$N_{\leq}$ are semigroups and satisfy $P_{\leq}^{-1}=N_{\leq}$ where
$P_{\leq}^{-1}:=\\{g^{-1}:g\in P_{\leq}\\}$. Reciprocally, given a partition
$G=P\sqcup H\sqcup N$ such that

  1. (1)

$P$ is a semigroup,

  2. (2)

$N=P^{-1}$,

  3. (3)

$H$ is a proper (possibly trivial) subgroup and

  4. (4)

$HPH\subseteq P$,

there exists a preorder $\leq\in\operatorname{\mathsf{LPO}}(G)$ such that
$P=P_{\leq}$, $H=[1]_{\leq}$ and $N=N_{\leq}$. See [decaup2019preordered] for
details.

When $H$ is a $\leq$-convex subgroup, the left coset space $G/H=\\{gH:g\in
G\\}$ defines a $\leq$-convex partition of $G$. The total order $<_{G/H}$
induced on $G/H$ by $\leq$ (see Remark 2.32) is preserved by left-
multiplication of $G$ on $G/H$. Given a $\leq$-convex subgroup $H\subseteq G$,
we define the _quotient preorder_ of $\leq$ under $H$ as the preorder
$\leq_{H}\in\operatorname{\mathsf{LPO}}(G)$ given by the pull-back
$\leq_{H}:=p^{\ast}(<_{G/H})$, where $p:G\to G/H$ is the coset projection.
Equivalently we can define $\leq_{H}$ by setting
$P_{\leq_{H}}=P_{\leq}\smallsetminus H$.

######  Remark 2.35.

After Remark 2.28, the identity map $\mathsf{id}:(G,\leq)\to(G,\leq_{H})$ is
order-preserving and equivariant with respect to the actions of $G$ by left-
multiplication.

####  2.3.3. Dynamical realizations of actions on totally ordered sets

One general principle that we often use is that for building actions of a
group $G$ on the line by homeomorphisms, one may start by finding a totally
ordered space $(X,<)$ and an action by order-preserving bijections
$\psi:G\to\operatorname{\mathsf{Aut}}(X,<)$. Then if the order topology on $X$
is nice enough, for instance when $X$ is countable, then the action $\psi$ can
be “completed” to an action
$\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ so that, there
exists an order-preserving map $i:X\to\mathbb{R}$ satisfying that
$\varphi(g)(i(x))=i(\psi(g)(x))$ for all $g\in G.$ Under some mild extra
conditions, we call such $\varphi$ a _dynamical realization_ of $\psi$. See
Definition 2.39.

######  Remark 2.36.

It is a classical fact that a countable group is left-orderable if and only if
it _embeds_ into $\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ (a fact that
we trace back to Conrad [Conrad] in the abelian setting, see [GOD, Ghys] for a
proof of the general case). In fact, one side of the proof consist in building
a dynamical realization of the left-regular representation of a countable
left-ordered group on itself. Analogously, one can show that a countable group
admits a left-invariant preorder if and only if it admits a (possibly not
faithful) non-trivial action by order-preserving homeomorphisms of the real
line [AntolinRivas].

We now proceed to formally define what we mean by dynamical realization.

######  Definition 2.37.

Let $(X,<)$ be a countable totally ordered set. We say that an injective
order-preserving map $i:X\to\mathbb{R}$ is a _good embedding_ if its image is
unbounded in both directions and every connected component $I$ of the
complement of $\overline{i(X)}$ satisfies $\partial I\subset i(X)$.

######  Remark 2.38.

Following the classical construction of the dynamical realization of a
countable left-ordered group (see [GOD, Ghys]), it follows that any countable
and totally ordered set $(X,<)$ has a good embedding and that, given two
different good embeddings $i_{1},i_{2}:X\to\mathbb{R}$, there exists
$h\in\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ so that $i_{2}=h\circ
i_{1}$.

######  Definition 2.39.

Assume that $\psi\colon G\to\operatorname{\mathsf{Aut}}(X,<)$ is a group
action. An action $\varphi\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ is said to be a _dynamical
realization_ of $\psi$ if there exists a good embedding $\iota\colon
X\to\mathbb{R}$ such that the following hold:

  1. (1)

$i$ is equivariant for $\varphi$ and $\psi$;

  2. (2)

for every connected component $I$ of the complement of $\overline{i(X)}$ the
(setwise) stabilizer $\operatorname{\mathsf{Stab}}^{\varphi}_{G}(I)$ of $I$ in
$G$ acts trivially on $I$.

