# A realisation result for moduli spaces of group actions on the line

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

###### Abstract.

Given a finitely generated group $G$, the possible actions of $G$ on the real
line (without global fixed points), considered up to semi-conjugacy, can be
encoded by the space of orbits of a flow on a compact space $(Y,\Phi)$
naturally associated with $G$ and uniquely defined up to flow equivalence,
that we call the _Deroin space_ of $G$. We show a realisation result: every
expansive flow $(Y,\Phi)$ on a compact metrisable space of topological
dimension 1, satisfying some mild additional assumptions, arises as the Deroin
space of a finitely generated group. This is proven by identifying the Deroin
space of an explicit family of groups acting on suspension flows of subshifts,
which is a variant of a construction introduced by the second and fourth
authors. This result provides a source of examples of finitely generated
groups satisfying various new phenomena for actions on the line, related to
their rigidity/flexibility properties and to the structure of (path-)connected
components of the space of actions.

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

###### Key words and phrases:

Group actions on the real line, semi-conjugacy of actions, Deroin space

All the authors acknowledge the support of the project MATH AMSUD, DGT –
Dynamical Group Theory (22-MATH-03). NMB and MT are partially supported by the
project ANR Gromeov (ANR19-CE40-0007). The work of NMB was supported by the
LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program
“‘France 2030” (ANR-11-IDEX-0007) operated by the French National Research
Agency (ANR). MT has been 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).

##  1\. Introduction

The topic of this paper is the study of actions of finitely generated groups
$G$ on the real line, that is, representations of $G$ into the group
$\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ of orientation-preserving
homeomorphisms of the line. We start with some basic terminology. A
representation $\rho\colon G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$
is said to be _irreducible_ if $\rho(G)$ has no fixed points in $\mathbb{R}$.
We will denote by
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ the space of
irreducible representations of a group $G$ in
$\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$. The latter has a natural
(Polish) topology, inherited from the compact-open topology on
$\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$. One says that two
representations
$\rho_{1},\rho_{2}\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$
are _semi-conjugate_ if there exists a non-decreasing $G$-equivariant map
$h\colon\mathbb{R}\to\mathbb{R}$ (here we strictly reserve this terminology to
the case of a non-decreasing map, and will always specify when two
representations are semi-conjugate only up to reversing the orientation). This
defines an equivalence relation on
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$. Considering
irreducible representations up to semi-conjugacy is a way to restrict the
attention to their minimal sets, which in a sense are the basic building
blocks of the dynamics for group actions in dimension 1.

Our main focus is the study of the semi-conjugacy classes in
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ from a
topological and dynamical perspective. One might think of the quotient of
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ by the semi-
conjugacy relation as a moduli space for actions of $G$ on the line;
unfortunately this quotient is often too ill-behaved to develop a meaningful
theory. In fact for many groups $G$ the semi-conjugacy relation is not
_smooth_ in the Borel sense, namely there is no Borel map from
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ to any
standard Borel space which is constant on semi-conjugacy classes and separates
them. This can be seen as an obstruction to find explicit invariants that
classify actions up to semi-conjugacy. This contrasts with the case of actions
on the circle, for which complete semi-conjugacy invariants are available (see
Ghys [Ghys] and Matsumoto [Matsumoto]) and the quotient by the semi-conjugacy
relation is a compact metrisable space (see Mann and Wolff [MW]).

For finitely generated groups, a solution to this problem is to consider an
intermediate quotient space, that we call the _Deroin space_ of the group $G$,
which is a canonical version of a construction of Deroin [Deroin]. It is a
compact metrisable space $(\mathcal{D},\Psi)$ endowed with a one-parameter
flow $\Psi:\mathbb{R}\times\mathcal{D}\to\mathcal{D}$ and a continuous
$G$-action preserving $\Psi$-orbits, which is uniquely associated to $G$ up to
$G$-equivariant flow equivalence (that is, homeomorphism sending flow-orbits
to flow-orbits in an orientation preserving way). The space $\mathcal{D}$ can
be abstractly defined as the quotient of
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ by the finer
equivalence relation of _pointed_ semi-conjugacy, namely semi-conjugacy
preserving a marked point on the line (see Definition 3.1), and the flow
$\Psi$ moves the marked point. By the results of Deroin, Kleptsyn, Navas, and
Parwani [DKNP], the space $\mathcal{D}$ can actually be realised as a subspace
of $\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ by
selecting a suitable system of representatives (namely the set of normalized
$\mu$-harmonic representations), in such a way that the flow $\Psi$ coincides
with the restriction to $\mathcal{D}$ of the translation action of
$(\mathbb{R},+)$ on
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$. (See §3 for
details).

The interest for Deroin spaces from the perspective described above is that
the quotient map
$r_{\mathcal{D}}\colon\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})\to\mathcal{D}$
is continuous, surjective, and has the property that two actions
$\rho_{1},\rho_{2}\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$
are semi-conjugate if and only if
$r_{\mathcal{D}}(\rho_{1}),r_{\mathcal{D}}(\rho_{2})$ belong to the same
$\Psi$-orbit. Therefore all the complexity of the semi-conjugacy relation in
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ is witnessed
by the dynamics of the flow $(\mathcal{D},\Psi)$. This is a strong restriction
even if one retains only the Borel structure of
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$, see the
discussion in our previous work [BMRT, §3.5] (in particular the semi-conjugacy
relation on $\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$
is smooth if and only if the orbit-equivalence relation of
$(\mathcal{D},\Psi)$ is so). Well beyond, the flow $(\mathcal{D},\Psi)$
carries topological information on how semi-conjugacy classes sit inside
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$.

Describing the Deroin space of a finitely generated group requires a global
understanding of its possible actions on the line up to semi-conjugacy. The
pool of examples for which such understanding is available gives rise to
rather degenerate flows; let us mention some. It has long been known that if
$G$ is a group of subexponential word-growth, then every
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ is
semi-conjugate to an action by translations (see Plante [Plante-measure]);
this implies that its Deroin space is either empty or homeomorphic to a sphere
$\mathbb{S}^{n}$ with trivial flow. Some groups admit finitely many actions on
the line up to semi-conjugacy, such as the solvable Baumslag–Solitar groups
(see the work of the third author [Rivas-conradian]), or central extensions of
some groups acting on the circle such as Thompson’s group $T$ (see the work of
the second and fourth authors [MB-T]), or the Fuchsian group $\Delta(2,3,7)$
(see Mann and the fourth author [Mann-Triestino]); in these cases the flow
($\mathcal{D},\Psi)$ can be easily computed and has only finitely many orbits
(e.g. in the last two examples it is the disjoint union of two circles, which
are $\Psi$-periodic orbits). Some more sources of examples come from our
previous works [BMRT, solvable], and give rise to flows with an uncountable
number of orbits but still very restricted dynamics: for example in the case
of Thompson’s group $F$, the Deroin space has a smooth orbit space and all
orbits converge towards some limit point (see [BMRT, Figure 1]); the same
holds for the wreath product $\mathbb{Z}\wr\mathbb{Z}$, relying on [solvable,
Proposition 6.3] (a more precise result for all finitely generated solvable
groups will be the object of a work in preparation). An example on the
opposite side of the spectrum is given by the non-abelian free groups
$\mathbb{F}_{n}$. In this case the flow $(\mathcal{D},\Psi)$ has a non-smooth
space of orbits, as one can show by exhibiting a family of representations of
$\mathbb{F}_{n}$ giving rise to a subflow with chaotic behaviour [BMRT, Remark
3.23]. An explicit description of the Deroin space of $\mathbb{F}_{n}$ is not
known (note that it would encode actions on the line of every $n$-generated
group).

Our main result shows that many flows can be realised as the Deroin space of a
finitely generated group. This is proven by identifying the Deroin space of an
explicit family of groups, which are relatives of the groups introduced by two
of the authors in [MB-T]. The class of flows that we consider are suspension
flows of subshifts, intrinsically characterised (by Bowen and Walters
[expansive]) as expansive flows on compact metrisable spaces of topological
dimension 1 (all these notions are recalled in §5.1). We say that a flow
$(Y,\Phi)$ is _freely reversible_ if it admits a fixed-point-free order-2
homeomorphism $\sigma\colon Y\to Y$ which is a flow equivalence between $\Phi$
and its time-reverse; the Deroin space of a finitely generated group always
has this property (Proposition 3.9).

######  Theorem 1.1.

Let $(Y,\Phi)$ be an expansive flow without fixed points on a compact
metrisable space of topological dimension 1. Assume that $\Phi$ is freely
reversible and topologically free (namely, the set of non-periodic points is
dense). Then, there exists a finitely generated group $G$ acting faithfully on
$Y$ by homeomorphisms, preserving each $\Phi$-orbit, such that $(Y,\Phi)$ can
be identified with the Deroin space of $G$ (up to $G$-equivariant flow
equivalence).

######  Remark 1.2.

Recall that a group is _left-orderable_ if it admits a total order which is
invariant under left multiplication. A countable group is left-orderable if
and only if it is isomorphic to a subgroup of
$\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$. Since we assume in Theorem
1.1 that $\Phi$ is topologically free, the group $G$ acts faithfully on the
union of non-periodic orbits, and thus it is left-orderable by a standard
argument (see [MB-T, Proposition 3.3]).

Theorem 1.1 provides the first explicit computation of Deroin spaces with
chaotic behaviour (in particular whose space of orbits is non-smooth). In fact
it produces examples such that the flow $(\mathcal{D},\Psi)$ has rather
arbitrary dynamical properties.

One direction of application of Theorem 1.1 is related to the study of
rigidity and flexibility of group actions on the line. In a broad sense, this
topic aims at understanding, given a group $G$, how its representations can be
modified (up to semi-conjugacy) by slight perturbations, which boils down to
understand how semi-conjugacy classes accumulate onto each other. This leads
to several tightly related notions of rigidity and flexibility for
representations, appearing in the literature on group actions on one-
manifolds, see e.g. [Mann-survey, Mann-Rivas, MW, KKMj]. We will use here the
following terminology, analogous to the one used by Mann and Wolff [MW] for
actions on the circle.

######  Definition 1.3.

A representation
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ is
_locally rigid_ if it is an interior point in its semi-conjugacy class. It is
_rigid_ if its whole semi-conjugacy class is open. A representation which is
not locally rigid is called _flexible_.

A representation
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ is
rigid if and only if its image $r_{\mathcal{D}}(\rho)$ in the Deroin space
$(\mathcal{D},\Psi)$ defines an isolated point in any local cross-section of
the flow $\Psi$, i.e. if its $\Psi$-orbit is an an open subset of
$\mathcal{D}$. Local rigidity is not witnessed by the Deroin space in general,
although it is equivalent to rigidity for minimal representations [BMRT,
Remark 3.5]. However for the groups that we construct to prove Theorem 1.1, we
additionally show that local rigidity and rigidity are equivalent for all
representations (see Corollary 6.3). Thus for this family of groups we have
complete control on which representations are (locally) rigid (clearly a flow
as in Theorem 1.1 may or may not admit open orbits). More generally, given
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$, we
obtain a precise characterisation of all representations that accumulate on
$\rho$, up to semi-conjugacy, in terms of the flow $(Y,\Phi)$, see Theorem
6.2.

A relevant special case of Theorem 6.2 arises by choosing the flow $(Y,\Phi)$
to be _minimal_ , i.e. with all orbits dense. In this case our construction
produces examples of groups with the following extreme flexibility property.

######  Corollary 1.4.

There exist finitely generated groups $G$ for which any semi-conjugacy class
is dense in $\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$
(and $\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G,\mathbb{R})$ contains
uncountably many semi-conjugacy classes).

In other words, a group $G$ as in Corollary 1.4 has the property that every
representation
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ admits
arbitrarily small perturbations which are semi-conjugate to any other
representation in
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$. We call such
a representation $\rho$ _universally flexible_ (see Definition 6.5). To our
knowledge, there were previously no groups known to admit any representation
with this property.

######  Remark 1.5.

Corollary 1.4 can be compared with the problem of whether there exists a
finitely generated left-orderable group acting minimally on its space of left
orders, see Navas [Navas-ICM, Question 6]. The conclusion in Corollary 1.4 is
the analogous property at the level of actions on the line; however there is
no formal connection: the groups constructed here do not act minimally on
their space of left-orders, and conversely this property does not seem to
imply _a priori_ the conclusion of Corollary 1.4 (because not all
representations in
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ on the line
are dynamical realisations of orders). However both properties imply that the
group $G$ acts minimally on its Deroin space, which can be identified with a
quotient of the space of left-orders [BMRT, §3].

Beyond the minimal case, different choices for the flow $(Y,\Phi)$ yield
groups with other prescribed rigidity/flexibility property, for example groups
which admit both universally flexible representations together with a given
number (finite or countably infinite) of rigid ones (see Corollary 6.7).

In a similar spirit, we consider a natural analogue of the Cantor–Bendixson
rank (CB-rank) for the space of semi-conjugacy classes in
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$, that we call
the _semi-conjugacy CB-rank_ of
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$, defined
according to a transfinite process in which the open semi-conjugacy classes
play a role analogous to the isolated points for the usual CB-rank (Definition
6.9). Then Theorem 1.1 produces examples of groups for which the semi-
conjugacy CB-rank of
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ can be
computed and can be an arbitrarily large countable ordinal, see Corollary
6.10. Again, this can be seen as a counterpart at the level of actions of a
question asked by Mann and the third author [Mann-Rivas] for spaces of left
orders, namely whether there exists a finitely generated left-orderable group
whose space of left orders has Cantor–Bendixson rank larger than 1.

As another application, we obtain examples of finitely generated groups that
admit many non semi-conjugate actions on $\mathbb{R}$, yet have a unique
_generic_ action (up to possibly reversing the orientation) in the following
strong sense.

######  Corollary 1.6 (Groups with a generic representation).

There exist finitely generated groups $G$ such that
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ contains
infinitely (countably or uncountably) many distinct semi-conjugacy classes,
and there is
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ such
that the semi-conjugacy classes of $\rho$ and of its conjugate by the
reflection $x\mapsto-x$ form a dense open set.

######  Remark 1.7.

Since we required semi-conjugacy to be orientation-preserving, it is not
possible for a group $G$ to have a representation
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ with
an open dense (or even co-meager) semi-conjugacy class, as its conjugate by
the reflection should have the same property.

Another general theme in the study of representation spaces is the
classification of their connected and path-connected components. See for
instance the works of Mann and Wolff [Mann, MW] for the case of surface group
actions on the circle (for which this problem remains open in general). For a
finitely generated group $G$, the semi-conjugacy classes in
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ are always
path connected, so the connected and path-connected components of
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ are in
correspondence with those of the Deroin space (Proposition 3.13). Again in
analogy with the terminology of [MW], one may call an action
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ _path-
rigid_ if its path-component coincides with its semi-conjugacy class (that is,
$\rho$ cannot be deformed _along a continuous path_ into any non-semi-
conjugate representation). Theorem 1.1 is a source of groups with a rather
extreme behaviour with regard to this notion. In particular, it shows that
path-continuous deformations of representations might be dramatically more
restricted than arbitrary small perturbations in the compact-open topology.

######  Corollary 1.8 (Path-continuous deformations vs general perturbations).

For every group $G$ satisfying the conclusion of Theorem 1.1, the path-
components of $\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$
coincide with the semi-conjugacy classes.

Combining with Corollary 1.4, there are finitely generated groups $G$ such
that all representations in
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ are
simoultaneously path-rigid and universally flexible, and the space
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ is connected
but not path-connected, and nowhere locally connected.

### On the group construction

The groups that we consider to prove Theorem 1.1 are based on a modification
of the groups $\mathsf{T}(\varphi)$ acting on the suspension space of subhifts
from [MB-T]. A subshift $(X,\varphi)$ is a closed shift-invariant subset $X$
of $A^{\mathbb{Z}}$, where $A$ is a finite alphabet, and $\varphi:X\to X$
denotes the restriction of the shift. Subshifts are characterised up to
topological conjugacy as expansive homeomorphisms of compact, totally
disconnected, metrisable spaces. The _suspension space_ of $(X,\varphi)$ is
the space $Y:=(X\times\mathbb{R})/_{(\varphi(x),t)\sim(x,t+1)}$. The space $Y$
is endowed with the _suspension flow_ $\Phi\colon\mathbb{R}\times Y\to Y$,
induced by the natural flow on $X\times\mathbb{R}$ which translates the
$\mathbb{R}$-coordinate. To take into account the restriction that the Deroin
space is freely reversible, we need to consider reversible subshifts, by which
we mean the data $(X,\varphi,\sigma)$ of a subshift additionally equipped with
an involution $\sigma\colon X\to X$ (an order-2 homeomorphism) such that
$\sigma\varphi\sigma=\varphi^{-1}$. Such an involution naturally induces an
involution $\hat{\sigma}\colon Y\to Y$ on the suspension space, which reverses
the suspension flow. To ensure that $\hat{\sigma}$ has no fixed points, we
need to further assume that $\sigma$ does not preserve any $\varphi$-orbit
(see Lemma 4.1). Every flow as in Theorem 1.1 can be realised as the
suspension flow of a subshift $(X,\varphi)$ with these properties, relying on
Bowen and Walters [expansive].