######  Lemma 2.40.

Consider an action $\psi:G\to\operatorname{\mathsf{Aut}}(X,<)$ where $(X,<)$
is a countable totally ordered set. Then a dynamical realization of $\psi$
exists and is unique up to positive conjugacy.

###### Sketch of proof.

Consider a good embedding $i:X\to\mathbb{R}$. Then, by transporting the action
$\psi$ through $i$ on $i(X)$ and extending it by continuity to the closure we
obtain an action
$\varphi_{0}:G\to\operatorname{\mathsf{Homeo}}_{0}(\overline{i(X)})$. For
every $g\in G$ denote by $\varphi(g)$ the extension of $\varphi_{0}(g)$ which
is affine on every connected component $I$ of $\overline{i(X)}$. It is direct
to check that $g\mapsto\varphi(g)$ is a dynamical realization of $\psi$.

Now, consider two actions
$\varphi_{1},\varphi_{2}:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$
which are dynamical realizations of $\psi$ with associated good embeddings
$i_{1},i_{2}:X\to\mathbb{R}$ respectively. Note that by Remark 2.38 there
exists $h\in\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ with $i_{2}=h\circ
i_{1}$ and after conjugating $\varphi_{1}$ by $h$ we can suppose that
$i_{1}=i_{1}=:i$. By equivariance $\varphi_{1}$ and $\varphi_{2}$ must
coincide on $\overline{i(X)}$. Let $\Omega$ be the set of connected components
of $\mathbb{R}\smallsetminus\overline{i(X)}$ and note that the $G$-actions on
$\Omega$ induced by $\varphi_{1}$ and $\varphi_{2}$ coincide, so that the set
of orbits $\Omega/G$ does not depend on the action. For $J\in\Omega$ we denote
by $\alpha(J)\in\Omega/G$ its $G$-orbit. Pick a system of representatives
$\\{I_{\alpha}\\}_{\alpha\in{\Omega/G}}$ of orbits. For $J\in\Omega$ choose
$g_{J}\in G$ such that $g_{J}(J)=I_{\alpha(J)}$, and for $i\in\\{1,2\\}$ set
$f_{i,J}=\varphi_{i}(g_{J})\restriction_{J}$. Note that each $f_{i,J}$ is a
homeomorphism from $J$ to $I_{\alpha(J)}$ which does not depend on the choice
of $g_{J}$ by the assumption (2) in the definition of dynamical realization.
Thus $f_{2,J}^{-1}f_{1,J}$ is a self-homeomorphism of $J$. Define a map
$q\colon\mathbb{R}\to\mathbb{R}$ which is the identity on $\overline{i(X)}$
and satisfies $q\restriction_{J}=f_{2,J}^{-1}f_{1,J}$ for every $J\in\Omega$.
Then one readily checks that $q$ conjugates $\varphi_{2}$ to $\varphi_{1}$. ∎

We proceed to give a sufficient condition for minimality of dynamical
realizations of actions on totally ordered sets. For this, we need the
following definition.

######  Definition 2.41.

Let $(X,<)$ be a totally ordered set. We say that a subgroup
$H\subseteq\operatorname{\mathsf{Aut}}(X,<)$ is of _homothetic type_ if the
following conditions are satisfied.

  1. (1)

There exists $x_{0}\in\mathsf{Fix}(H)$.

  2. (2)

For every $x\in X\smallsetminus\\{x_{0}\\}$, there exists a sequence of
elements $(h_{n})\subset H$ such that $h_{n}(x)\to+\infty$ if $x>x_{0}$, and
$h_{n}(x)\to-\infty$ if $x<x_{0}$.

When the cyclic subgroup $\langle g\rangle$ is of homothetic type we say that
$g$ is a _homothety_.

######  Lemma 2.42 (Minimality criterion).

Let $(X,<)$ be a countable totally ordered set and consider an action
$\psi:G\to\operatorname{\mathsf{Aut}}(X,<)$. Assume that for every four points
$x_{1},x_{2},y_{1},y_{2}$ of $X$ with

$x_{1}<y_{1}<y_{2}<x_{2},$

there exists $g\in G$ such that

$g.y_{1}<x_{1}<x_{2}<g.y_{2}.$

Then the dynamical realization of $\psi$ is minimal.