Given $(X,\varphi)$, the group $\mathsf{T}(\varphi)$ is defined as a group of
homeomorphisms of the its suspension space $Y$ analogous to Thompson’s groups,
whose action in the flow direction is by dyadic PL homeomorphisms, and such
that the displacement along each flow-orbit is locally constant in the
transverse direction. For a reversible subshift $(X,\varphi,\sigma)$ one can
define an analogous group $\mathsf{T}(\varphi,\sigma)$ as the centralizer of
$\hat{\sigma}$ in $\mathsf{T}(\varphi)$ (this definition appears in the work
by Le Boudec and the second author [confined]); a special case of the groups
$\mathsf{T}(\varphi,\sigma)$ are the groups defined by Hyde and Lodha in [HL].
The definition of the groups $\mathsf{T}(\varphi)$ was largely inspired by the
notion of Deroin space, and since their introduction the question naturally
arose whether such constructions could be used to prove a realisation result
as Theorem 1.1, by identifying their Deroin space completely (see Question 1
in [MB-T]). With the original definition of the groups $\mathsf{T}(\varphi)$
or $\mathsf{T}(\varphi,\sigma)$, our attempts to do so have run into technical
difficulties, related to the fact that certain subgroups playing an important
role (namely the subgroups supported in dyadic charts) have huge
abelianisation (free abelian of infinite rank). This complicates the analysis
of actions on the line. This technical point is an artifact of the use of PL
maps in the definition of the groups $\mathsf{T}(\varphi)$. A starting point
of this paper is the idea of replacing PL maps by a larger pseudogroup of
transformations; namely we consider dyadic PL transformations where we allow a
non-discrete set of discontinuity points for the derivative, which accumulate
onto some isolated “higher order” singularities with a controlled behaviour,
required to locally commute with the doubling map. Certain groups acting by
such countably singular PL maps were already considered in our previous work
[BMRT, §12.2]. On the interval $[0,1]$, this definition yields a finitely
generated group $\mathscr{F}$ sharing many features with Thompson’s group $F$,
except that it is _perfect_. Accordingly we define groups
$\mathscr{T}(\varphi)$ and $\mathscr{T}(\varphi,\sigma)$ associated with any
reversible subshift $(X,\varphi,\sigma)$. The main content of this work is the
identification of the Deroin space of the groups $\mathscr{T}(\varphi,\sigma)$
with the suspension flow of $(X,\varphi)$, from which Theorem 1.1 follows.

######  Remark 1.9.

Beyond finite generation, the other properties established for the groups
$\mathsf{T}(\varphi)$ in [MB-T] remain valid for the groups
$\mathscr{T}(\varphi)$ and $\mathscr{T}(\varphi,\sigma)$, with similar proofs.
For instance, these groups are simple if and only if $\varphi$ is minimal
[MB-T, Theorem B], they do not have Kazhdan’s property ($T$) [MB-T, Theorem
F], and they are not finitely presented if $\varphi$ is not a subshift of
finite type [MB-T, Theorem G]. We avoid a more detailed discussion here, as
these properties are not relevant for the main object of the paper.

It would be interesting to identify similar constructions of groups acting on
flows over spaces of higher topological dimension, and use them to extend
Theorem 1.1. Thus we conclude this introduction by proposing the following
general question.

######  Question 1.10.

Which compact spaces $(Y,\Phi)$ endowed with a flow can be realised as the
Deroin space of a finitely generated group?

### Acknowledgements

We thank Ville Salo for sharing useful comments on the CB-rank of subshifts.

##  2\. General preliminaries on actions on the line

Recall that given a group $G$, we call a representation $\rho\colon
G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ irreducible if it has no
fixed points. The space of all irreducible representations is denoted by
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$. It is
endowed with the topology induced from the product topology on
$\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})^{G}$, where
$\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ has the compact-open topology.
We say that
$\rho_{1},\rho_{2}\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$
are _semi-conjugate_ if there exists a non-decreasing map
$h\colon\mathbb{R}\to\mathbb{R}$ which intertwines $\rho_{1}$ and $\rho_{2}$.
Semi-conjugacy is an equivalence relation.

Throughout the text, by a _minimal set_ for a representation
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ we
mean a closed, non-empty, minimal invariant subset $\Lambda\subset\mathbb{R}$.
Every such set satisfies one of the following possibilities:

  1. (1)

either $\Lambda$ is a closed discrete orbit, in which case the action of $G$
on $\Lambda$ factors through a cyclic quotient of $G$, or

  2. (2)

$\Lambda$ is a perfect set and it is the unique minimal set of $\rho$.
Furthermore, in this case we have either $\Lambda=\mathbb{R}$, or $\Lambda$
has empty interior.

It is also well known that when $G$ is _finitely generated_ , every
irreducible action
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ admits
a minimal set, see for instance Navas [Navasbook, Proposition 2.1.12].
(However such a set need not exist if the finite generation assumption is
dropped.)

This discussion implies that if $G$ is a finitely generated group, then every
semi-conjugacy class in
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ contains a
representative which is either minimal, or _cyclic_ (namely which factors
through an epimorphism of $G$ onto a cyclic group of translations). Such a
representative is unique up to conjugacy by an element of
$\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$. Moreover any map implementing
a semi-conjugacy between minimal actions is automatically a homeomorphism, and
hence a conjugacy (see for instance Kim, Koberda, and Mj [KKMj, Lemma A.4]).

We also record the following.

######  Proposition 2.1.

Let $G$ be a finitely generated group. Then every semi-conjugacy class in
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ is path-
connected.

###### Proof.

Let
$\mathcal{C}\subset\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$
be a semi-conjugacy class, and choose a representative $\rho\in\mathcal{C}$
which is minimal or cyclic. It is enough to see that there exists a continuous
path from any $\tilde{\rho}\in\mathcal{C}$ to ${\rho}$ inside $\mathcal{C}$.
Assume first that $\rho$ is minimal. Let $h:\mathbb{R}\to\mathbb{R}$ be a
semi-conjugacy between $\tilde{\rho}$ and $\rho$ and remark that as ${\rho}$
is minimal, the map $h$ is continuous (see [KKMj, Lemma A.4]). For
$t\in[0,1]$, write $h_{t}=(1-t)\mathsf{id}+th$, and observe that for
$t\in[0,1)$ this gives a homeomorphism because it is the convex sum of an
increasing map with a non-decreasing map. For $t\in(0,1)$, define
$\tilde{\rho}_{t}:=h_{t}\tilde{\rho}h_{t}^{-1}$. This gives the desired
continuous path. Assume now that ${\rho}$ is cyclic. This implies that
$\tilde{\rho}$ has a closed discrete orbit $\Lambda\subset\mathbb{R}$. Let
$\tilde{\rho}^{\prime}\in\mathcal{C}$ be the action obtained by blowing up
each point of $\Lambda$ to an interval. Then there is a semi-conjugacy from
$\rho^{\prime}$ to $\tilde{\rho}$ which is implemented by a continuous map
(obtained by collapsing back the blown-up intervals). There also exists a
semi-conjugacy from $\tilde{\rho}^{\prime}$ to ${\rho}$ through a continuous
map (obtained by collapsing the complement of the closure of the blown-up
intervals). Repeating the argument for the minimal case, we may find
continuous paths connecting $\tilde{\rho}$ first to $\tilde{\rho}^{\prime}$
and then to ${\rho}$. ∎

######  Remark 2.2.

When studying topological properties of
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G,\mathbb{R})$, such as the
space of path-components and the separation properties of semi-conjugacy
classes, it is crucial that we restrict the attention to irreducible
representations. For example it is not difficult to see that the space of
_all_ representations of $G$ into
$\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ always deformation retracts
onto the trivial representation (using a version of the classical “Alexander
trick”), moreover every semi-conjugacy class accumulates on the trivial
representation (see Mann and Wolff [MW, Proposition 2.12]).

We finally recall the fundamental classification of minimal group actions on
the line into three types according to their centralizer. Given an action
$\rho\colon G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$, we denote by
$\mathcal{Z}(\rho)$ the centraliser of $\rho(G)$ in
$\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$. We say that two points
$x,y\in\mathbb{R}$ are _proximal_ for $\rho$ if there exists a sequence
$(g_{n})\subset G$ such that $\rho(g_{n})(x)$ and $\rho(g_{n})(y)$ converge to
the same point of $\mathbb{R}$. The action $\rho$ is _proximal_ if all pairs
of points are proximal, and _locally proximal_ if every point is contained in
some open interval whose endpoints are proximal. The following result is
proven in Malyutin [Malyutin].

######  Theorem 2.3.

Let $G$ be any group and
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ a
minimal action. Then one of the following holds.

  1. (I)

Either $\rho$ is conjugate to an action by translation, and
$\mathcal{Z}(\rho)$ is isomorphic to $(\mathbb{R},+)$; or

  2. (II)

$\rho$ is locally proximal, but not proximal, $\mathcal{Z}(\rho)$ is conjugate
to an infinite cyclic group of translations, and the action of $G$ on the
circle $\mathbb{R}/\mathcal{Z}(\rho)\cong\mathbb{S}^{1}$ is proximal; or

  3. (III)

$\rho$ is proximal, and $\mathcal{Z}(\rho)$ is trivial.

In particular, in all cases, $\mathcal{Z}(\rho)$ is abelian.

##  3\. The Deroin space

###  3.1. Definition of the Deroin space

Let $G$ be a discrete group and consider the space
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$. Given
$s\in\mathbb{R}$, denote by
$T_{s}\in\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ the translation by
$s$. We define the _translation flow_
$\Psi:\mathbb{R}\times\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})\to\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$,
given by $\Psi^{s}(\rho)=T_{-s}\circ\rho\circ T_{s}$. We also have an
involution $\hat{\iota}$ on
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ defined by
conjugation by the reflection $x\mapsto-x$. Together with $\Psi$, this gives
an action of the group $\mathsf{Isom}(\mathbb{R})$ of isometries of
$\mathbb{R}$ on
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$, which is
continuous with respect to the compact-open topology. We also define the
_translation action_
$G\times\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})\to\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$
by $g.\rho=\Psi^{\rho(g)(0)}(\rho)$. A proof that this is a well-defined
action can be found in Deroin [Deroin, pp. 187–188]. It is clear that the
translation action is continuous and preserves every $\Psi$-orbit.

In other words, one can think of an element
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ as an
action on the real line with a marker on the point $0$. The translation flow
corresponds to moving the marker by a translation, while the translation
action of $G$ moves the marker according to the underlying $G$-action. With
this point of view in mind, it is useful to introduce the following
terminology.

######  Definition 3.1.

Consider irreducible actions
$\rho_{1},\rho_{2}\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$.

  1. (1)

We say that $\rho_{1}$ and $\rho_{2}$ are _pointed conjugate_ if there exists
an orientation-preserving conjugacy $h\colon\mathbb{R}\to\mathbb{R}$ between
$\rho_{1}$ and $\rho_{2}$ such that $h(0)=0$.

  2. (2)

We say that $\rho_{1}$ and $\rho_{2}$ are _pointed semi-conjugate_ if there
exist
$\rho_{\ast}\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$,
either minimal or cyclic, and semi-conjugacies $h_{i}$ between $\rho_{i}$ and
$\rho_{\ast}$ for $i\in\\{1,2\\}$, such that $h_{1}(0)=h_{2}(0)$.

######  Remark 3.2.

Notice that pointed semi-conjugacy is an equivalence relation, which is
(strictly) finer than semi-conjugacy. Also, notice that if $\rho_{1}$ and
$\rho_{2}$ are minimal or cyclic, then they are pointed semi-conjugate if and
only if they are pointed conjugate. When $\rho_{1}$ and $\rho_{2}$ are cyclic,
this is clear, and when they are minimal it follows from the fact that a semi-
conjugacy between minimal actions is automatically a conjugacy.

Deroin showed in [Deroin] that if $G$ is finitely generated, then every
representation
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ is
conjugate to one which belongs to some compact $\Psi$-invariant subset of
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$. A more
universal version of this construction follows from the results in [DKNP] and
allows to find a single such set which intersects all semi-conjugacy classes,
see [BMRT, §3]. The following definition captures the main properties of this
construction which will be important for our purposes.

######  Definition 3.3.

Let $G$ be a finitely generated group. We say that a subset
$\mathcal{D}\subset\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$
is a _Deroin space_ for $G$ if it satisfies:

  1. (1)

$\mathcal{D}$ is compact, and invariant under the translation flow $\Psi$;

  2. (2)

every $\rho\in\mathcal{D}$ is either minimal or cyclic;

  3. (3)

every $\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$
is pointed semi-conjugate to a unique element of $\mathcal{D}$.

######  Remark 3.4.

As a consequence of the definition, any Deroin space is $G$-invariant for the
translation action of $G$, for the latter preserves orbits of the translations
flow. Moreover the $G$-orbit of every $\rho\in\mathcal{D}$ is dense in its
$\Psi$-orbit (when $\rho$ is cyclic this is trivial since the translation flow
fixes $\rho$, else it follows from the minimality of $\rho$ by the definition
of translation action).

Given a Deroin space
$\mathcal{D}\subset\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$,
we can define a map

$r_{\mathcal{D}}:\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})\to\mathcal{D}$

that associates to each irreducible representation its unique representative
up to pointed semi-conjugacy in $\mathcal{D}$.

######  Proposition 3.5.

Let $\mathcal{D}$ be a Deroin space for a finitely generated group $G$. The
map $r_{\mathcal{D}}$ is a $G$-equivariant continuous retraction, which
preserves pointed-semi-conjugacy classes. Two actions
$\rho_{1},\rho_{2}\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$
are semi-conjugate if and only if $r_{\mathcal{D}}(\rho_{1})$ and
$r_{\mathcal{D}}(\rho_{2})$ belong to the same $\Psi$-orbit in $\mathcal{D}$.

###### Proof.

The proof that the map $r_{\mathcal{D}}$ is a continuous retraction is given
in [BMRT, Theorem 3.2]. (This result is stated there for a specific
realisation of Deroin space, namely the space of normalised $\mu$-harmonic
actions, but its proof only relies on the abstract properties in Definition
3.3). It remains to prove $G$-equivariance of $r_{\mathcal{D}}$. For this,
take
$\rho_{0}\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$
and let $\rho_{1}:=r_{\mathcal{D}}(\rho_{0})$ be its projection to
$\mathcal{D}$. Then, there exist a minimal or cyclic model
$\rho_{\ast}\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$
and semi-conjugacies $h_{i}$ between $\rho_{i}$ and $\rho_{\ast}$ for
$i\in\\{0,1\\}$, so that $h_{0}(0)=h_{1}(0)$. Thus we have

$h_{0}(\rho_{0}(g)(0))=\rho_{\ast}(g)(h_{0}(0))=\rho_{\ast}(g)(h_{1}(0))=h_{1}(\rho_{1}(g)(0)),$

which implies that $g.\rho_{0}$ and $g.\rho_{1}$ are pointed semi-conjugate.
Since Deroin spaces intersect each pointed-semi-conjugacy class in a single
point, we have that $r_{\mathcal{D}}(g.\rho_{0})=g.\rho_{1}$. To show the last
sentence, note that if $r_{\mathcal{D}}(\rho_{1})$ and
$r_{\mathcal{D}}(\rho_{2})$ are actions in $\mathcal{D}$ that belong to the
same orbit of the translation flow, then they are evidently conjugate, so that
$\rho_{1}$ and $\rho_{2}$ are semi-conjugate. Conversely if $\rho_{1}$ and
$\rho_{2}$ are semi-conjugate, then so are $r_{\mathcal{D}}(\rho_{1})$ and
$r_{\mathcal{D}}(\rho_{2})$, and since actions in $\mathcal{D}$ are minimal or
cyclic, this implies that they are actually conjugate. So some conjugate of
$r_{\mathcal{D}}(\rho_{1})$ by a translation is actually pointed conjugate to
$r_{\mathcal{D}}(\rho_{2})$, and by uniqueness this is possible only if it is
equal, so $r_{\mathcal{D}}(\rho_{1})$ and $r_{\mathcal{D}}(\rho_{2})$ belong
to the same $\Psi$-orbit. ∎

######  Definition 3.6.

Let $(Y_{1},\Phi_{1}),(Y_{2},\Phi_{2})$ be spaces with a flow. A _flow
equivalence_ is a homeomorphism $\mathfrak{h}\colon Y_{1}\to Y_{2}$ such that
for every $y\in Y_{1}$, there exists an orientation-preserving homeomorphism
$\tau\colon\mathbb{R}\to\mathbb{R}$ fixing $0$ such that
$\mathfrak{h}(\Phi_{1}^{s}(y))=\Phi_{2}^{\tau(s)}(\mathfrak{h}(y))$ for every
$s\in\mathbb{R}$.

######  Theorem 3.7 (Existence and uniqueness [DKNP, BMRT]).

Every finitely generated group has a Deroin space.

Moreover, such a space is unique in the following sense: if
$\mathcal{D}_{1},\mathcal{D}_{2}\subset\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$
are Deroin spaces of $G$, there exists a $G$-equivariant flow equivalence
$\mathfrak{h}:(\mathcal{D}_{1},\Psi|_{\mathcal{D}_{1}})\to(\mathcal{D}_{2},\Psi|_{\mathcal{D}_{2}})$.