###### Proof.

Let $\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ be a dynamical
realization of $\psi$ with its associated good embedding $i:X\to\mathbb{R}$
(Lemma 2.40). We first claim that $i$ has dense image. Suppose by
contradiction that it is not the case and take a connected component
$(\eta_{1},\eta_{2})$ of the complement of $\overline{i(X)}$. Since $i$ is a
good embedding we have that $\\{\eta_{1},\eta_{2}\\}\subset i(X)$. Choose two
points $\xi_{1}$ and $\xi_{2}$ in $i(X)$ such that
$(\eta_{1},\eta_{2})\subsetneq(\xi_{1},\xi_{2})$; by our assumption, we can
find an element $g\in G$ such that $(\eta_{1},\eta_{2})\supsetneq
g.(\xi_{1},\xi_{2})$, contradicting the $\varphi(g)$-invariance of $i(X)$.
This shows that $i$ has dense image.

Suppose again by contradiction that there exists a proper closed
$\varphi(G)$-invariant subset $\Lambda\subset\mathbb{R}$. Take a connected
component $(\eta_{1},\eta_{2})$ of $\mathbb{R}\smallsetminus\Lambda$. By
density of $i(\mathbb{R})$, we can find four points
$\xi_{1},\xi_{2},\zeta_{1},\zeta_{2}$ in $i(X)$ such that

$\zeta_{1}<\eta_{1}<\xi_{1}<\xi_{2}<\eta_{2}<\zeta_{2}.$

After our assumption, we can find an element $g\in G$ such that
$(\eta_{1},\eta_{2})\supsetneq g.(\zeta_{1},\zeta_{2})$. This contradicts the
$\varphi(g)$-invariance of $\Lambda$ showing that $\varphi$ is minimal, as
desired. ∎

The condition in Lemma 2.42 is met in the following situation.

######  Proposition 2.43.

Let $(X,<)$ be a countable totally ordered set and consider an action
$\psi:G\to\operatorname{\mathsf{Aut}}(X,<)$. If for every $x\in X$ there
exists a subgroup $H\subseteq G$ such that $\psi(H)$ is a subgroup of
homothetic type fixing $x$, the dynamical realization of $\psi$ is minimal.

###### Proof.

Consider four points $x_{1},x_{2},y_{1},y_{2}$ of $X$ with
$x_{1}<y_{1}<y_{2}<x_{2}$. By assumption, we can find an element $h_{1}\in
H_{y_{1}}$ such that $y_{1}<h_{1}.y_{2}<y_{2}$. Consider the point
$x_{\ast}=h_{1}.y_{2}$, so that there exists $h_{\ast}\in H_{x_{\ast}}$ such
that $h_{\ast}.y_{1}<x_{1}<x_{2}<h_{\ast}.y_{2}$. Thus Lemma 2.42 applies. ∎

###  2.4. Bieri–Strebel groups

To illustrate our results we will often consider examples of locally moving
groups arising as groups of piecewise linear (PL) homeomorphisms of the line.
Let us briefly fix the terminology and recall a large family of such groups,
following Bieri and Strebel [BieriStrebel].

We say that a homeomorphism $f\colon X\to X$ of an interval
$X\subset\mathbb{R}$ (bounded or unbounded) is _piecewise linear_ (PL, for
short) if there exists a discrete subset $\Sigma\subset X$ such that in
restriction to $X\smallsetminus\Sigma$, the map $f$ is locally affine, that
is, of the form $x\mapsto\lambda x+a$, with $\lambda>0$ and $a\in\mathbb{R}$.
We denote by $\operatorname{\mathsf{BP}}(f)$ the minimal subset $\Sigma$
satisfying such condition, and points of $\operatorname{\mathsf{BP}}(f)$ will
be the _breakpoints_ of $f$. When $\operatorname{\mathsf{BP}}(f)$ is finite,
we say that $f$ is _finitary_ PL. Note that with this definition, a PL
homeomorphism always preserves the orientation.

######  Definition 2.44.

Given a non-trivial multiplicative subgroup
$\Lambda\subseteq\mathbb{R}_{+}^{*}$, and a non-trivial
$\mathbb{Z}[\Lambda]$-submodule $A\subset\mathbb{R}$, the _Bieri–Strebel group
$G(X;A,\Lambda)$_ is the group of finitary PL homeomorphisms $f:X\to X$ with
the following properties:

  1. (1)

breakpoints of $f$ are in $A$,

  2. (2)

in restriction to $X\smallsetminus\operatorname{\mathsf{BP}}(f)$, the map $f$
is locally of the form $\lambda x+a$, with $\lambda\in\Lambda$ and $a\in A$.

For example, Thompson’s $F$ is $G([0,1];\mathbb{Z}[1/2],\langle
2\rangle_{*})$. Other interesting examples are provided by the so-called
Thompson–Brown–Stein groups.

######  Definition 2.45.

Let $1<n_{1}<\dots<n_{k}$ be natural numbers such that the multiplicative
subgroup $\Lambda=\langle n_{i}\rangle\subseteq\mathbb{R}_{+}^{*}$ is an
abelian group of rank $k$. Denote by $A$ the ring
$\mathbb{Z}\left[1/m\right]$, where $m$ is the least common multiple of the
$n_{i}$. The group

$F_{n_{1},\ldots,n_{k}}:=G([0,1];A,\Lambda)$

is the corresponding _Thompson–Brown–Stein group_.

The group $F_{2}$ is simply Thompson’s group $F$. For every $n\geq 2$, the
group $F_{n}$ is isomorphic to a subgroup of $F$, and these groups were first
considered by Brown in [Brown], inspired by the so-called Higman–Thompson
groups. Later Stein [Stein] started the investigation of the more general
class of groups $F_{n_{1},\ldots,n_{k}}$. She proved that every
Thompson–Brown–Stein group is finitely generated and even finitely presented.
Moreover, given any $m$-adic interval $I\subset[0,1]$ (that is, an interval
with endpoints in $\mathbb{Z}[1/m]$), the subgroup
$\left(F_{n_{1},\ldots,n_{k}}\right)_{I}$ is isomorphic to
$F_{n_{1},\ldots,n_{k}}$.

We refer to the classical monograph by Bieri and Strebel [BieriStrebel] for an
extensive investigation of the groups $G(X;A,\Lambda)$.

######  Remark 2.46.

It would be interesting to see how the results of this text apply to other
groups of piecewise projective homeomorphisms, such as Monod’s groups $H(A)$
[Monod] (we will not pursue this task).

##  3\. Deroin spaces and preorders

Recall that we denote by
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
the space of representations of $G$ without fixed points. The following notion
is classical in dynamical systems.

######  Definition 3.1.

An action $\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ without
fixed points is _locally rigid_ if there exists a neighborhood $\mathcal{U}$
of $\varphi$ in
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$
consisting of representations all positively semi-conjugate to $\varphi$.

In other terms, every sufficiently small perturbation of the action $\varphi$
gives a positively semi-conjugate action. The main goal of this section is to
characterize local rigidity in terms of the topology of the orbits along the
translation flow in the Deroin space (see §2.2 for a definition of this
space). This will be a consequence of the following result.

######  Theorem 3.2.

Let $G$ be a finitely generated group, and consider a symmetric probability
measure $\mu$ whose support is finite and generates $G$. There exists a
continuous retraction

$r:\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))\to\operatorname{\mathsf{Der}}_{\mu}(G)$

which preserves positive semi-conjugacy classes.

######  Definition 3.3.

The map
$r:\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))\to\operatorname{\mathsf{Der}}_{\mu}(G)$
from Theorem 3.2 is called the _harmonic retraction_ of the representation
space
$\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$.

######  Corollary 3.4 (Local rigidity criterion).

Let $G$ be a finitely generated group, and consider a symmetric probability
measure $\mu$ whose support is finite and generates $G$. For
$\varphi\in\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))$,
let $\psi\in\operatorname{\mathsf{Der}}_{\mu}(G)$ be a normalized
$\mu$-harmonic action which is positively semi-conjugate to $\varphi$. If the
orbit along the translation flow of $\psi$ is open in
$\operatorname{\mathsf{Der}}_{\mu}(G)$, then $\varphi$ is locally rigid. The
converse hold provided $\varphi$ is minimal.