###### Proof.

Existence follows from the results of Deroin, Kleptsyn, Navas, Parwani [DKNP];
see [BMRT, Proposition 2.19 and Theorem 2.22]. For uniqueness, consider the
map $\mathfrak{h}:\mathcal{D}_{1}\to\mathcal{D}_{2}$ defined as the
restriction of $r_{\mathcal{D}_{2}}$ to the Deroin space $\mathcal{D}_{1}$. By
Proposition 3.5 this map is $G$-equivariant, continuous, and preserves
pointed-semi-conjugacy classes. Since each pointed-semi-conjugacy class has
exactly one representative in each Deroin space, the map $\mathfrak{h}$ is
bijective. It remains to check that $\mathfrak{h}$ is a flow equivalence. For
this, consider $\rho_{1}\in\mathcal{D}_{1}$ and
$\rho_{2}:=\mathfrak{h}(\rho_{1})$. Since $\rho_{1}$ and $\rho_{2}$ are
pointed semi-conjugate, and they are minimal or cyclic, by Remark 3.2, there
exists an orientation preserving homeomorphism $\tau:\mathbb{R}\to\mathbb{R}$
which conjugates $\rho_{1}$ and $\rho_{2}$, and sends $0$ to itself. Thus,
since $\mathfrak{h}$ preserves pointed-semi-conjugacy classes, it must hold
that $\mathfrak{h}(\Psi^{s}(\rho_{1}))=\Psi^{\tau(s)}(\rho_{2})$. ∎

In view of the previous theorem, we can abuse terminology and speak of _the_
Deroin space of $G$ as a compact space with a flow $(\mathcal{D},\Psi)$ and a
$G$-action preserving $\Psi$-orbits, which is well defined up to
$G$-equivariant flow equivalence. Note that the space $\mathcal{D}$ can be
more abstractly described, up to homeomorphism, as the quotient of
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ by the
pointed-semi-conjugacy relation. It can also be identified with a quotient of
the space of left-invariant preorders on $G$, see [BMRT, Theorem 3.20].

###  3.2. First properties

We conclude with some basic properties of the Deroin space, which follow
easily from the definition and Proposition 3.5.

######  Definition 3.8.

A flow $(Y,\Phi)$ is _freely reversible_ if there is a fixed-point-free
involution (that is, a homeomorphism of order 2) $\iota\colon Y\to Y$ which
establishes a flow equivalence between the flow $(Y,\Phi)$ and its time-
reverse, denoted $(Y,\Phi^{-1})$.

If $G$ is a finitely generated group, let
$\hat{\iota}\colon\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})\to\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$
be the conjugation map by the reflection $x\to-x$. If $\mathcal{D}$ is a
Deroin space, then $\hat{\iota}$ descends to a map
$\iota\colon\mathcal{D}\to\mathcal{D}$, given by
$\iota=r_{\mathcal{D}}\circ\hat{\iota}$.

######  Proposition 3.9.

Let
$\mathcal{D}\subset\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$
be a Deroin space for $G$. Then the map $\iota\colon\mathcal{D}\to\mathcal{D}$
is a fixed-point-free involution which establishes a $G$-equivariant flow-
equivalence between $(\mathcal{D},\Psi)$ and $(\mathcal{D},\Psi^{-1})$. In
particular, $(\mathcal{D},\Psi)$ is freely reversible.

###### Proof.

The fact that $\iota$ is a $G$-equivariant flow equivalence between
$(\mathcal{D},\Psi)$ and $(\mathcal{D},\Psi^{-1})$ follows from Proposition
3.5. To observe that it is fixed-point free, assume by contradiction that
$\iota(\rho)=\rho$ for some $\rho\in\mathcal{D}$. This implies that every
element $\rho(g)$ commutes with an orientation-reversing homeomorphism
$h\colon\mathbb{R}\to\mathbb{R}$ fixing $0$. But $0$ is the unique fixed point
of $h$, so it must also be fixed by $\rho(G)$, contradicting that $\rho$ is
irreducible. ∎

The next proposition gives a precise dictionary with the statement of Theorem
2.3. It will not be used elsewhere in this paper, but we include it for
completeness.

######  Proposition 3.10 (Structure of orbits and $G$-invariant measures).

Let $\mathcal{D}$ be a Deroin space for the finitely generated group $G$ with
flow $\Psi$. Fix $\rho\in\mathcal{D}$, and write $\ell\subset\mathcal{D}$ for
its $\Psi$-orbit. Then we have the following alternative.

  1. (I)

The orbit $\ell$ is a point if and only if $\rho$ is semi-conjugate to an
action by translation. In particular, the set of fixed points of $\Psi$ is
homeomorphic to the sphere $\mathbb{S}^{b_{1}(G)-1}$, where
$b_{1}(G)=\mathsf{rk}(H^{1}(G,\mathbb{Z}))$ (with
$\mathbb{S}^{-1}=\varnothing$).

  2. (II)

The orbit $\ell$ is a topological circle if and only if $\rho$ is locally
proximal, but not proximal.

  3. (III)

The orbit $\ell$ is not closed if and only if $\rho$ is proximal.

Moreover, every $G$-invariant probability measure on $\mathcal{D}$ must be
supported on the set of fixed points of $\Psi$.

###### Proof.

By definition, $\rho$ is fixed by $\Psi$ if and only $\rho(G)$ commutes with
all translations, which happens if and only $\rho$ is an action by
translations. Conversely, assume that $\rho\in\mathcal{D}$ is semi-conjugate
to an action by translations, and thus actually conjugate to it (since all
actions in $\mathcal{D}$ are minimal or cyclic). Then all $\Psi$-translates of
$\rho$ are pointed conjugate to it. Since $\mathcal{D}$ contains a unique
representative of each pointed-semi-conjugacy class, we have that $\rho$ is
actually fixed by $\Psi$. This shows the first part of (I), and a similar
argument gives (II), and thus (III), using Theorem 2.3. Now, the set of fixed
points is homeomorphic to the space of non-trivial homomorphisms of $G$ to
$(\mathbb{R},+)$ up to positive constant, which is either empty or
homeomorphic to $\mathbb{S}^{n}$, where $n$ is the torsion-free rank of the
abelianisation of $G$ minus 1.

To show the last statement, let $\mu$ be a $G$-invariant probability measure
on $\mathcal{D}$. Then its support $\operatorname{\mathsf{Supp}}\mu$ is a
closed $G$-invariant subset, and since the $G$-orbit of every
$\rho\in\mathcal{D}$ is dense in its $\Psi$-orbit,
$\operatorname{\mathsf{Supp}}\mu$ is also $\Psi$-invariant. Assume by
contradiction that $\rho\in\operatorname{\mathsf{Supp}}\mu$ is not a fixed
point, then it is a locally proximal action by Theorem 2.3. Hence we can find
$g\in G$ and an interval $I=(-a,a)$ and $0<\varepsilon<a$ such that
$\rho(g)(\overline{I})\subset J:=(-a+\varepsilon,a-\varepsilon)$. Then
$U=\\{\rho^{\prime}\colon\rho^{\prime}(\overline{I})\subset J\\}$ is an open
neighbourhood of $\rho$ in $\mathcal{D}$. Let $U_{I}=\bigcup_{t\in
I}\Psi^{t}(U)$, and $U_{J}$ be defined similarly. Then by definition of the
translation action we have $g^{n}.U_{I}\subset U_{J}$ for every $n\geq 0$, and
thus $\mu(U_{I}\smallsetminus\overline{U_{J}})=0$. Hence
$U_{I}\smallsetminus\overline{U_{J}}$ is an open subset avoiding the support
of $\mu$. However, it contains points in the orbit of $\rho$, namely
$\Phi^{t}(\rho)$ for every $t\in I\smallsetminus\overline{J}$. This
contradicts the $\Psi$-invariance of $\operatorname{\mathsf{Supp}}\mu$. ∎

######  Remark 3.11.

The last statement in Proposition 3.10 implies the theorem of Witte Morris
[Witte], which states that a finitely generated amenable group has a non-
trivial representation to $\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ if
and only if it has a non-trivial homomorphism to $(\mathbb{R},+)$. This proof
is close to the one given by Deroin [Deroin] using compact $\Psi$-invariant
subsets of $\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$,
but avoids the machinery of disintegration of measures along $\Psi$-orbits.

######  Remark 3.12.

From Proposition 3.10, one can also deduce that a finitely generated group has
the property that its Deroin space contains no closed $\Psi$-orbit if and only
if any action on the circle has a fixed point. The groups
$\mathsf{T}(\varphi)$ and $\mathsf{T}(\varphi,\sigma)$ (for minimal subshifts
$\varphi$) were the first known groups displaying the latter property [MB-T,
MR4177294].

Finally we record the following straightforward consequence of Propositions
3.5 and 2.1, which further clarifies the relevance of Deroin space from the
perspective described in the introduction.

######  Proposition 3.13.

Let $(\mathcal{D},\Psi)$ be a Deroin space for a finitely generated group $G$
and
$r_{\mathcal{D}}\colon\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})\to\mathcal{D}$
the associated retraction.

  1. (1)

The open semi-conjugacy-saturated subsets of
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ are precisely
the preimages under $r_{\mathcal{D}}$ of the open $\Psi$-invariant subsets of
$\mathcal{D}$. In particular a representation
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ is
rigid if and only if $r_{\mathcal{D}}(\rho)$ belongs to an open $\Psi$-orbit.

  2. (2)

The connected and the path-connected components of
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ coincide with
their corresponding preimages of those of $\mathcal{D}$.

##  4\. Finitely generated groups of homeomorphisms of suspension spaces

###  4.1. Groups of homeomorphisms of suspension spaces

Let $X$ be a topological space, and $\varphi:X\to X$ a homeomorphism. Recall
that we denote by $Y$ the suspension space of $(X,\varphi)$, namely
$Y=(X\times\mathbb{R})/\mathbb{Z}$, where $\mathbb{Z}$ acts on
$X\times\mathbb{R}$ by $n\cdot(x,t)=(\varphi^{n}(x),t-n)$. We let
$\pi_{Y}\colon X\times\mathbb{R}\to Y$ be the quotient projection, and denote
by $\Phi$ the suspension flow on $Y$.

Assume that $(X,\varphi)$ is reversible, with associated involution
$\sigma:X\to X$ such that $\sigma\varphi\sigma=\varphi^{-1}$. The pair
$(\varphi,\sigma)$ then determines an action of the infinite dihedral group
$D_{\infty}$ on $X$, given by $(n,j)\cdot x=\varphi^{n}\sigma^{j}(x)$. Here we
identify $D_{\infty}$ with the semi-direct product
$\mathbb{Z}\rtimes(\mathbb{Z}/2\mathbb{Z})$, and write accordingly its
elements as pairs $(n,j)$ with $n\in\mathbb{Z}$ and
$j\in\mathbb{Z}/2\mathbb{Z}$. We can then consider a smaller suspension space
(a “mapping Klein bottle”). To see this, consider the standard isometric
action of $D_{\infty}$ on $\mathbb{R}$ given by $(n,j)\cdot t=(-1)^{j}x-n$.
Then the quotient $Z:=(X\times\mathbb{R})/D_{\infty}$ with respect to the
associated diagonal action is called the _dihedral suspension_ of
$(\varphi,\sigma)$. We will denote by $\pi_{Z}\colon X\times\mathbb{R}\to Z$
the quotient projection. The space $Z$ is a 2-to-1 quotient of the suspension
$Y$ of $\varphi$. More precisely, we have $Z=Y/\langle\hat{\sigma}\rangle$,
where $\hat{\sigma}:Y\to Y$ is the involution given by

$\hat{\sigma}\,\colon\,\pi_{Y}(x,t)\,\mapsto\,\pi_{Y}(\sigma(x),-t).$

Writing $p\colon Y\to Z$ for the quotient map, we have a commutative diagram:

${X\times\mathbb{R}}$${Y}$${Z=Y/\langle\hat{\sigma}\rangle}$$\scriptstyle{\pi_{Z}}$$\scriptstyle{\pi_{Y}}$$\scriptstyle{p}$

Note that since the involution $\sigma$ normalizes the cyclic group
$\langle\varphi\rangle$, it must necessarily send $\varphi$-orbits to
$\varphi$-orbits. The next elementary result clarifies the assumption that we
need to put on the involution $\sigma$.

######  Lemma 4.1.

Let $(X,\varphi,\sigma)$ be as above. Then, the following are equivalent:

  1. (1)

$\sigma$ does not preserve any $\varphi$-orbit;

  2. (2)

$\sigma(x)\notin\\{x,\varphi(x)\\}$ for every $x\in X$;

  3. (3)

every element of order 2 in $D_{\infty}$ acts on $X$ without fixed points;

  4. (4)

the map $\hat{\sigma}\colon Y\to Y$ has no fixed point;

  5. (5)

the diagonal action of $D_{\infty}$ on $X\times\mathbb{R}$ is free.

###### Proof.

It is clear that (1)$\Rightarrow$(2). Every element of order 2 in $D_{\infty}$
is conjugate to either $\sigma$ or $\sigma\varphi$, so (2)$\Rightarrow$ (3).
The point $\pi_{Y}(x,t)$ is fixed by $\hat{\sigma}$ if and only if there
exists $n\in\mathbb{Z}$ such that $\varphi^{n}\sigma$ fixes $x$ and $t=-n/2$,
so (3)$\Leftrightarrow$(4). Since elements of infinite order already act
without fixed points on $\mathbb{R}$, we have (3)$\Rightarrow$(5). Finally if
(1) does not hold, then there exist $x\in X$ and $n\in\mathbb{Z}$ such that
$\varphi^{-n}\sigma(x)=x$, and thus the element $\gamma=(-n,1)\in D_{\infty}$
fixes the point $(x,-n/2)$, so that (5)$\Rightarrow$(1). ∎

From now on, we will work in the following setting.

######  Assumption 4.2.

We let $X$ be a totally disconnected, metrisable, compact space, and
$(\varphi,\sigma)$ homeomorphisms of $X$ such that $\sigma^{2}=\mathsf{id}$
and $\sigma\varphi\sigma=\varphi^{-1}$. We further assume that:

  * •

the action of $\mathbb{Z}$ on $X$ determined by $\varphi$ is topologically
free, that is, the subset of points which are not periodic for $\varphi$ is
dense in $X$;

  * •

the homeomorphism $\sigma$ satisfies the equivalent conditions in Lemma 4.1.

As above, we always denote by $Y$ the suspension of $\varphi$, with suspension
flow $\Phi$, and by $Z$ the dihedral suspension of $(\varphi,\sigma)$.

######  Example 4.3.

As a basic example, start with any homeomorphism $\varphi_{0}\colon X_{0}\to
X_{0}$ of a compact totally disconnected metrisable space. Let $X$ be the
disjoint union of two copies of $X$, and define $\varphi$ to be equal to
$\varphi_{0}$ on one copy and to $\varphi_{0}^{-1}$ on the other. Finally let
$\sigma$ exchange the two copies of $X_{0}$. This class of examples is
somewhat degenerate because $\varphi$ preserves a partition into two clopen
sets (in particular it can never be topologically transitive).

######  Example 4.4.

Let $\varphi$ be an irrational rotation on $\mathbb{R}/\mathbb{Z}$, and
$\sigma$ the reflection $x\mapsto-x$. Then the pair $(\varphi,\sigma)$ defines
an action of $D_{\infty}$ on $\mathbb{R}/\mathbb{Z}$. The fixed points of all
involutions in $D_{\infty}$ belong to exactly 4 orbits (as there are 2
conjugacy classes of reflections and each has two fixed points). Blow up these
orbits to obtain a minimal invariant Cantor set $X$. Then $(X,\varphi,\sigma)$
satisfies Assumption 4.2.

######  Example 4.5.

The following source of examples is inspired from Hyde and Lodha [HL], see
[confined, Remark 4.19]. Take a finite alphabet $A$ of even cardinality
$|A|\geq 4$, and assume that $A$ is endowed with a map $a\mapsto a^{-1}$ which
is an involution without fixed points. Let $\varphi$ be the shift map on
$A^{\mathbb{Z}}$, and consider also the map
$\sigma((a_{n})_{n})=(a_{-(n+1)}^{-1})_{n}$ (namely the formal inverse of a
bi-infinite word in $A$). It is easy to verify that $\sigma^{2}=\mathsf{id}$
and $\sigma\varphi\sigma=\varphi^{-1}$, so $(\varphi,\sigma)$ determines an
action of $D_{\infty}$ on $\Omega$. This action does not satisfy the
conditions in Assumption 4.2, as $\sigma$ has fixed points. To avoid this
problem, we can consider the subshift of finite type
$X_{\text{red}}=\\{(a_{n})_{n}\in A^{\mathbb{Z}}:a_{n+1}\neq a^{-1}_{n}\\}$ of
_reduced_ words in $A$, which is closed and $D_{\infty}$-invariant. Then,
Assumption 4.2 is satisfied by $(X_{\text{red}},\varphi,\sigma)$. One can then
obtain many more examples as $D_{\infty}$-invariant closed subsets $X\subset
X_{\text{red}}$.

The set $Z$ is naturally partitioned into subsets of the form
$\ell=\pi_{Z}(\\{x\\}\times\mathbb{R})$, that we call the _leaves_ of $Z$. As
$X$ is totally disconnected, these coincide with the path components of $Z$.
By Lemma 4.1, the map $p\colon Y\to Z$ is a local homeomorphism, and is
injective in restriction to each $\Phi$-orbit. Take now a clopen subset
$C\subset X$, and an open interval $I\subset\mathbb{R}$. If the restriction of
$\pi_{Y}$ to $C\times I$ is injective, we denote by $Y_{C,I}$ its image. Then
the map

$\pi_{Y}\colon C\times I\to Y_{C,I}\subset Y$

is a homeomorphism, and is called a _chart_ of $Y$. Similarly, if the
restriction of $\pi_{Z}$ to $C\times I$ is injective, we denote its image by
$Z_{C,I}$ and call the map

$\pi_{Z}\colon C\times I\to Z_{C,I}\subset Z$

a chart of $Z$. Most of the time, we will just refer to the sets $Y_{C,I}$ and
$Z_{C,I}$ as charts, with an implicit identification with $C\times I$. Since
the actions of $\mathbb{Z}$ and $D_{\infty}$ on $X\times\mathbb{R}$ are free
and properly discontinuous, both maps $\pi_{Y}$ and $\pi_{Z}$ are local
homeomorphisms, and thus the spaces $Y$ and $Z$ can be covered by charts.