###### Proof.

Let $\mathcal{I}=\\{\Phi^{t}(\psi):t\in\mathbb{R}\\}$ be the orbit of $\psi$
in $\operatorname{\mathsf{Der}}_{\mu}(G)$ and let
$r\colon\operatorname{\mathsf{Hom}_{\mathrm{irr}}}(G,\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R}))\to\operatorname{\mathsf{Der}}_{\mu}(G)$
be the harmonic retraction (Theorem 3.2). Then $r(\varphi)$ is an element of
$\operatorname{\mathsf{Der}}_{\mu}(G)$ which is positively semi-conjugate to
$\psi$ and thus belongs to $\mathcal{I}$, i.e. $\varphi\in
r^{-1}(\mathcal{I})$. If $\mathcal{I}$ is open, then $r^{-1}(\mathcal{I})$ is
an open neighborhood of $\varphi$ consisting only of actions positively semi-
conjugate to $\psi$, so the claim follows. Conversely, assume that $\varphi$
is minimal. Since local rigidity is preserved under conjugacy we can assume
that $\varphi=\psi$. Thus if $\mathcal{I}$ is not open, using the translation
flow we find that $\varphi$ can be approximated by $\mu$-harmonic actions in
different $\Phi$-orbits. Since actions in
$\operatorname{\mathsf{Der}}_{\mu}(G)$ belonging to different $\Phi$-orbits
are not positively semi-conjugate (Corollary 2.24), this implies that
$\varphi$ is not locally rigid. ∎

######  Remark 3.5.

The requirement that $\varphi$ be minimal in the second part of Corollary 3.4
is essential. To see this consider the action
$\varphi:\mathbb{F}_{2}\times\mathbb{Z}\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$
obtained as the lift of the ping-pong action
$\varphi_{0}:\mathbb{F}_{2}\to\mathsf{PSL}(2,\mathbb{R})\subset\operatorname{\mathsf{Homeo}}_{0}(\mathbb{S}^{1})$,
given by the action of the fundamental group of a hyperbolic _one-holed_ torus
on the boundary at infinity of the hyperbolic plane; to make this more
concrete, one can consider the subgroup of $\mathsf{PSL}(2,\mathbb{R})$
generated by the two homographies $\begin{bmatrix}5&3\\\ 3&2\end{bmatrix}$ and
$\begin{bmatrix}1&1\\\ 1&2\end{bmatrix}$. The action $\varphi_{0}$ is then
non-minimal and locally rigid so the same holds for $\varphi$. However, the
canonical model of $\varphi_{0}$ corresponds to the action of the fundamental
group of a hyperbolic _one-punctured_ torus on the boundary at infinity of the
hyperbolic plane, and so this canonical model has a parabolic element (the
commutator of the generators). In particular, this minimal action and its
corresponding lift to the line are not locally rigid. Thus the representative
of $\varphi$ in
$\operatorname{\mathsf{Der}}_{\mu}(\mathbb{F}_{2}\times\mathbb{Z})$ has non-
isolated orbit along the translation flow.

In order to prove Theorem 3.2, we obtain an order-theoretic description of the
Deroin space. Note that not every group action is the dynamical realization of
some order, so we are forced to consider the much more flexible notion of
preorder. However, different preorders may correspond to the same conjugacy
class of a minimal action, and we have to translate this into an equivalence
relation on preorders. The appropriate equivalence relation on preorders is
introduced in §3.1. In §3.2, we make explicit the relations between properties
of preorders and their dynamical realizations. In §3.3 we investigate the
topology of the corresponding quotient space. Finally, in §3.4 we identify
such quotient space with the Deroin space and provide the proof of Theorem
3.2. As a by-product of this, we will get that the topology of
$\operatorname{\mathsf{Der}}_{\mu}(G)$ does not depend on the choice of the
probability measure $\mu$.

###  3.1. Equivalence of preorders

We start with the following definition which gives an order-theoretic analogue
of continuous positive semi-conjugacies between representations in
$\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$. Part of our discussion is
close to the exposition of Decaup and Rond [decaup2019preordered].

######  Definition 3.6.

Let $G$ be a group and let $\leq,\preceq\in\operatorname{\mathsf{LPO}}(G)$ be
two preorders. We say that $\leq$ _dominates_ $\preceq$ if the identity map
$\mathsf{id}:(G,\preceq)\to(G,\leq)$ is order-preserving.