We now consider the group $\mathsf{H}_{0}(\varphi)$ (respectively,
$\mathsf{H}_{0}(\varphi,\sigma)$) of all homeomorphisms of $Y$ (respectively,
$Z$) isotopic to the identity. Note that this condition forces all elements of
$\mathsf{H}_{0}(\varphi)$ (respectively, $\mathsf{H}_{0}(\varphi,\sigma)$) to
preserve each $\Phi$-orbit in $Y$ (respectively, each leaf of $Z$), since
these are exactly the path components of the corresponding space. By [BCE,
Theorem 3.1], the group $\mathsf{H}_{0}(\varphi)$ is exactly the group of
homeomorphisms $h\colon Y\to Y$ for which there exists a continuous function
$\tau_{h}\colon Y\to\mathbb{R}$ such that

(4.1) $h(y)=\Phi^{\tau_{h}(y)}(y).$

It follows from Assumption 4.2 that such a function $\tau_{h}$ is necessarily
unique. Indeed, the value $\tau_{h}(y)$ is uniquely determined by $h(y)$
provided $y$ does not belong to a $\Phi$-periodic orbit. By our assumption
such points are dense, and thus $\tau_{h}$ is uniquely determined everywhere.
The function $\tau_{h}$ will be called the _translation cocycle_ of $h$.

######  Lemma 4.6.

Let $(X,\varphi,\sigma)$ be as in Assumption 4.2. Then, the action of
$\mathsf{H}_{0}(\varphi,\sigma)$ on $Z$ lifts through the map $p\colon Y\to Z$
to a unique action on $Y$ which preserves every $\Phi$-orbit. This action
identifies $\mathsf{H}_{0}(\varphi,\sigma)$ with the centralizer of
$\hat{\sigma}$ in $\mathsf{H}_{0}(\varphi)$, which consists of all elements
$h$ such that $\tau_{h}\circ\hat{\sigma}=-\tau_{h}$.

###### Proof.

By Lemma 4.1, the map $p$ sends each $\Phi$-orbit bijectively onto a leaf
$\ell$ of $Z$. Hence, at least at a set-theoretic level, the action of
$\mathsf{H}_{0}(\varphi,\sigma)$ on $Z$ can be leafwise lifted to a unique
action on $Y$ which preserves every $\Phi$-orbit, and such that the map $p$ is
equivariant. To check that this action is continuous, fix a finite cover by
charts $Z=\bigcup_{i=1}^{r}Z_{C_{i},I_{i}}$; each chart $Z_{C_{i},I_{i}}$
lifts to a pair of disjoint charts in $Y$, mapped homeomorphically onto it by
$p$. Take $\varepsilon>0$ such that every arc of length at most $\varepsilon$
in a leaf of $Z$ is contained in one of the charts $Z_{C_{i},I_{i}}$. If
$g\in\mathsf{H}_{0}(\varphi,\sigma)$ is an element that displaces any point of
$Z$ by a distance less than $\varepsilon$ on the corresponding leaf, then one
can check on charts that the lift of $g$ to $Y$ defined above is continuous.
But such elements generate $\mathsf{H}_{0}(\varphi,\sigma)$, as all elements
are isotopic to the identity. This implies that the lift of any element is
continuous. By the same reasoning, every isotopy of an element
$g\in\mathsf{H}_{0}(\varphi,\sigma)$ to the identity in the group of
homeomorphism of $Z$ lifts to an isotopy in the group of homeomorphisms of
$Y$. Therefore $\mathsf{H}_{0}(\varphi,\sigma)$ lifts to a subgroup of
$\mathsf{H}_{0}(\varphi)$. Moreover, its action clearly commutes with
$\hat{\sigma}$. Conversely, assume that $h\in\mathsf{H}_{0}(\varphi)$ commutes
with $\hat{\sigma}$. It is easily checked that this is equivalent to
$\tau_{h}\circ\hat{\sigma}=-\tau_{h}$. Choose the isotopy $h_{s}$ of $h$ to
the identity given by $h_{s}(y)=\Phi^{(1-s)\tau_{h}(y)}(y)$. Then each element
$h_{s}$ commutes with $\hat{\sigma}$ and thus descends to a homeomorphism of
$Z$. This shows that $h$ defines indeed an element of
$\mathsf{H}_{0}(\varphi,\sigma)$. ∎

###  4.2. Perfect germ extensions of (solenoidal) Thompson’s groups

Here we consider variations of the classical Thompson’s groups. We will use
some of the standard properties of these groups (such as finite generation)
that the reader can find explained in standard references, such as the text by
Cannon, Floyd, and Parry [CFP].

Let $I,J\subset\mathbb{R}$ be open dyadic intervals. We say that a
homeomorphism $f:I\to J$ is _dyadic piecewise linear (PL)_ if there exists a
finite subset $\Sigma\subset I$ so that:

  * •

$\Sigma\subset\mathbb{Z}[\frac{1}{2}]$, and

  * •

$f$ is locally dyadic affine outside $\Sigma$, that is, on each connected
component of $I\smallsetminus\Sigma$, the map is defined as $x\mapsto
2^{n}x+b$ for some $n\in\mathbb{Z}$ and $b\in\mathbb{Z}[\frac{1}{2}]$.

We are interested in the family of maps which are locally dyadic PL outside a
finite subset, and satisfy an extra local condition. To define this properly,
given a dyadic rational, let $h_{x_{0}}:\mathbb{R}\to\mathbb{R}$ be the map
defined by $h_{x_{0}}(x)=2(x-x_{0})+x_{0}$.

######  Definition 4.7.

Say that a map $f:I\to J$ is _of type_ ${\mathfrak{D}}$ if there exists a
finite subset $\mathsf{BP}^{2}(f)\subset I$ such that

  * •

every $x\in I\smallsetminus\mathsf{BP}^{2}(f)$ has an open neighborhood $U$ so
that $f|_{U}$ is dyadic PL, and

  * •

every $x_{0}\in\mathsf{BP}^{2}(f)$ has a neighborhood $U$ so that $f\circ
h_{x_{0}}=h_{f(x_{0})}\circ f$ for $x\in U$.

Given an interval $I\subset\mathbb{R}$, we denote by $\mathscr{F}_{I}$ the
group of homeomorphisms of $\overline{I}$ of type ${\mathfrak{D}}$, and
$\mathscr{F}:=\mathscr{F}_{(0,1)}$. These are all isomorphic if $I$ is dyadic.
Note that the subgroup of all elements $f\in\mathscr{F}$ such that
$\operatorname{\mathsf{BP}}^{2}(f)=\varnothing$ is the standard Thompson’s
group $F$. Given a dyadic $x\in[0,1]$, denote by $\mathcal{D}_{x}^{+}$
(respectively, $\mathcal{D}_{x}^{-}$) the group of right (respectively, left)
germs of $\mathscr{F}$ at $x$. Recall that these are defined as the quotient
of the stabiliser of $\mathscr{F}$ by the normal subgroup of elements acting
trivially on a right (respectively, left) neighbourhood. Write
$\tilde{T}\subseteq\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ for the
$\mathbb{Z}$-central extension of Thompson’s group $T$, obtained by lifting
the action of $T$ on the circle. Explicitly, $\tilde{T}$ is the group of all
(locally) dyadic PL homeomorphisms of $\mathbb{R}$ commuting with the unit
translation $t_{1}\colon x\mapsto x+1$. The proof of the next lemma is very
similar to that of [BMRT, Lemma 12.22].

######  Lemma 4.8.

The groups $\mathcal{D}^{+}_{x}$ and $\mathcal{D}^{-}_{x}$ are isomorphic to
$\tilde{T}$ for every $x\in\mathbb{Z}[\frac{1}{2}]\cap[0,1]$.

###### Proof.

Take $x\in(0,1)$ dyadic and notice that the map $h_{x}|_{(-\infty,x)}$ is
conjugate to the translation $t_{1}$ by a dyadic PL homeomorphism
$f\colon(-\infty,x)\to\mathbb{R}$. This establishes an isomorphism between
$\mathcal{D}_{x}^{-}$ and the group of germs of $\tilde{T}$ at $+\infty$,
which is isomorphic to $\tilde{T}$ itself. ∎

######  Lemma 4.9.

The group $\mathscr{F}$ is finitely generated and perfect.

###### Proof.

Fix a dyadic $x\in(0,1)$. Notice that the group of germs at $x$, that we
denote by $\mathcal{D}_{x}$, is isomorphic to the direct product
$\mathcal{D}_{x}^{-}\times\mathcal{D}_{x}^{+}$. Thus, by Lemma 4.8, we have
that $\mathcal{D}_{x}$ is isomorphic to $\tilde{T}\times\tilde{T}$. In
particular it is finitely generated, so we can choose a finite subset
$S\subset\mathscr{F}$ that generates $\mathcal{D}_{x}$. Moreover, we can ask
every $f\in S$ to satisfy $\mathsf{BP}^{2}(f)=\\{x\\}$. Thus, given
$g\in\mathscr{F}$, we can find $g_{\ast}$ in the group $\langle
F^{\prime},S\rangle$ so that
$\mathsf{BP}^{2}(gg_{\ast})\cap(0,1)=\varnothing$. Analogously, we can also
consider finite subsets $S_{0}$ and $S_{1}$ generating $\mathcal{D}_{0}^{+}$
and $\mathcal{D}_{1}^{-}$, respectively. Therefore we can write

$\mathscr{F}=\langle F^{\prime},S,S_{0},S_{1}\rangle=\langle
F,S,S_{0},S_{1}\rangle.$

Since Thompson’s group $F$ is finitely generated, we deduce that so is
$\mathscr{F}$. We next prove that $\mathscr{F}$ is perfect. For this, notice
that, since $\tilde{T}$ is perfect, we can find $S$, $S_{0}$ and $S_{1}$
consisting of products of commutators. Finally, since $F^{\prime}$ is also
perfect it follows that $\mathscr{F}$ is perfect. ∎

We now define two countable subgroups of the groups $\mathsf{H}_{0}(\varphi)$
and $\mathsf{H}_{0}(\varphi,\sigma)$, respectively, which are
type-${\mathfrak{D}}$ analogues of the group $\mathsf{T}(\varphi)$ defined in
[MB-T].

######  Definition 4.10.

We let $\mathscr{T}(\varphi)$ be the group of homeomorphisms $g$ of $Y$ such
that for every $y\in Y$, there exist charts $Y_{C,I},Y_{C,J}$ containing $y$
and $g(y)$, respectively, and a homeomorphism $f\colon I\to J$ of type
${\mathfrak{D}}$ such that $g(Y_{C,I})=Y_{C,J}$, and $g|_{Y_{C,I}}$ is given
by

$g(\pi_{Y}(x,t))=\pi_{Y}(x,f(t))\quad\text{for any }(x,t)\in C\times Y.$

Similarly, we let $\mathscr{T}(\varphi,\sigma)$ be the group of homeomorphisms
$g$ of $Z$ such that for every $y\in Y$, there exist charts $Z_{C,I},Z_{C,J}$
containing $y$ and $g(y)$, respectively, and a homeomorphism $f\colon I\to J$
of type ${\mathfrak{D}}$ such that $g(Z_{C,I})=Z_{C,J}$, and $g|_{Z_{C,I}}$ is
given by

$g(\pi_{Z}(x,t))=\pi_{Z}(x,f(t))\quad\text{for any }(x,t)\in C\times Y.$

######  Lemma 4.11.

We have $\mathscr{T}(\varphi)\leq\mathsf{H}_{0}(\varphi)$ and
$\mathscr{T}(\varphi,\sigma)\leq\mathsf{H}_{0}(\varphi,\sigma)$. The group
$\mathscr{T}(\varphi,\sigma)$ coincides with the centralizer of $\hat{\sigma}$
in $\mathscr{T}(\varphi)$.

###### Proof.

For $g\in\mathscr{T}(\varphi)$, one can define a continuous function
$\tau_{g}$ as in (4.1) locally around any point $y\in Y$, by choosing charts
$Y_{C,I},Y_{C,J}$, and $f\colon I\to J$ as in the definition, and setting
$\tau_{g}(p(x,t))=f(t)-t$. This definition must agree on overlapping charts,
since $\tau_{g}(y)$ satisfying (4.1) is uniquely determined for $y$ in a dense
set. Thus $\mathscr{T}(\varphi)\leq\mathsf{H}_{0}(\varphi)$. The rest of the
proof is similar to that of Lemma 4.6. ∎

The following result characterizes when the group
$\mathscr{T}(\varphi,\sigma)$ is finitely generated. condition is satisfied
exactly when $\varphi:X\to X$ is conjugate to a subshift (or equivalently,
when it is expansive).

######  Theorem 4.12.

Let $(X,\varphi,\sigma)$ be a reversible subshift satisfying Assumption 4.2.
Then, $\mathscr{T}(\varphi,\sigma)$ is finitely generated.

This is the analogue of [MB-T, Theorem A]. The argument in the proof is
essentially the same but needs to be slightly modified, as it makes use of the
existence of elements in $\mathsf{T}(\varphi)$ that move all points in the
suspension space $Y$ strictly in the direction of the flow. This is not
possible here as there is no globally well-defined direction on the leaves of
$Z$. We provide details in Appendix A.

######  Remark 4.13.

For any $(X,\varphi)$, the group $\mathscr{T}(\varphi)$ can be identified with
$\mathscr{T}(\varphi_{1},\sigma_{1})$, where $(X_{1},\varphi_{1},\sigma_{1})$
is obtained from $(X,\varphi)$ through the doubling construction in Example
4.3. Thus any result on the groups $\mathscr{T}(\varphi,\sigma)$ can be
translated to the groups $\mathscr{T}(\varphi)$.

##  5\. Actions on the line of $\mathscr{T}(\varphi,\sigma)$

###  5.1. Identifying the Deroin space

In this section we analyse actions on the line of the group
$G=\mathscr{T}(\varphi,\sigma)$, where $(X,\varphi,\sigma)$ is as in
Assumption 4.2. Our ultimate goal is to show that, when $G$ is finitely
generated, the Deroin space of $G$ can be identified with $(Y,\Phi)$. While
finite generation is necessary for the Deroin space to be well-defined, the
reader will see that the main result of the section does not require that $G$
be finitely generated: we will prove that any irreducible action of $G$ on the
line, admitting a minimal set, comes from $Y$. Thus we do not assume here that
$(X,\varphi)$ is conjugate to a subshift.

Recall that given $g\in G$, we use the notation $\tau_{g}\colon
Y\to\mathbb{R}$ for the translation cocycle of $g$. Given $y\in Y$, let
$\rho_{y}\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ be
the action given by

(5.1) $\rho_{y}(g)(t)=t+\tau_{g}(\Phi^{t}(y)).$

When $y$ is not periodic for $\Phi$, this is simply the action of $G$ on the
$\Phi$-orbit of $y$, if we identify the orbit with $\mathbb{R}$ using the
flow, in such a way that $0$ corresponds to the point $y$. When $y$ is
periodic, its orbit is homeomorphic to a circle, and the action $\rho_{y}$ is
a lift of the action of $G$ under the covering map
$\mathbb{R}\to\mathbb{S}^{1}$ determined by the flow and which maps $0$ to
$y$. Then we have the following.

######  Theorem 5.1.

Let $(X,\varphi,\sigma)$ be as in Assumption 4.2, and write
$G:=\mathscr{T}(\varphi,\sigma)$. Let
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ be a
minimal action. Then, there exists $y\in Y$ such that $\rho$ and $\rho_{y}$
are pointed conjugate.

Before proving Theorem 5.1, let us deduce Theorem 1.1, which is based on the
following consequence.

######  Corollary 5.2.

Let $(X,\varphi,\sigma)$ be as in Assumption 4.2. Consider the map $q\colon
Y\to\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ given by
$q\colon y\mapsto\rho_{y}$. Then $q$ is a $G$-equivariant homeomorphism onto
its image, which conjugates the flow $\Phi$ to the translation flow $\Psi$ on
$q(Y)$. In particular if $(X,\varphi)$ is a subshift, then $(Y,\Phi)$
identifies with a Deroin space for $G$.

###### Proof.

From the explicit formula defining the action $\rho_{y}$, it is clear that $q$
is continuous. Pick two distinct points $y_{1},y_{2}\in Y$. If
$y_{1}\neq\hat{\sigma}(y_{2})$, then also their projections to $Z$ are
distinct. It follows that they have distinct stabilisers in $G$. As a
consequence, the actions $\rho_{y_{1}}$ and $\rho_{y_{2}}$ are not pointed
conjugate, and since they are minimal, they are not pointed semi-conjugate
either (see Remark 3.2). If $y_{1}=\hat{\sigma}(y_{2})$, then every element
$g\in G$ such that $\rho_{y_{1}}(g)(0)>0$, must satisfy
$\rho_{y_{2}}(g)(0)<0$, and thus also the actions $\rho_{y_{i}}$ are not
pointed semi-conjugate. In particular the map $q$ is injective and thus a
homeomorphism onto its image. It is straightforward that $q$ conjugates $\Phi$
to the translation flow and is $G$-equivariant.

For the last statement, note that after Theorem 4.12, the group $G$ is
finitely generated, so that any irreducible action admits a minimal set.
Moreover, since the group $G$ is perfect, it does not admit any cyclic action.
It thus follows from Theorem 5.1 that $q(Y)$ contains exactly one
representative for each pointed-semi-conjugacy class.∎

Corollary 5.2 implies Theorem 1.1, using the results of Bowen and Walters
[expansive]. We first recall all the necessary terminology.

######  Definition 5.3.

Let $(Y,\Phi)$ be a flow on a compact metrisable space, and $d$ a metric on
$Y$. Then $\Phi$ is _expansive_ if for every $\varepsilon>0$ there exists
$\delta>0$ with the property that if $x,y\in Y$ satisfy
$d(\Phi^{t}(x),\Phi^{s(t)})<\delta$ for every $t\in\mathbb{R}$ and for some
continuous map $s\colon\mathbb{R}\to\mathbb{R}$, then $y=\Phi^{t}(x)$ with
$|t|<\varepsilon$.

This property does not depend on the choice of the metric $d$. Recall also
that a space $Y$ has topological dimension $d$ if every open cover
$\mathcal{U}$ has a refinement $\mathcal{U^{\prime}}$ such that every point of
$Y$ is contained in at most $d+1$ elements of $\mathcal{U}^{\prime}$, and $d$
is the least integer with this property.

###### Proof of Theorem 1.1.

Let $(Y,\Phi)$ be as in Theorem 1.1. It is shown in [expansive] that $\Phi$
admits a cross-section $X\subset Y$ (namely a closed subset which intersects
every $\Phi$-orbit in a discrete set of times) and that for every such cross-
section, the first-return map $\varphi$ to $X$ is conjugate to a subshift, so
that $(Y,\Phi)$ is flow-equivalent to the suspension flow of $(X,\varphi)$.
The only minor point to address is that we want $(X,\varphi)$ to be reversible
and to satisfy Assumption 4.2. Since $(Y,\Phi)$ is freely reversible, there
exists a fixed-point-free involution $\hat{\sigma}\colon Y\to Y$ sending any
$\Phi$-orbit to a $\Phi$-orbit in an orientation preserving way. Let $X_{0}$
be any cross-section and set $X=X_{0}\cup\hat{\sigma}(X_{0})$. Then $X$ is
still a cross-section. Denoting by $\varphi$ the first-return map and
$\sigma=\hat{\sigma}|_{X}$, then $(X,\varphi,\sigma)$ is a reversible
subshift. Moreover $\hat{\sigma}$ cannot preserve any $\Phi$-orbit (or it
would have a fixed point), so neither does $\sigma$ and by Lemma 4.1 we have
that $(X,\varphi,\sigma)$ satisfies Assumption 4.2. ∎

The rest of the section is devoted to the proof of Theorem 5.1.

###  5.2. Finding minimal sets for non-finitely generated groups

The first step is to show Proposition 5.6 below. We will make repeated use of
the following consequence of Theorem 2.3.

######  Corollary 5.4.

Let $G$ be a group and
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ an
irreducible action admitting a perfect minimal set $\Lambda$. Then, every
perfect subgroup of $\mathcal{Z}(\rho)$ pointwise fixes $\Lambda$.

###### Proof.

Since $\Lambda$ is the unique minimal set for $\rho$, it must be preserved by
$\mathcal{Z}(\rho)$. Moreover, by collapsing connected components of the
complement of $\Lambda$, we can semi-conjugate $\rho$ to a minimal action
$\hat{\rho}:H\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$. Since
$\mathcal{Z}(\rho)$ preserves $\Lambda$, it projects to
$\mathcal{Z}(\hat{\rho})$. By Theorem 2.3, we get that any perfect subgroup of
$\mathcal{Z}(\rho)$ projects to the trivial group, implying that it pointwise
fixes $\Lambda$. ∎

Given a group $H$ and an integer $n\geq 1$, write
$H^{n}=H_{1}\times\cdots\times H_{n}$ for the direct product of $n$ copies of
$H$. Denote by $\pi_{i}:H^{n}\to H_{i}$ the projection from $H^{n}$ onto
$H_{i}$.

######  Lemma 5.5.

Let $H$ be a finitely generated perfect group. Consider also an action
$\rho:H^{n}\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ and a subgroup
$K\subseteq H^{n}$ for which the projections $\pi_{i}|_{K}$ are surjective for
any $i\in\\{1,\ldots,n\\}$. Then we have the following.

  1. (1)

$\operatorname{\mathsf{Fix}}(\rho|_{K})=\operatorname{\mathsf{Fix}}(\rho)$.

  2. (2)

If $\rho$ is irreducible, then $\rho|_{K}$ and $\rho$ have the same unique
perfect minimal set.

###### Proof.

We will first prove (2). Notice that there is $m\in\\{1,\ldots,n\\}$ so that
$\rho|_{H_{m}}$ is irreducible, and without loss of generality we can assume
$m=n$. Indeed, if this was not the case, $\rho$ would have a fixed point,
contradicting its irreducibility. Moreover, since $H$ is finitely generated,
the action $\rho|_{H_{n}}$ preserves a minimal set that we denote by
$\Lambda$. This set cannot be a single closed orbit, since $H_{n}$ is perfect,
and thus $\Lambda$ is a perfect set. Thus, by Corollary 5.4, we have that the
subgroup $H_{1}\times\cdots\times H_{n-1}$ must fix $\Lambda$ pointwise. This
implies that the action of $\rho$ on $\Lambda$ (that we denote by
$\rho_{\ast}:H^{n}\to\operatorname{\mathsf{Homeo}}(\Lambda)$) factors through
the projection $\pi_{n}$, that is, $\rho_{\ast}=\Psi_{\ast}\circ\pi_{n}$. On
the other hand, by hypothesis $\pi_{n}|_{K}$ surjects to $H_{n}$ and therefore
$\rho_{\ast}|_{K}:K\to\operatorname{\mathsf{Homeo}}(\Lambda)$ has the same
image as $\rho_{\ast}$, implying that it is also a minimal action. This
implies that $\Lambda$ is a minimal set of $\rho|_{K}$, as desired.

We next prove (1). For this, take a connected component $J$ of
$\mathbb{R}\smallsetminus\operatorname{\mathsf{Fix}}(\rho)$. By part (2) the
actions of $H^{n}$ and $K$ on $J$ have the same minimal set, and in particular
$J$ is a connected component of the support of $\rho|_{K}$. Since the
component $J$ was arbitrary this implies that the support of $\rho$ is
contained in that of $\rho|_{K}$, which taking complements gives the reverse
inclusion
$\operatorname{\mathsf{Fix}}(\rho|_{K})\subseteq\operatorname{\mathsf{Fix}}(\rho)$.
Since the other inclusion is trivial, equality in (1) follows. ∎

Given a space $X$ and a group $H$, denote by $\mathcal{C}(X,H)$ the group of
continuous functions from $X$ to $H$. Note that this group is not finitely
generated in general thus its actions on the line may a priori fail to admit a
minimal set. However, this is the case when $X$ is as in Assumption 4.2, and
$H$ is perfect and finitely generated:

######  Proposition 5.6.

Let $X$ be a metrisable, compact, totally disconnected space, and $H$ a
finitely generated perfect group. Then, every action
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(\mathcal{C}(X,H);\mathbb{R})$
admits a perfect minimal set.

Before proving this result, let us introduce some further notation. Assume $X$
is a compact, totally disconnected space. For a clopen subset $C\subseteq X$,
let $H_{C}\leq\mathcal{C}(X,H)$ be the subgroup (isomorphic to $H$) of
functions $\sigma\colon X\to H$ which are constants on $C$ and equal to the
identity on $X\smallsetminus C$. Also, if $\mathcal{P}$ is a clopen partition
of $X$, write
$H^{(\mathcal{P})}:=\langle\bigcup_{C\in\mathcal{P}}H_{C}\rangle$, which is
naturally isomorphic to the direct product $H^{|\mathcal{P}|}$. Recall also
that if $\mathcal{P}_{1},\mathcal{P}_{2}$ are clopen partitions of $X$, one
says that $\mathcal{P}_{2}$ is _finer_ than $\mathcal{P}_{1}$ if for every
$C\in\mathcal{P}_{2}$, there exists $D\in\mathcal{P}_{1}$ so that $C\subseteq
D$.

######  Remark 5.7.

When $\mathcal{P}_{2}$ is finer that $\mathcal{P}_{1}$, there exists a natural
injective morphism $\iota:H^{(\mathcal{P}_{1})}\to H^{(\mathcal{P}_{2})}$, and
this morphism satisfies that $\pi\circ\iota$ is surjective for every
projection $\pi:H^{(\mathcal{P}_{2})}\to H_{C}$ given by any clopen subset
$C\in\mathcal{P}_{2}$. This holds in particular when $\mathcal{P}_{1}=\\{X\\}$
is the trivial partition.

###### Proof of Proposition 5.6.

As $X$ is metrisable, we can find an increasingly finer sequence of partitions
$\mathcal{P}_{n}$ of $X$ that separates points. Then we can write
$\mathcal{C}(X,H)$ as an increasing union

(5.2) $\mathcal{C}(X,H)=\bigcup_{n\in\mathbb{N}}H^{(\mathcal{P}_{n})}.$

By Remark 5.7, for every projection $\pi:H^{(\mathcal{P}_{n})}\to H_{C}$
associated with every $C\in\mathcal{P}_{n}$, the restriction
$\pi|_{H_{X}}:H_{X}\to H_{C}$ is surjective. Write
$\rho_{n}:=\rho|_{H^{(\mathcal{P}_{n})}}$. Thus, applying the first part of
Lemma 5.5 (with $\rho=\rho_{n}$ and $K=H_{X}$), we have the equality

(5.3)
$\operatorname{\mathsf{Fix}}(\rho_{n}|_{H_{X}})=\operatorname{\mathsf{Fix}}(\rho_{n}).$

Since $\rho_{n}|_{H_{X}}=\rho|_{H_{X}}$, (5.3) gives
$\operatorname{\mathsf{Fix}}(\rho|_{H_{X}})=\operatorname{\mathsf{Fix}}(\rho_{n})$
for every $n\in\mathbb{N}$, whereas (5.2) gives
$\operatorname{\mathsf{Fix}}(\rho)=\bigcap_{n\in\mathbb{N}}\operatorname{\mathsf{Fix}}(\rho_{n})$.
As $\rho$ is irreducible, we deduce that
$\operatorname{\mathsf{Fix}}(\rho|_{H_{X}})=\varnothing$, and thus, by (5.3)
again, also $\rho_{n}$ is irreducible for any $n\in\mathbb{N}$. Therefore, we
can apply the second part of Lemma 5.5 (applied to $\rho=\rho_{n}$ and
$K=H_{X}$ again), and obtain that $\rho_{n}$ has a unique perfect minimal set
$\Lambda$, which does not depend on $n\in\mathbb{N}$, so, by (5.2), $\Lambda$
is a minimal set for $\rho$. ∎

######  Remark 5.8.

We will apply Proposition 5.6 to the group $H=\mathscr{F}$, the perfect germ
extension of Thompson’s group $F$ introduced in §4.2. We note that if instead
we take $H$ to be the usual Thompson’s group $F$, then the conclusion of
Proposition 5.6 is far from true. Indeed the abelianisation of the group
$\mathcal{C}(X,F)$ is free abelian of infinite rank; one can use this to
construct a wealth of faithful actions of $\mathcal{C}(X,F)$ without a minimal
set, by defining orders on $\mathcal{C}(X,F)$ through lexicographic
constructions similar to the one in [BMRT, §5.3.1]. This is the reason why we
consider a germ extension of PL homeomorphisms to define the groups
$\mathscr{T}(\varphi,\sigma)$.

###  5.3. Building an equivariant map to a leaf

The key step in proving Theorem 5.1 is the following result, which says that
any minimal action of $G$ comes from an action on a leaf of $Z$.

######  Proposition 5.9.

With assumptions as in Theorem 5.1, there exists a map $h\colon\mathbb{R}\to
Z$ which is $G$-equivariant (with respect to the $\rho$-action on $\mathbb{R}$
and the natural action on $Z$), locally injective, and takes values in a
single leaf.

The main difference from the statement of Theorem 5.1 is that the map is to a
leaf of $Z$ (instead of $Y$), and moreover the leaf can be closed. In the
latter case, we want to prove that the minimal action $\rho$ is conjugate to
the lift of the action on the leaf to its universal cover. For this, we
introduce the group of circle homeomorphisms $\mathscr{T}$, defined to be the
perfect germ extension of Thompson’s group $T$ acting on the circle, namely
the group of all homeomorphisms of $\mathbb{S}^{1}$ of type ${\mathfrak{D}}$.
We further let $\tilde{\mathscr{T}}$ denote its lift to $\mathbb{R}$, namely
the group of all homeomorphisms of $\mathbb{R}$ which are locally of type
${\mathfrak{D}}$ and commute with integer translations. Thus,
$\tilde{\mathscr{T}}$ has an infinite cyclic center
$\mathcal{Z}(\tilde{\mathscr{T}})$ and
$\tilde{\mathscr{T}}/\mathcal{Z}(\tilde{\mathscr{T}})=\mathscr{T}$. We will
need the following lemma.

######  Lemma 5.10.

Let $f\colon\tilde{\mathscr{T}}\to\tilde{\mathscr{T}}$ be a group homomorphism
such that
$f(\mathcal{Z}(\tilde{\mathscr{T}}))\subset\mathcal{Z}(\tilde{\mathscr{T}})$
and such that $f$ induces in the quotient the identity map
$\mathscr{T}\to\mathscr{T}$. Then $f$ is the identity.

###### Proof.

The analogous statement for the usual Thompson’s group $T$ this is proven at
the end of the proof of [MB-T, Theorem 8.7]. The same proof works for
$\mathscr{T}$. ∎

We will also use the fact that the group $G$ is perfect. For this, given an
element $g\in G$, we call the set of points of $Z$ (respectively, $Y$) that
are moved by $g$ the _support_ of $g$, and denote it by
$\operatorname{\mathsf{Supp}}_{Z}(g)$ (respectively,
$\operatorname{\mathsf{Supp}}_{Y}(g)$). Note that the support is an open set.
For an open subset $U\subset Z$ we let $G_{U}$ be the subgroup of elements
with support contained in $U$; if $U=Z_{C,I}$ is a chart, we also denote
$G_{U}$ by $G_{C,I}$, and use analogous notation for the action on $Y$. In
order to prove that $G$ is perfect, we prove that it is generated by the
perfect subgroups $G_{C,I}$ (see the proof of Lemma 5.12 below), provided that
we take charts $Z_{C,I}$ covering $Z$. More precisely, we have the following
fundamental property, which is the analogue of [MB-T, Lemma 4.8]. The proof is
discussed in Appendix A.

######  Proposition 5.11 (Fragmentation lemma).

Let $(X,\varphi,\sigma)$ be as in Assumption 4.2, and set
$G=\mathscr{T}(\varphi,\sigma)$. For any element $g\in G$ and open cover
$\mathcal{U}$ of $\overline{\operatorname{\mathsf{Supp}}_{Z}(g)}$, we have
$g\in\left\langle\bigcup_{U\in\mathcal{U}}G_{U}\right\rangle$.

In particular, the group $G$ is perfect.

Let us see now how to deduce Theorem 5.1, assuming Proposition 5.9.

###### Proof of Theorem 5.1.

Let $h\colon\mathbb{R}\to Z$ be the locally injective map provided by
Proposition 5.9 and set $z:=h(0)$. Since the quotient map $q\colon Y\to Z$
identifies each $\Phi$-orbit to a leaf, and maps exactly two $\Phi$-orbits to
each leaf, the map $h$ admits two lifts
$\tilde{h}_{1},\tilde{h}_{2}\colon\mathbb{R}\to Y$ taking values in two
different $\Phi$-orbits, with
$\tilde{h}_{i}\circ\hat{\sigma}=\tilde{h}_{1-i}$. Moreover, since
$\hat{\sigma}$ reverses the time of the flow $\Phi$, exactly one of these two
lifts is orientation preserving with respect to the orientation of orbits
given by the flow. Rename this lift by $\tilde{h}$ and set $y=\tilde{h}(0)$.
If $y$ is not a periodic orbit, the map $\tilde{h}$ is a bijection of
$\mathbb{R}$ onto the orbit, and it follows that $\rho$ is pointed conjugate
to $\rho_{y}$.