######  Remark 3.7.

The direction in the definition of domination could appear counterintuitive,
but in fact it is justified when thinking in terms of preorders (it is even
clearer in terms of dynamical realization). Indeed, take an element $g\in G$,
and suppose that we want to know whether $g\succneq 1$. Then we first check
whether $g\gneq 1$ (or $g\lneq 1$), and only in the case $g\in[1]_{\leq}$, we
take the preorder $\preceq$ into consideration.

Note that when $\leq$ dominates $\preceq$, the subgroup $H=[1]_{\leq}$ is
$\preceq$-convex, thus the quotient preorder $\preceq_{H}$ is well-defined.
The following lemma characterizes domination in terms of positive cones.

######  Lemma 3.8.

Let $G$ be a group and let $\leq,\preceq\in\operatorname{\mathsf{LPO}}(G)$ be
two preoders. Then, the following are equivalent.

  1. (1)

$\leq$ dominates $\preceq$.

  2. (2)

$P_{\leq}\subseteq P_{\preceq}$.

  3. (3)

The subgroup $H=[1]_{\leq}$ is $\preceq$-convex, and
$P_{\leq}=P_{\preceq}\smallsetminus H$ (so that $\leq$ coincides with the
quotient preorder $\preceq_{H}$).

###### Proof.

We prove that (1) implies (3). As $\mathsf{id}:(G,\preceq)\to(G,\leq)$ is
order-preserving, the subgroup $H=[1]_{\leq}$, which coincides with its
preimage, is $\preceq$-convex. Moreover, we have the following inclusions:

$P_{\preceq}\subseteq
P_{\leq}\sqcup[1]_{\leq},\quad[1]_{\preceq}\subseteq[1]_{\leq},\quad
N_{\preceq}\subseteq[1]_{\leq}\sqcup N_{\leq}.$

As $G=P_{\preceq}\sqcup[1]_{\preceq}\sqcup N_{\preceq}$, we deduce
$P_{\preceq}=P_{\leq}\smallsetminus[1]_{\leq}$. Clearly (3) implies (2).
Finally, by left-invariance, we have $g\preceq h$ if and only if $h^{-1}g\in
G\smallsetminus P_{\preceq}$, and clearly the same holds for the preorder
$\leq$. This gives that (1) and (2) are equivalent. ∎

Let $G$ be a group. For a given a preorder
$\leq\in\operatorname{\mathsf{LPO}}(G)$, the collection of proper
$\leq$-convex subgroups is totally ordered by inclusion. The subgroup
$[1]_{\leq}$ is always the least such subgroup, however a maximal proper
$\leq$-convex subgroup may not exist. This issue is the order-theoretic
analogue of the problem of existence of a minimal invariant set (Remark 2.9).
To simplify the discussion, we will systematically assume finite generation in
the rest of the section, in which case there exists a maximal proper
$\leq$-convex subset, that we will denote by $H_{\leq}$. This leads to the
notion of minimal model of a preorder which is an order-theoretic analogue of
canonical models for actions on the line.

######  Definition 3.9.

Let $G$ be a finitely generated group. The _minimal model_ of a preorder
$\leq\in\operatorname{\mathsf{LPO}}(G)$ is the quotient preorder $\leq_{H}$,
where $H=H_{\leq}$ is the maximal proper $\leq$-convex subgroup. The minimal
model will be denoted by $\leq^{\ast}$. When the preorder $\leq$ coincides
with $\leq^{\ast}$, we say that $\leq\in\operatorname{\mathsf{LPO}}(G)$ is a
minimal model.

######  Remark 3.10.

Minimal preorders are exactly those whose structure of convex subgroups is the
simplest possible. Indeed, after Lemma 3.8, the minimal model dominates the
original preorder. Therefore, $H_{\leq}$ is the unique proper
$\leq^{*}$-subgroup of $(G,\leq^{*})$, and thus $H_{\leq}=[1]_{\leq^{*}}$.
Reversely, if $[1]_{\leq}=H_{\leq}$ is the unique proper $\leq$-convex
subgroup, then $\leq$ is a minimal model.

It should be not surprising that the minimal model is unique, in the following
sense.