The case where $y$ is a periodic orbit requires some further inspection.
Assume that the period of $y$ is $m\in\mathbb{N}$. Let $H$ be the subgroup of
$G$ of elements that act trivially on the leaf $\ell$ of $z$, and let
$H_{0}\leq H$ be the subgroup consisting of elements $g$ such that
$\overline{\operatorname{\mathsf{Supp}}_{Z}(g)}\cap\ell=\varnothing$. Note
that both are normal subgroups of $G$: the group $H$ coincides with the group
of elements $g\in G$ such that $\rho_{y}(g)$ is a translation by an integer
multiple of $m$, while $H_{0}$ is the kernel of $\rho_{y}$, using that non-
periodic orbits of $\Phi$ are dense. The action of $G$ on $\ell$ identifies
$G/H$ with the natural copy of $\mathscr{T}$ acting on
$\ell=\mathbb{R}/m\mathbb{Z}\cong\mathbb{S}^{1}$. Accordingly, the action
$\rho_{y}$ induces an isomorphism of $G/{H_{0}}$ to the natural copy of
$\tilde{\mathscr{T}}$ conjugated by the dilation $x\mapsto mx$. Now the map
$h\colon\mathbb{R}\to\ell$ is a covering map, equivariant with respect to the
action $\rho$ and the natural action on $\ell$. Upon conjugating $\rho$, we
can assume that $h$ coincides with the natural covering map $\mathbb{R}$ to
$\mathbb{R}/m\mathbb{Z}\cong\ell$, so that $\rho$ also takes values in the
$m$-dilated copy of $\tilde{\mathscr{T}}$. Since $H$ acts trivially on $\ell$,
we deduce that $\rho(H)$ is central in $\rho(G)$, and in particular it is
abelian. However, the subgroup $H_{0}$ is perfect, as a consequence of
Proposition 5.11. Therefore $\rho(H_{0})=\\{\mathsf{id}\\}$, and thus $\rho$
descends to a map $\bar{\rho}\colon
G/H_{0}\cong\tilde{\mathscr{T}}\to\tilde{\mathscr{T}}$ that satisfies the
assumption of Lemma 5.10. We conclude that $\bar{\rho}$ is the identity and
that $\rho=\rho_{y}$. ∎

The rest of the section is devoted to the proof of Proposition 5.9. The proof
requires some preliminary work. We keep assumptions as in the statement, and
we introduce further notation. Let $H$ be a subgroup of $G$, and take
$\xi\in\mathbb{R}$. When $\xi\in\operatorname{\mathsf{Supp}}(\rho|_{H})$, we
denote by $I^{\rho}(H,\xi)$ the component of the support of $\rho(H)$
containing $\xi$. When $\xi\in\operatorname{\mathsf{Fix}}(\rho|_{H})$, we
simply define $I^{\rho}(H,\xi)=\\{\xi\\}$. In the case that $H=G_{C,I}$ for
some chart $Z_{C,I}$, we write $I^{\rho}(C,I,\xi)$ instead of
$I^{\rho}(G_{C,I},\xi)$. When there is no risk of confusion, we simply write
$I(H,\xi)$ or $I(C,I,\xi)$. We then have the following lemma, which is the
main application of Proposition 5.6.

######  Lemma 5.12.

With assumptions as in Theorem 5.1, given any dyadic chart $Z_{C,I}$ and
$\xi\in\mathbb{R}$, we have that the $\rho$-action of $G_{C,I}$ on
$I(C,I,\xi)$ admits a perfect minimal set.

###### Proof.

When $\xi\in\operatorname{\mathsf{Fix}}(\rho|_{G_{C,I}})$ there is nothing to
prove. So we can assume that $I(C,I,\xi)$ is a non-empty open interval. The
group $G_{C,I}$ is naturally identified with the group
$\mathcal{C}(C,\mathscr{F}_{I})$. Since $\mathscr{F}_{I}\cong\mathscr{F}$ is
finitely generated and perfect (Lemma 4.9), Proposition 5.6 gives the desired
conclusion. ∎

Given points $z\in Z$, $\xi\in\mathbb{R}$, and a decreasing sequence of charts
$\big{(}Z_{C_{n},I_{n}}\big{)}_{n\in\mathbb{N}}$ satisfying
$\bigcap_{n\in\mathbb{N}}Z_{C_{n},I_{n}}=\\{z\\}$ (by metrisability of $X$,
for any point $z\in Z$ we can find such a sequence), we write
$I(z,\xi):=\bigcap_{n\in\mathbb{N}}I(C_{n},I_{n},\xi)$, which is the
intersection of a nested sequence of intervals, and therefore an interval
which contains $\xi$ (possibly reduced to the singleton $\\{\xi\\}$). Notice
that the interval $I(z,\xi)$ does not depend on the decreasing sequence of
charts considered.

######  Lemma 5.13.

With assumptions as in Theorem 5.1, we have $I(z,\xi)=\\{\xi\\}$ for every
$z\in Z$ and $\xi\in\mathbb{R}$.

###### Proof.

Consider the family of intervals $\mathfrak{I}:=\\{I(z,\xi):z\in
Z,\xi\in\mathbb{R}\\}$. We first show that inclusion defines a partial order
relation on $\mathfrak{I}$. For this, given intervals $I,J\subset\mathbb{R}$
(possibly singletons), we say that $I$ and $J$ are _linked_ if their interiors
are neither disjoint, nor nested.

######  Claim 1.

The family $\mathfrak{I}$ has no linked elements.

###### Proof of claim.

We distinguish two cases. Suppose first that $z_{1}=z_{2}=z$. In order to
construct the intervals $I(z,\xi_{1})$ and $I(z,\xi_{2})$, we can use the same
decreasing sequence $Z_{C_{n},I_{n}}$ converging to $z$. Then, there are two
possibilities:

  * •

either $I(C_{n},I_{n},\xi_{1})=I(C_{n},I_{n},\xi_{2})$ for every
$n\in\mathbb{N}$, in which case $I(z,\xi_{1})=I(z,\xi_{2})$, or

  * •

for some $n\in\mathbb{N}$ it holds that $I(C_{n},I_{n},\xi_{1})\cap
I(C_{n},I_{n},\xi_{2})=\varnothing$, in which case we get that $I(p,\xi_{1})$
and $I(p,\xi_{2})$ are disjoint.

In both cases we deduce that the corresponding intervals are unlinked. Assume
now that $z_{1}\neq z_{2}$, and consider nested sequences of charts
$(Z_{C_{n},I_{n}})_{n\in\mathbb{N}}$ and $(Z_{D_{n},J_{n}})_{n\in\mathbb{N}}$
converging to $z_{1}$ and $z_{2}$, respectively. Suppose by contradiction that
$I(z_{1},\xi_{1})$ and $I(z_{2},\xi_{2})$ are linked. Then, for sufficiently
large $k\in\mathbb{N}$, we have that $I(C_{k},I_{k},\xi_{1})$ and
$I(D_{k},J_{k},\xi_{2})$ are linked, while $Z_{C_{k},I_{k}}$ and
$Z_{D_{k},J_{k}}$ are disjoint. The second condition implies that the
subgroups $G_{C_{k},I_{k}}$ and $G_{D_{k},J_{k}}$ commute. This gives a
contradiction since connected components of the support of commuting subgroups
are pairwise unlinked. ∎

We set now

$\mathfrak{I}_{0}:=\left\\{\mathsf{Int}\left(\bigcup_{\alpha}J_{\alpha}\right):(J_{\alpha})\text{
is a maximal chain of }(\mathfrak{I},\subseteq)\right\\}.$

By Claim 1, elements of $\mathfrak{I}_{0}$ are pairwise disjoint open
intervals.

######  Claim 2.

Every element of $\mathfrak{I}_{0}$ is a bounded interval.

###### Proof of claim.

Let $J_{n}=I(z_{n},\xi)$ be an increasing sequence of intervals in
$\mathfrak{I}$. Up to extracting a subsequence we can suppose that $(z_{n})$
converges to some point $z\in Z$. Then for every chart $Z_{C,I}$ containing
$z$, we eventually have $I(z_{n},\xi)\subseteq I(C,I,\xi)$, so it is enough to
prove that the latter is bounded for some chart $Z_{C,I}$. Choose a chart
small enough so that we can find $h\in G$ such that $h(Z_{C,I})\cap
Z_{C,I}=\varnothing$. Assume by contradiction that $I(C,I,\xi)$ is unbounded,
and without loss of generality, we assume that it accumulates at $+\infty$. By
Lemma 5.12, $G_{C,I}$ admits a perfect minimal set $\Lambda\subset
I(C,I,\xi)$, which also accumulate at $+\infty$. Since $I(C,I,\xi)$ is
unbounded, every element in the centralizer of $G_{C,I}$ must preserve it. For
$h\in G$ such that $h(Z_{C,I})\cap Z_{C,I}=\varnothing$, the subgroups
$G_{C,I}$ and $hG_{C,I}h^{-1}$ commute. We deduce that $hG_{C,I}h^{-1}$
preserves $I(C,I,\xi)$, and, since it is perfect, we can apply Corollary 5.4
and deduce that it fixes $\Lambda$ pointwise. This implies that
$hG_{C,I}h^{-1}$, and therefore also the conjugate $G_{C,I}$, have fixed
points accumulating at $+\infty$. This gives the desired contradiction. ∎

Finally, we observe that for every $z\in Z$, $\xi\in\mathbb{R}$ and $g\in G$
we have that $\rho(g)(I(z,\xi))=I(g(z),\rho(g)(\xi))$, from which we deduce
that the family $\mathfrak{I}$ is $\rho$-invariant, and therefore so is
$\mathfrak{I}_{0}$. Putting all of this together, we have that the set given
by the union of all the intervals in $\mathfrak{I}_{0}$ is a proper and open
$\rho$-invariant subset. This implies that $\mathfrak{I}_{0}$ is empty, as
desired. ∎

######  Lemma 5.14.

With assumptions as in Theorem 5.1, take disjoint charts $Z_{C,I}$, $Z_{D,J}$,
and a point $\xi\in\operatorname{\mathsf{Supp}}(\rho|_{G_{C,I}})$. Then,
$G_{D,J}$ acts as the identity on $I(C,I,\xi)$.

###### Proof.

As charts are disjoint, we have that the subgroups $G_{C,I}$ and $G_{D,J}$
commute. Thus, by Corollary 5.4, it is enough to prove that the action of
$G_{C,I}$ on $I(C,I,\xi)$ is minimal. To see this, we know by Lemma 5.12 that
$G_{C,I}$ admits a perfect minimal set $\Lambda\subset I(C,I,\xi)$. Suppose by
contradiction that $\Lambda\neq I(C,I,\xi)$. Note that
$I(C,I,\xi)=I(C,I,\eta)$ for every $\eta\in I(C,I,\xi)$, so it is not
restrictive to assume that $\xi\notin\Lambda$. Assuming so, take the connected
component $U$ of $I(C,I,\xi)\smallsetminus\Lambda$ containing $\xi$. By the
choice of $U$, every element in $G_{C,I}$ either preserves $U$ or maps it
disjointly. Therefore, for every family of subgroups
$\\{H_{\alpha}\\}_{\alpha\in A}$ generating $G_{C,I}$, there must exist some
$\alpha\in A$ such that $U\subset I(H_{\alpha},\xi)$, otherwise we would have
that $U$ is preserved by $G_{C,I}$, contradicting minimality of its action on
$\Lambda$. Consider now a family of charts
$\\{Z_{D_{\alpha},J_{\alpha}}\\}_{\alpha\in A}$ contained in $Z_{C,I}$,
covering $Z_{C,I}$. By the fragmentation lemma for $G$ (Proposition 5.11), we
have that $G_{C,I}$ is generated by the subgroups
$\\{G_{D_{\alpha},J_{\alpha}}\\}_{\alpha\in A}$. Therefore, we can find
$\alpha\in A$ such that $U\subset I(D_{\alpha},J_{\alpha},\xi)$. Starting the
argument again with the chart $Z_{D_{\alpha},J_{\alpha}}$, and proceeding by
induction, we can construct a nested sequence of charts
$(Z_{C_{n},I_{n}})_{n\in\mathbb{N}}$ such that
$\bigcap_{n\in\mathbb{N}}Z_{C_{n},I_{n}}$ is a singleton $\\{z\\}$, and such
that $U\subset I(C_{n},I_{n},\xi)$ for every $n\in\mathbb{N}$. We conclude
that $I(z,\xi)$ contains $U$, but this contradicts Lemma 5.13. ∎

######  Lemma 5.15.

With assumptions as in Theorem 5.1, we have that for every $\xi\in\mathbb{R}$,
there exists a unique $z_{\xi}\in Z$ such that
$\xi\in\operatorname{\mathsf{Supp}}(\rho|_{G_{C,I}})$ for every chart
$Z_{C,I}$ containing $z_{\xi}$.

###### Proof.

Suppose by contradiction that for every $z\in Z$, there exists a chart
$Z_{C_{z},I_{z}}$ containing $z$, and such that
$\xi\in\operatorname{\mathsf{Fix}}(\rho|_{G_{C_{z},I_{z}}})$. Thus, by
compactness of $Z$, we can find finitely many points $z_{1},\ldots,z_{n}\in Z$
whose associated charts cover $Z$. On the other hand, by the fragmentation
lemma (Proposition 5.11), we have that $G$ is generated by the subgroups
$G_{C_{z_{1}},I_{z_{1}}},\ldots,G_{C_{z_{n}},I_{z_{n}}}$. This implies that
$\xi\in\operatorname{\mathsf{Fix}}(\rho)$, which is a contradiction. Thus,
there must be at least one $z\in Z$ such that
$\xi\in\operatorname{\mathsf{Supp}}(\rho|_{G_{C,I}})$ for every chart
$Z_{C,I}$ containing $z$. In order to check uniqueness, suppose by
contradiction that there exist $z_{1}\neq z_{2}$ with this property. Take
disjoint charts $Z_{C_{1},I_{1}}$ and $Z_{C_{2},I_{2}}$ containing $z_{1}$ and
$z_{2}$, respectively. By our assumption on $z_{1}$, Proposition 5.14 gives
that $G_{C_{2},I_{2}}$ acts as the identity on $I(C_{1},I_{1},\xi)$. This
contradicts our assumption on $z_{2}$. ∎

We are ready to prove Proposition 5.9.

###### Proof of Proposition 5.9.

Consider the map $h:\mathbb{R}\to Z$ that associates to each
$\xi\in\mathbb{R}$ the element $z_{\xi}\in Z$ given by Lemma 5.15. It follows
directly from the definition that $h$ is equivariant.

######  Claim.

The map $h$ is continuous.

###### Proof of claim.

Take $\xi\in\mathbb{R}$ and a chart $Z_{C,I}$ containing $z_{\xi}$. Notice
that, in order to show continuity of $h$, it is enough to show that
$z_{\eta}\in Z_{C,I}$ for every $\eta\in I(C,I,\xi)$. Suppose by contradiction
this is not the case. and take a point $\eta\in I(C,I,\xi)$ and a chart
$Z_{D,J}$ containing $z_{\eta}$, and disjoint from $Z_{C,I}$. By definition of
$z_{\eta}$, we have $\eta\in\operatorname{\mathsf{Supp}}(\rho|_{G_{D,J}})$,
but Lemma 5.14 implies that $G_{D,J}$ acts trivially on $I(C,I,\xi)$. This
provides the desired contradiction. ∎

We deduce from the claim that the image of $\mathbb{R}$ is contained in a
single leaf $\ell=\pi_{Z}(\\{x\\}\times\mathbb{R})\subset Z$. It remains to
prove that $h$ is locally injective. Arguing by way of contradiction, fix a
point $\xi_{0}$ such that for any open interval $U\ni\xi_{0}$, the restriction
$h|_{U}$ is not injective. By continuity of $h$, we can choose $U$
sufficiently small, so that $h(U)$ is a proper subset of the leaf $\ell$.
Consider two points $\xi_{1}<\xi_{2}$ in $U$, with $h(\xi_{1})=h(\xi_{2})=:z$.
Notice that there must exist $\xi\in(\xi_{1},\xi_{2})$ such that $h(\xi)\neq
z$, because otherwise equivariance of $h$ would allow us to construct a
proper, non-empty, open invariant subset of $\mathbb{R}$, contradicting
minimality of $\rho$. Since $h([\xi_{1},\xi_{2}])$ does not cover $\ell$, we
can take a chart $Z_{C,I}$ with the following properties:

$z\notin Z_{C,I},\quad\pi_{Z}(\\{x\\}\times I)\subset Z_{C,I},\quad
h(\xi)\in\pi_{Z}(\\{x\\}\times I).$

With such a choice, we have that the orbit $\rho(G_{C,I})(h(\xi))$ is dense in
$\pi_{Z}(\\{x\\}\times I)$, and therefore $\rho(G_{C,I})(h(\xi))\not\subseteq
h((\xi_{1},\xi_{2}))$. On the other hand, since $z\notin Z_{C,I}$, we can
repeat the argument in the proof of the claim, and get that
$\\{\xi_{1},\xi_{2}\\}\in\operatorname{\mathsf{Fix}}(\rho|_{G_{C,I}})$. By
equivariance of $\rho$, this implies that $\rho(G_{C,I})(h(\xi))\subset
h((\xi_{1},\xi_{2}))$, contradicting the choice of the chart $Z_{C,I}$. ∎

##  6\. From the Deroin space to the space of irreducible representations

Throughout this section we let $(X,\varphi,\sigma)$ be a reversible subshift
satisfying Assumption 4.2, and set again $G=\mathscr{T}(\varphi,\sigma)$. In
this section we further study the space
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ and prove the
corollaries stated in the introduction.

###  6.1. Lifting converging sequences

Recall that for every $y\in Y$ we denote by
$\rho_{y}\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$
the associated action given by (5.1). By Corollary 5.2, the map
$y\mapsto\rho_{y}$ identifies $(Y,\Phi)$ with a Deroin space for $G$. We keep
this identification as implicit, and denote by

$r_{Y}\colon\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})\to
Y$

the retraction map as in Proposition 3.5, namely $r_{Y}(\rho)=y$ if $y$ is the
unique point such that $\rho$ is pointed semi-conjugate to $\rho_{y}$.

In order to derive stronger conclusions on
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ than the
general ones that follow from Propositions 3.5 and 3.13, we first need to give
a closer look at properties of the map $r_{Y}$ when we specialize the
discussion to the groups $G=\mathscr{T}(\varphi,\sigma)$. In this case we have
the following, which says that the map $r_{Y}$ behaves like an open map in the
direction transversal to the flow.

######  Proposition 6.1.

Let $\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$,
and assume that $y_{0}:=r_{Y}(\rho)$ can be written in coordinates as
$y_{0}=\pi_{Y}(x_{0},t_{0})$ for $(x_{0},t_{0})\in X\times\mathbb{R}$. Let
$\mathcal{U}$ be an open neighbourhood of $\rho$. Then there exists an open
neighbourhood $V\subset X$ of $x_{0}$ such that $r_{Y}(\mathcal{U})$ contains
$\pi_{Y}(V\times\\{t_{0}\\})$.

###### Proof.

Let $S\subset G$ be a finite generating subset, and take a neighborhood
$\mathcal{U}$ of
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$. By
definition of the compact-open topology, we can find $\varepsilon>0$ and a
compact interval $\tilde{K}$ containing $0$ such that if
$\rho^{\prime}\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$
is such that

(6.1) $\sup_{\gamma\in
S}|\rho(\gamma)(t)-\rho^{\prime}(\gamma)(t)|<\varepsilon\quad\forall
t\in\tilde{K},$

then $\rho^{\prime}\in\mathcal{U}$. One can simply ask that (6.1) is satisfied
for a sufficiently dense, finite collection of points $t\in\tilde{K}$. More
precisely, we are going to use the following property.

######  Claim.

There exists a finite subset $\tilde{D}\subset\mathbb{R}$ containing $0$ such
that the following holds. Let
$\rho^{\prime}\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$
be such that $\rho^{\prime}(\gamma)(t)=\rho(\gamma)(t)$ for every $\gamma\in
S$ and $t\in\tilde{D}$. Then, $\rho^{\prime}\in\mathcal{U}$.

###### Proof of claim.

Fix $\delta>0$ such that for any $\gamma\in S$ and $s,t\in\tilde{K}$
satisfying $|s-t|<\delta$, we have
$|\rho(\gamma)(s)-\rho(\gamma)(t)|<\varepsilon$, and choose a finite subset
$\tilde{D}\subset\mathbb{R}$ which is $\delta/2$-dense in $\tilde{K}$. Given
$t\in\tilde{K}$, take $t_{-},t_{+}\in\tilde{D}$ such that $t_{-}<t<t_{+}$, and
$|t_{+}-t_{-}|<\delta$. As actions preserve the orientation, for any
$\gamma\in S$ the images $\rho(\gamma)(t)$ and $\rho^{\prime}(\gamma)(t)$ are
contained in the interval
$(\rho(\gamma)(t_{-}),\rho(\gamma)(t_{+}))=(\rho^{\prime}(\gamma)(t_{-}),\rho^{\prime}(\gamma)(t_{+}))$,
whose length does not exceed $\varepsilon$. This implies that
$|\rho(\gamma)(t)-\rho^{\prime}(\gamma)(t)|<\varepsilon$ for every $\gamma\in
S$, as wanted. ∎

In what follows, for $x\in X$, we will shorten notation by writing $\rho_{x}$
instead of $\rho_{\pi_{Y}(x,t_{0})}$ for the action given by (5.1). Let
$h:\mathbb{R}\to\mathbb{R}$ be the pointed semi-conjugacy between $\rho$ and
$\rho_{x_{0}}$ (which is a continuous map), and write $K=h(\tilde{K})$.

By construction, there exists a clopen neighborhood $V$ of $x_{0}$ such that
the translation cocycle $\tau_{\gamma}$ is constant on
$\pi_{Y}(V\times\\{t_{0}+t\\})$ for every $t\in K$ and $\gamma\in S$. Hence
for any $x\in V$, the $\rho_{x}$-action of every $\gamma\in S$ on $K$ is the
same as its $\rho_{x_{0}}$-action on $K$.

We set $\tilde{\Omega}:=\rho(G)(\tilde{D})$ and
$\Omega_{x}:=\rho_{x}(G)(h(\tilde{D}))$ for every $x\in V$, which we consider
as ordered subsets of $(\mathbb{R},<)$. Note that both these sets contain $0$,
which we think of as a marked point. We let $<_{x}$ denote the lexicographic
ordering of $\Omega_{x}\times\tilde{\Omega}$: given $s_{1},s_{2}\in\Omega_{x}$
and $t_{1},t_{2}\in\tilde{\Omega}$, one has

$(s_{1},t_{1})<_{x}(s_{2},t_{2})\Leftrightarrow\left\\{\begin{array}[]{l}s_{1}<s_{2},\text{
or}\\\\[5.0pt] s_{1}=s_{2}\text{ and }t_{1}<t_{2}.\end{array}\right.$

Notice that the following equivalence holds for every
$\gamma_{1},\gamma_{2}\in G$ and every $t\in\mathbb{R}$:

(6.2)
$\rho(\gamma_{1})(t)<\rho(\gamma_{2})(t)\Leftrightarrow\Big{(}h(\rho(\gamma_{1})(t)),\rho(\gamma_{1})(t)\Big{)}<_{x_{0}}\Big{(}\rho(\gamma_{2})(t)),\rho(\gamma_{2})(t)\Big{)}$

The set $\Omega_{x}\times\tilde{\Omega}$ is also naturally marked at $(0,0)$.
The order $<_{x}$ is invariant under the action $\psi_{x}$ of $G$ defined by
$\psi_{x}(g):(s,t)\mapsto(\rho_{x}(g)(s),\tilde{\rho}(g)(t))$. We can then
consider the dynamical realisation
$\rho^{\prime}_{x}:G\to\operatorname{\mathsf{Homeo}}_{0}(\mathbb{R})$ of the
action $\psi_{x}$, which comes with an equivariant order-preserving embedding
$j_{x}:(\Omega_{x}\times\tilde{\Omega},<_{x})\to(\mathbb{R},<)$ (see [BMRT,
Lemma 2.40]), which we assume to send the marked point to $0$. Because of the
choice of $V$, for any $x\in V$, $t\in\tilde{D}$, and $\gamma\in S$, we have

(6.3) $h(\rho(\gamma)(t))=\rho_{x_{0}}(\gamma)(h(t))=\rho_{x}(\gamma)(h(t)).$

Therefore putting together (6.2) and (6.3), we get that for any $x\in V$,
$t_{1},t_{2}\in\tilde{D}$, and $\gamma_{1},\gamma_{2}\in S$, the following
inequalities are equivalent:

$\displaystyle\rho(\gamma_{1})(t_{1})<\rho(\gamma_{2})(t_{2})$
$\displaystyle\Leftrightarrow\,$
$\displaystyle(\rho_{x}(\gamma_{1})(h(t_{1})),\rho(\gamma_{1})(t_{1}))<_{x}(\rho_{x}(\gamma_{2})(h(t_{2})),\rho(\gamma_{2})(t_{2}))$
$\displaystyle\Leftrightarrow\,$
$\displaystyle\psi_{x}(\gamma_{1})(h(t_{1}),t_{1})<_{x}\psi_{x}(\gamma_{2})(h(t_{2}),t_{2})$
$\displaystyle\Leftrightarrow\,$
$\displaystyle\rho_{x}^{\prime}(\gamma_{1})(j_{x}(h(t_{1}),t_{1}))<\rho_{x}^{\prime}(\gamma_{2})(j_{x}(h(t_{2}),t_{2})).$

Thus, for given $x\in V$, we can take an action $\rho^{\prime\prime}_{x}$
pointed conjugate to $\rho_{x}^{\prime}$ such that
$\rho(\gamma)(t)=\rho^{\prime\prime}_{x}(\gamma)(t)$ for every $\gamma\in S$
and $t\in\tilde{D}$. We deduce from the claim that
$\rho^{\prime\prime}_{x}\in\mathcal{U}$ for every $x\in V$. We claim that
$r_{Y}(\rho^{\prime\prime}_{x})=\pi_{Y}(x,t_{0})$, i.e. that $\rho_{x}$ and
$\rho^{\prime\prime}_{x}$ are pointed semi-conjugate. To see this, it is
enough to see that $\rho_{x}$ and $\rho_{x}^{\prime}$ are. This comes from the
fact that $\rho_{x}$ is pointed conjugate to the dynamical realisation of the
induced action on $\Omega_{x}$ (this follows from [BMRT, Lemma 2.40]), and the
latter is pointed semi-conjugate to $\rho_{x}^{\prime}$ (by considering the
order-preserving projection $\Omega_{x}\times\tilde{\Omega}\to\Omega_{x}$). ∎

The following result allows to completely describe the closure of semi-
conjugacy classes in
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ (or in other
words which actions can be reached by perturbations) in terms of the closure
of $\Phi$-orbits in $(Y,\Phi)$.

######  Theorem 6.2.

Let
$\mathcal{F}\subset\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$
be a family of representations, and fix
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$. Then
the following are equivalent.

  1. (1)

$\rho$ is accumulated by representations which are semi-conjugate to elements
of $\mathcal{F}$;

  2. (2)

$r_{Y}(\rho)$ belongs to the closure of the union of $\Phi$-orbits of points
in $r_{Y}(\mathcal{F})$.

###### Proof.

The implication (1) $\Rightarrow$ (2) is a general consequence of Proposition
3.5. The converse follows from Proposition 6.1, indeed if (2) holds, and if
$r_{Y}(\rho)=\pi_{Y}(x_{0},t_{0})$, then for every neighbourhood $V$ of
$x_{0}$ the local transversal $\pi_{Y}(V\times\\{t_{0}\\})$ intersects the
orbit of some point in $\mathcal{F}$, and hence every neighbourhood of $\rho$
intersects the semi-conjugacy class of an element of $\mathcal{F}$. ∎

In the rest of the section we describe some applications of our results.

###  6.2. Rigidity and flexibility

######  Corollary 6.3.

The following conditions on a representation
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ are
equivalent

  1. (1)

$\rho$ is locally rigid;

  2. (2)

$\rho$ is rigid;

  3. (3)

the $\Phi$-orbit of $r_{Y}(\rho)$ is an open subset of $Y$.

###### Proof.

The equivalence between (2) and (3) is a general consequence of Proposition
3.13, and it is clear that (2) implies (1) . If (1) holds, then by Theorem
6.2, $r_{Y}(\rho)$ is an interior point of its $\Phi$-orbit, hence its whole
$\Phi$-orbit is open. ∎

######  Corollary 6.4.

A representation
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ has a
dense semi-conjugacy class if and only if $r_{Y}(\rho)$ has a dense orbit. In
particular if $(X,\varphi)$ is minimal, then all semi-conjugacy classes are
dense in $\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$.

###### Proof.

Apply Theorem 6.2 to $\mathcal{F}=\\{\rho\\}$ to see that if $r_{Y}(\rho)$ has
a dense orbit, then the semi-conjugacy class of $\rho$ accumulates on every
$\rho^{\prime}\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$.
∎

Next we introduce the following strong form of flexibility of representations.

######  Definition 6.5.

A representation
$\varphi\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ is
_universally flexible_ if every neighbourhood $U$ of $\varphi$ intersects non-
trivially every semi-conjugacy class in
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$.

We have the following.

######  Corollary 6.6.

A representation
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ is
universally flexible if and only if $(Y,\Phi)$ (equivalently $(X,\varphi)$)
has a unique non-empty closed minimal invariant subset and $r_{Y}(\rho)$
belongs to it.

###### Proof.

If $r_{Y}(\rho)$ belongs to a unique minimal subset of $(Y,\Phi)$, then the
orbit-closure of every point in $Y$ contains it, so that by Theorem 6.2,
$\rho$ is accumulated by representations in any given semi-conjugacy class.
That this condition is necessary follows from Proposition 3.5. ∎

These results can be used to produce examples with various prescribed
properties, by constructing subshifts $(X,\varphi,\sigma)$ with the desired
corresponding properties (for instance among closed invariant subshifts of the
subshift of reduced words $X_{\text{red}}\subset A^{\mathbb{Z}}$ as in Example
4.5). As an example we include the following.

######  Corollary 6.7 (Groups with rigid and universally flexible
representations).

Fix $n\in 2\mathbb{N}\cup\\{\infty\\}$. Then there exists a finitely generated
group $G$ such that
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ contains
uncountably many semi-conjugacy classes and the following hold.

  1. (1)

There are exactly $n$ semi-conjugacy classes of rigid representations.

  2. (2)

Every representation which is not rigid is universally flexible.

###### Proof.

By Corollaries 6.3 and 6.6, it is enough to find for each
$n\in\mathbb{N}\cup\\{\infty\\}$ a reversible subshift $(X,\varphi,\sigma)$ so
that $(X,\varphi)$ has a unique minimal closed invariant subset which is
infinite, and exactly $2n$ orbits of isolated points. This is routine, we
sketch a construction below. Consider the subshift of reduced words
$X_{\text{red}}\subset A^{\mathbb{Z}}$ from Example 4.5.

Choose first an infinite minimal subshift $M\subset X_{\text{red}}$ which is
invariant under $\sigma$. Choose also a sequence $x\in M$. Using that $M$ is
infinite and minimal, we can write $M$ as the intersection of a strictly
decreasing sequence $M_{0}\supsetneq M_{1}\supsetneq M_{2}\supsetneq\cdots$ of
irreducible subshifts of finite type, which can be assumed to be
$\sigma$-invariant.

By properties of subshifts of finite type, we can find for each $n$ a sequence
$x_{n}\in M_{n}$ which has some infinite prefix and suffix which coincide with
some prefix and suffix of $x$, respectively, and such that $x\notin M_{n+1}$
(for this it is enough that between the desired prefix and suffix there is
some finite word which is allowed in $M_{n}$ but not in $M_{n+1}$). Note that
every accumulation point of the $\varphi$-orbit of $x_{n}$ and of
$\sigma(x_{n})$ belongs to $M$. Then for $n\geq 0$ let

$X_{n}=X\cup\\{\varphi^{i}(x_{j}),\varphi^{i}(\sigma(x_{j}))\colon
i\in\mathbb{Z},j\leq n\\}$

and $X_{\infty}=\bigcup X_{n}$. These satisfy the desired conclusion,
respectively for finite $n$ and for $n=\infty$. ∎

######  Remark 6.8.

The restriction that $n$ be even is necessary in Corollary 6.7 since here we
do not allow semi-conjugacies to reverse the orientation, so that rigid semi-
conjugacy classes always come in pairs.

###  6.3. Cantor–Bendixson rank

As another application, let us consider a notion of Cantor–Bendixson rank for
the space of semi-conjugacy classes in
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$. To this end,
given a finitely generated group $G$ define a decreasing transfinite sequence
of subspaces
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}_{\alpha}(G;\mathbb{R})$
invariant under semi-conjugacy, as follows:

  * •

if $\alpha=\beta+1$ is a successor, we define
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}_{\alpha}(G;\mathbb{R})$ by
removing from
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}_{\beta}(G;\mathbb{R})$ all
semi-conjugacy classes that are open in
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}_{\beta}(G;\mathbb{R})$;

  * •

if $\alpha$ is a limit ordinal we define
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}_{\alpha}(G;\mathbb{R})=\bigcap_{\beta<\alpha}\operatorname{\mathsf{Rep}_{\operatorname{irr}}}_{\beta}(G;\alpha)$.

If $(\mathcal{D},\Psi)$ is a Deroin space for $G$, then a similar process
yields a decreasing sequence of closed subsets $\mathcal{D}_{\alpha}$,
obtained by successively removing open orbits, with
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}_{\alpha}(G)=r_{\mathcal{D}}^{-1}(\mathcal{D}_{\alpha})$.
These sequences must stabilize at some countable ordinal, see [Kechris,
Theorem 6.9].

######  Definition 6.9.

Let $G$ be a finitely generated group. The smallest ordinal $\alpha$ such that
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}_{\alpha}(G;\mathbb{R})=\operatorname{\mathsf{Rep}_{\operatorname{irr}}}_{\alpha+1}(G;\mathbb{R})$
(equivalently $\mathcal{D}_{\alpha}=\mathcal{D}_{\alpha+1}$) is called the
_semi-conjugacy CB-rank_ of
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$.

Note that if $\alpha$ is the semi-conjugacy CB-rank, then
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}_{\alpha}(G;\mathbb{R})=\varnothing$
if and only if
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ contains only
countably many semi-conjugacy classes.

Recall that the usual Cantor–Bendixson rank (CB-rank) of a compact space is
defined in a similar way, but by successively removing its isolated points.
Then for the groups $G=\mathscr{T}(\varphi,\sigma)$ we have the following.

######  Corollary 6.10.

The semi-conjugacy CB-rank of
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ is equal to
the usual CB-rank of $X$.