######  Lemma 3.11.

Let $G$ be a finitely generated group and consider preorders
$\leq,\preceq\in\operatorname{\mathsf{LPO}}(G)$ such that $\leq$ is a minimal
model and dominates $\preceq$. Then, $\leq$ is the minimal model of $\preceq$.

###### Proof.

As $\leq$ dominates $\preceq$, the subgroup $H=[1]_{\leq}$ is
$\preceq$-convex, so $H\subseteq H_{\preceq}$. This gives $P_{\leq}\supseteq
P_{\preceq^{*}}$, so by Lemma 3.8, the map

$\mathsf{id}:(G,\leq)\to(G,\preceq^{*})$

is order-preserving. Using Remark 3.10, as $H_{\preceq}=[1]_{\preceq^{*}}$, we
deduce that $H_{\preceq}$ is $\leq$-convex, and thus that $H_{\preceq}=H$. ∎

We next introduce the following equivalence relation on preorders.

######  Definition 3.12.

Let $G$ be a finitely generated group. We say two preorders are _equivalent_
if they have the same minimal model. We denote by $[\leq]$ the equivalence
class of $\leq\in\operatorname{\mathsf{LPO}}(G)$ and we write
$[\operatorname{\mathsf{LPO}}](G)=\\{[\leq]:\leq\in\operatorname{\mathsf{LPO}}(G)\\}$
for the corresponding quotient.

It is immediate to verify that this is indeed an equivalence relation. We have
the following result.

######  Proposition 3.13.

Let $G$ be a finitely generated group and consider preorders
$\leq_{1},\leq_{2}\in\operatorname{\mathsf{LPO}}(G)$. Then, there exists a
preorder $\leq$ that dominates $\leq_{1}$ and $\leq_{2}$, if and only if the
equivalence classes $[\leq_{1}]$ and $[\leq_{2}]$ are the same.

###### Proof.

We have that the minimal order $\leq^{*}$ dominates $\leq$ and thus both
$\leq_{1}$ and $\leq_{2}$. From Lemma 3.11, we conclude that $\leq^{*}$ is the
minimal model of both $\leq_{1}$ and $\leq_{2}$. The converse statement
follows directly from Remark 3.10.∎

###  3.2. Relations between preorders and actions on the real line

We can now make explicit the relation between minimal models and their
dynamical counterpart. For this, note that if $G$ is a countable group and
$\leq\in\operatorname{\mathsf{LPO}}(G)$ is a preorder, we can consider a
_dynamical realization_ $\varphi_{\leq}$ for the action of $G$ on the totally
ordered set $(G/H,<_{G/H})$, which exists by Lemma 2.40. Conversely, given an
action $\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ without
fixed points, we denote by $\leq_{\varphi}\in\operatorname{\mathsf{LPO}}(G)$
the preorder induced by the $\varphi(G)$-orbit of $0$.

######  Proposition 3.14.

Let $G$ be a finitely generated group and let
$\varphi:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ be a canonical
model. Then the preorder $\leq_{\varphi}$ is a minimal model.

###### Proof.

Suppose by way of contradiction that there exists a proper
$\leq_{\varphi}$-convex subgroup $H\neq[1]_{\leq_{\varphi}}$. Then, for each
$gH\in G/H$, denote by $I_{gH}\subset\mathbb{R}$ the interior of the convex
hull of $\varphi(gH)(0)$. Since $H$ is $\leq_{\varphi}$-convex, we get that
$\mathbb{R}\smallsetminus\bigcup_{g\in G}I_{gH}$ is a proper closed
$\varphi(G)$-invariant subset. On the other hand, since $H$ strictly contains
$[1]_{\leq_{\varphi}}$ the $\varphi(G)$-stabilizer of each $I_{g}$ acts non-
trivially on $I_{g}$, and this implies that the action $\varphi$ is not
cyclic, a contradiction. ∎

######  Proposition 3.15.

Let $G$ be a finitely generated group and consider a preorder
$\leq\in\operatorname{\mathsf{LPO}}(G)$. Then the dynamical realization
$\varphi_{\leq}$ is a canonical model if and only if $\leq$ is a minimal
model.

Moreover, the dynamical realization $\varphi_{\leq^{\ast}}$ is a canonical
model for $\varphi_{\leq}$.

The rest of the subsection is devoted to the proof of Proposition 3.15. We
will need a preliminary result.

######  Lemma 3.16.

Let $G$ be a group. Consider preorders
$\leq_{1},\leq_{2}\in\operatorname{\mathsf{LPO}}(G)$ so that $\leq_{2}$
dominates $\leq_{1}$. Then the dynamical realizations $\varphi_{\leq_{1}}$ and
$\varphi_{\leq_{2}}$ are positively semi-conjugate.

###### Proof.

For $i\in\\{1,2\\}$ write $H_{i}:=[1]_{\leq_{i}}$ and
$\iota_{i}:G/H_{i}\to\mathbb{R}$ for the corresponding good embeddings. As
$\mathsf{id}:(G,\leq_{1})\to(G,\leq_{2})$ is order-preserving, we have
$H_{1}\subseteq H_{2}$ and the quotient map

$i_{0}:(G/H_{1},<_{1})\to(G/H_{2},<_{2})$

is order-preserving and equivariant (here $<_{i}$ denotes the total ordered
induced by $\leq_{i}$ for $i\in\\{1,2\\}$). Write $X=\iota_{1}(G/H_{1})$ and
define $j_{0}:X\to\mathbb{R}$ as

$j_{0}(\iota_{1}(gH_{1}))=\iota_{2}(i_{0}(gH_{1}))=\iota_{2}(gH_{2}).$

It follows directly from the definitions that $X$ if
$\varphi_{\leq_{1}}(G)$-invariant and that $j_{0}$ is order-preserving and
equivariant. We conclude using Lemma 2.7. ∎

Let us immediately point out the following conclusion.

######  Corollary 3.17.

For finitely generated groups, equivalent preorders have positively semi-
conjugate dynamical realizations.

###### Proof.

After Lemma 3.16, the dynamical realizations of equivalent preorders are
positively semi-conjugate to the dynamical realization of their minimal model.
As positive semi-conjugacy is an equivalence relation, the conclusion follows.
∎

###### Proof of Proposition 3.15.

Suppose first that $\varphi_{\leq}$ is not a canonical model. Then
$\varphi_{\leq}$ is neither minimal nor cyclic and so $\varphi_{\leq}$ has an
exceptional minimal invariant set $\Lambda\subset\mathbb{R}$ or
$\varphi_{\leq}$ has a closed discrete orbit but its image is not cyclic. We
write $X=\iota(G/[1]_{\leq})$ for the image of the good embedding
$\iota:G/[1]_{\leq}\to\mathbb{R}$ associated with $\varphi_{\leq}$, and remark
that $X$ consists of a single $\varphi_{\leq}(G)$-orbit.

######  Case I.

$\varphi_{\leq}$ has an exceptional minimal invariant set.

We first show the following.

######  Claim.

$X$ is contained in $\mathbb{R}\smallsetminus\Lambda$.

###### Proof of claim.

We assume by contradiction that $X\cap\Lambda\neq\varnothing$. As $X$ consists
of a single orbit, by invariance of $\Lambda$ we get that $X\subset\Lambda$
and thus $\Lambda=\overline{X}$ by minimality of $\Lambda$. Consider a
connected component $I=(\xi,\eta)$ of $\mathbb{R}\smallsetminus\Lambda$ and
note that, by Definition 2.37 of good embedding, we have
$\\{\xi,\eta\\}\subseteq X$ and thus the points $\xi$ and $\eta$ are in the
same orbit. This shows that $\Lambda$ is discrete, which is a contradiction. ∎

Consider now the connected component $U$ of $\mathbb{R}\smallsetminus\Lambda$
that contains $\xi_{0}=\iota([1]_{\leq})$ and write
$H=\operatorname{\mathsf{Stab}}^{\varphi_{\leq}}_{G}(U)$. Note that
$H=\iota^{-1}(U)$ is a $\leq$-convex subgroup. We will show that $H$ strictly
contains $[1]_{\leq}$, which by Remark 3.10, implies that $\leq$ is not a
minimal model. Equivalently, we need to show that $U\cap X$ strictly contains
$\xi_{0}$. Assume that $\xi_{0}$ is an isolated point of $X$, otherwise there
is nothing to prove. Let $\lambda\in\Lambda$ be the rightmost point of $U$.
Note that $\overline{X}$ is $\varphi_{\leq}(G)$-invariant, so we must have