###### Proof.

It is clear that open orbits of the suspension flow $(Y,\Phi)$ are removed
according to the usual Cantor–Bendixson derivative process in $X$. ∎

######  Remark 6.11.

We could not locate a reference stating which ordinals are realizable as the
CB-rank of subshifts (although there are various such results for special
classes of subshifts). However Ville Salo kindly communicated us a proof that
the CB-rank of countable susbhifts are exactly the finite ordinals and the
countable ordinals of the form $\beta+2$. The following slightly simpler
version of his construction allows to realize all countable ordinals of the
form $\beta+3$. On an alphabet of the form $A=B\sqcup\\{\ast\\}$, let $x$ be
the constant sequence of $\ast$. Choose a countable closed subset $C\subset
B^{\mathbb{N}}$, and let $(C_{\alpha})$ be its sequence of CB-derivatives.
Denote by $\alpha_{0}$ the CB-rank of $C$ and notice that
$C_{\alpha_{0}}=\varnothing$ since $C$ is countable. We proceed to show how to
construct a subshift with rank $\alpha_{0}+2$. The set $C$ can be chosen so
that $\alpha_{0}=\beta+1$ for any given countable ordinal $\beta$ (for
instance choosing $C$ to be homeomorphic to the ordinal $\beta+1$ with the
order topology); this will show that any countable ordinal of the form
$\beta+3$ can be the CB-rank of a subshift.

For each $c=(b_{n})\in C$, let $x_{c}$ be the sequence in $A^{\mathbb{Z}}$
obtained by replacing the letter at position $2^{n}$ of $x$ with $b_{n}$ for
$n\geq 0$. Write $X_{1}=\\{\sigma^{n}(x_{c}):c\in C,\text{ }n\in\mathbb{Z}\\}$
and let $X\subset A^{\mathbb{Z}}$ be the subshift given by the closure of
$\\{x\\}\cup X_{1}$. It is not difficult to see, by construction of $X_{1}$,
that $X$ is contained in the union $\\{x\\}\sqcup X_{1}\sqcup X_{2}$ where
$X_{2}$ consists of those sequences with at most one letter in $B$. Notice
also that the set $\\{x_{c}:c\in B\\}\subset X_{1}$ is a clopen subset of $X$
since it can be defined by looking at the cylinder associated to positions
$1$, $2$, and $4$. Thus, it follows that the points $x_{c}$ are removed from
$X$ according to the Cantor–Bendixson derivative process in $C$ and therefore
$X_{\alpha_{0}}\subset\\{x\\}\cup X_{2}$. Moreover, it is direct to see that
$X_{\alpha_{0}}$ meets both $\\{x\\}$ and $X_{2}$. Since the points of $X_{2}$
are isolated points of $X_{\alpha_{0}}$ that accumulate on $\\{x\\}$, it
follows that $X_{{\alpha_{0}}+1}=\\{\ast\\}$. Therefore the CB-rank of $X$ is
${\alpha_{0}}+2$ as desired.

Notice that $X$ can be turned into a subshift satisfying Assumption 4.2
through the doubling construction in Example 4.3.

###  6.4. Connectedness properties

Recall from the introduction that we say that a representation
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ is
_path-rigid_ if its path component in
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ coincides
with its semi-conjugacy class.

######  Corollary 6.12.

The following property hold.

  1. (1)

All representations
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ are
path-rigid.

  2. (2)

The space $\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ is
connected if and only if the only $\varphi$-invariant clopen subsets of $X$
are $\varnothing$ and $X$.

  3. (3)

If $X$ has no isolated point, then
$\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$ is nowhere
locally connected.

###### Proof.

(1) follows from Proposition 3.13. To show (2) note that every partition of
$Y$ into two open subsets must consist of $\Phi$-invariant subsets, and thus
corresponds to a partition of $X$ into $\varphi$-invariant clopen sets. To
show (3), suppose that $X$ has no isolated point, fix
$\rho\in\operatorname{\mathsf{Rep}_{\operatorname{irr}}}(G;\mathbb{R})$, and
let $\mathcal{U}$ be a neighbourhood of $\rho$ whose image is contained in a
chart $Y_{C,I}$, with $C\subset X$ clopen. Then Proposition 6.1 implies that
the composition of $r_{Y}|_{\mathcal{U}}$ with the projection $Y_{C,I}\to C$
is open. Since $C$ is totally disconnected and has no isolated point, it
follows that no open subset of $\mathcal{U}$ is connected. ∎

##  Appendix A Properties of the groups $\mathscr{T}(\varphi,\sigma)$

Here we discuss some generating properties of the group
$\mathscr{T}(\varphi,\sigma)$. These results and proof are largely analogous
to the results on the groups $\mathsf{T}(\varphi)$ from [MB-T]. The main
technical difference does not come from considering type-${\mathfrak{D}}$
maps, but rather from the fact that we have to deal with charts in the
dihedral suspension; we will detail the proofs when differences are relevant.
Throughout the section, $(X,\varphi,\sigma)$ is as in Assumption 4.2, and we
set $G=\mathscr{T}(\varphi,\sigma)$.

###  A.1. More about charts

Say that a chart $Z_{C,I}$ is _extendable_ if it is contained in a chart
$Z_{C,J}$, with $\overline{I}\subset J$.

######  Lemma A.1.

Let $x\in X$. If $x$ is periodic, of minimal period $n$, and $I$ is an open
interval such that $|I|<n$, then there exists a clopen neighbourhood $C$ of
$x$ such that $Z_{C,I}$ is an extendable chart. When $x$ is not periodic, the
same conclusion holds for every bounded interval $I\subset\mathbb{R}$.

###### Proof.

The proof is similar to that of [confined, Lemma 4.10]. We consider the case
where $x$ is periodic of minimal period $n$, the other case can be treated
similarly. Take an interval $I\subset\mathbb{R}$ satisfying $|I|<n$. Consider
the collection

$\Gamma=\\{\gamma\in
D_{\infty}\smallsetminus\\{\mathsf{id}\\}\colon\gamma(I)\cap
I\neq\varnothing\\},$

which consists of finitely many elements. It $\gamma\in\Gamma$ is a
translation, then it must be by some integer $k$, with $|k|\leq|I|<n$. As $n$
is chosen as the minimal period of $x$, we deduce that for any translation
$\gamma\in\Gamma$, we have $\gamma(x)\neq x$. On the other hand, if
$\gamma\in\Gamma$ is a reflection, then Assumption 4.2 guarantees that
$\gamma(x)\neq x$ (this is one of the equivalent conditions in Lemma 4.1). By
continuity, we can find a clopen neighbourhood $C\subset X$ of $x$ such that
$\gamma(C)\cap C=\varnothing$ for any $\gamma\in\Gamma$. With this choice, we
have that $\gamma(C\times I)\cap(C\times I)=\varnothing$ for all $\gamma\in
D_{\infty}\smallsetminus\\{\mathsf{id}\\}$. Therefore $\pi_{Z}(C\times
I)=Z_{C,I}$ is a chart. Notice that if $J\supset\overline{I}$ is another open
interval such that $|J|<n$, the same argument shows that $Z_{C,J}$ is a chart,
which implies that $Z_{C,I}$ is an extendable chart. ∎

If $Z_{C,I}$ is an extendable chart with $I=(a,b)$, we have that
$\partial{Z_{C,I}}$ is the disjoint union of the subsets
$\pi_{Z}(C\times\\{a\\})$ and $\pi_{Z}(C\times\\{b\\})$. We refer to these as
the _sides_ of ${Z_{C,I}}$.

######  Lemma A.2 (Chart decomposition).

The space $Z$ can be written as a finite union
$Z=\bigcup_{i=1}^{k}\overline{Z_{C_{i},I_{i}}}$, where $Z_{C_{i},I_{i}}$ are
extendable charts, such that for every distinct $i,j\in\\{1,\ldots,k\\}$, the
intersection $\overline{Z_{C_{i},I_{i}}}\cap\overline{Z_{C_{j},I_{j}}}$ is
contained in a single side of ${Z_{C_{i},I_{i}}}$ and of ${Z_{C_{j},I_{j}}}$.

###### Proof.

Consider the dyadic intervals $I_{j}=(j/4,(j+1)/4)$ with $j\in\\{0,1,2,3\\}$.
By Lemma A.1, for every $x\in X$, there exists a clopen subset $C\subset X$
such that $Z_{C,I_{j}}$ is an extendable chart for $j\in\\{0,1,2,3\\}$. Thus,
there exists a clopen partition $X=C_{1}\sqcup\cdots\sqcup C_{m}$ such that
$Z_{i,j}:=Z_{C_{i},I_{j}}$ is an extendable chart for any
$i\in\\{1,\ldots,m\\}$ and $j\in\\{0,1,2,3\\}$. Finally, since the action of
$D_{\infty}$ on $\mathbb{R}$ is isometric and the intervals $I_{j}$ have
length $1/4$, which is smaller than $1/2$, which is the length of a
fundamental interval for this action, we conclude that whenever
$(i,j)\neq(i^{\prime},j^{\prime})$, the intersection
$\overline{Z_{i,j}}\cap\overline{Z_{i^{\prime},j^{\prime}}}$ is contained in a
single side of ${Z_{i,j}}$ and of ${Z_{i^{\prime},j^{\prime}}}$. ∎

###  A.2. Fragmentation property

The goal of this subsection is to prove the fragmentation lemma (Proposition
5.11) for $\mathscr{T}(\varphi,\sigma)$. We start with the fragmentation lemma
for the group $\mathscr{F}$.

######  Lemma A.3 (Fragmentation lemma for $\mathscr{F}$).

Let $I,I_{1},\ldots,I_{n}$ be open dyadic intervals so that
$I=\bigcup_{i=1}^{n}I_{i}$. Then
$\mathscr{F}_{I}=\langle\bigcup_{i=1}^{n}\mathscr{F}_{I_{i}}\rangle$.

###### Proof.

Let us detail the proof in the case $n=2$, the general case follows by
induction. Assume, without loss of generality, that $I_{1}=(a,c)$ and
$I_{2}=(b,d)$, with $a<b<c<d$, and take a dyadic rational $x\in(b,c)$. As in
the proof of Lemma 4.9, write

(A.1) $\mathscr{F}_{I}=\langle F^{\prime}_{I},S_{x},S_{a},S_{d}\rangle,$

where $S_{x}\subset\mathscr{F}_{(b,c)}$, $S_{a}\subset\mathscr{F}_{(a,c)}$,
and $S_{d}\subset\mathscr{F}_{(b,d)}$ are finite subsets generating the groups
of germs $\mathcal{D}_{x}$, $\mathcal{D}_{a}^{+}$, and $\mathcal{D}_{b}^{-}$,
respectively. Analogously, we can write $\mathscr{F}_{I_{1}}=\langle
F^{\prime}_{I_{1}},S_{x},S_{a},S_{c}\rangle$ and $\mathscr{F}_{I_{2}}=\langle
F^{\prime}_{I_{2}},S_{x},S_{b},S_{d}\rangle$. Thus, we have that

$\langle\mathscr{F}_{I_{1}},\mathscr{F}_{I_{2}}\rangle=\langle
F^{\prime}_{I_{1}},F^{\prime}_{I_{2}},S_{x},S_{a},S_{b},S_{c},S_{d}\rangle.$

Since $F^{\prime}_{I}=\langle F^{\prime}_{I_{1}},F^{\prime}_{I_{2}}\rangle$,
the previous line and (A.1) imply that
$\mathscr{F}_{I}=\langle\mathscr{F}_{I_{1}},\mathscr{F}_{I_{2}}\rangle$, as
desired. ∎

We next need a result on homeomorphisms of $Z$, whose proof is a refinement of
the argument given for Lemma 4.6.

######  Lemma A.4.

Let $h\in\mathsf{H}_{0}(\varphi,\sigma)$. Then, there exists an isotopy
$(h_{s})_{s\in[0,1]}$ between $h$ and the identity which satisfies
$\overline{\operatorname{\mathsf{Supp}}_{Z}(h_{s})}=\overline{\operatorname{\mathsf{Supp}}_{Z}(h)}$
for every $s\in[0,1)$.

###### Proof.

For the isotopy, we will consider the one defined in the proof of Lemma 4.6.
Explicitly, we take a lift $f\in\mathsf{H}_{0}(\varphi)$ of $h$, consider the
isotopy $(f_{s})_{s\in[0,1]}\subset\mathsf{H}_{0}(\varphi)$ between $f$ and
the identity that satisfies $\tau_{f_{s}}=(1-s)\tau_{f}$, and then project it
to an isotopy $(h_{s})_{s\in[0,1]}\subset\mathsf{H}_{0}(\varphi,\sigma)$
between $h$ and the identity. In order to prove the equality between supports
of the elements in the isotopy we need the following.

######  Claim.

Given $g\in\mathsf{H}_{0}(\varphi)$, write $T_{g}=\\{y\in Y:\tau_{f}(y)\neq
0\\}$. Then $\overline{T_{g}}=\overline{\operatorname{\mathsf{Supp}}_{Y}(g)}$.

###### Proof of claim.

Notice that if the leaf through $y\in T_{g}$ is non-closed, then
$y\in\operatorname{\mathsf{Supp}}_{Y}(g)$. On the other hand, Assumption 4.2
implies that the points of $T_{g}$ with non-closed leaves are dense. Since
$T_{g}$ is open, this implies that
$\overline{T_{g}}\subseteq\overline{\operatorname{\mathsf{Supp}}_{Y}(g)}$. The
reverse inclusion is trivial. ∎

Take now $h\in\mathsf{H}_{0}(\varphi,\sigma)$, $f\in\mathsf{H}_{0}(\varphi)$,
and $(f_{s})_{s\in[0,1]}$ as above. By the choice of the isotopy, we have that
$T_{f_{s}}=T_{f}$ for every $s\in[0,1)$. Thus, the claim implies that
$\overline{\operatorname{\mathsf{Supp}}_{Y}(f_{s})}=\overline{\operatorname{\mathsf{Supp}}_{Y}(f)}$
for every $s\in[0,1)$. This implies
$\overline{\operatorname{\mathsf{Supp}}_{Z}(h_{s})}=\overline{\operatorname{\mathsf{Supp}}_{Z}(h)}$
for every $s\in[0,1)$, as desired. ∎

We can now prove the fragmentation lemma for $\mathscr{T}(\varphi,\sigma)$.

###### Proof of Proposition 5.11.

The proof is somehow analogue to the discussion in [MB-T, Appendix A]. We will
use that $\mathsf{H}_{0}(\varphi)$ is a topological group, whose topology can
be defined by looking at displacement of elements along leaves (more
precisely, by looking at the uniform norm on the translation cocycles).
Moreover, using Lemma 4.6, we can identify $\mathsf{H}_{0}(\varphi,\sigma)$ to
a closed subgroup of $\mathsf{H}_{0}(\varphi)$.

We first construct a family of charts subordinated to $\mathcal{U}$ that is
well suited for our purposes. For this, choose a decomposition
$Z=\bigcup_{i=1}^{k}\overline{Z_{C_{i},I_{i}}}$ as in Lemma A.2. After
subdividing these into smaller charts if needed, we can suppose that each
$\\{Z_{C_{i},I_{i}}\\}_{i\in\mathcal{I}}$ is contained in some element of
$\mathcal{U}$. Slightly enlarge each chart $Z_{C_{i},I_{i}}$ to a dyadic chart
$Z_{C_{i},J_{i}}$, where $J_{i}$ is an $\varepsilon$-neighbourhood of $I_{i}$
satisfying:

  1. (1)

$Z_{C_{i},J_{i}}$ is still contained in an element of $\mathcal{U}$ for every
$i\in\mathcal{I}$;

  2. (2)

whenever $i,j\in\mathcal{I}$, with $i\neq j$, are such that
$\overline{Z_{C_{i},J_{i}}}\cap\overline{Z_{C_{j},J_{j}}}\neq\varnothing$,
then $Z_{C_{i},J_{i}}\cap Z_{C_{j},J_{j}}$ is a chart of the form
$Z_{D_{ij},L_{ij}}$, where $L_{ij}$ is an interval of the form
$L_{i}=(t_{ij}-\varepsilon,t_{ij}+\varepsilon)$, so that
$\overline{Z_{C_{i},I_{i}}}\cap\overline{Z_{C_{j},I_{j}}}=\pi_{Z}(D_{ij}\times\\{t_{ij}\\})$.

Set

$\mathcal{I}=\left\\{i\in\\{1,\ldots,k\\}:\overline{Z_{C_{i},I_{i}}}\cap\overline{\operatorname{\mathsf{Supp}}_{Z}(g)}\neq\varnothing\right\\},$

and notice that $\\{Z_{C_{i},J_{i}}\\}_{i\in\mathcal{I}}$ is an open cover of
$\overline{\operatorname{\mathsf{Supp}}_{Z}(G)}$ subordinate to $\mathcal{U}$,
and the charts $\\{Z_{D_{ij},L_{ij}}\\}_{i,j\in\mathcal{I},i\neq j}$ (which
are also subordinate to $\mathcal{U}$) are pairwise disjoint and cover all
sides of the charts $\\{{Z_{C_{i},I_{i}}\\}}$. To get the statement, it is
therefore enough to show that the subgroup

$G_{g}:=\left\\{h\in
G:\overline{\operatorname{\mathsf{Supp}}_{Z}(h)}\subseteq\overline{\operatorname{\mathsf{Supp}}_{Z}(g)}\right\\}$

is contained in the subgroup

$K:=\left\langle\bigcup_{i\in\mathcal{I}}G_{C_{i},J_{i}}\right\rangle.$

For this, we will prove that the elements in $G_{g}$ with small displacement
belong to $K$, and then that every element in $G_{g}$ can be written as a
product of elements in $G_{g}$ with small displacement.

For the first step, take $h\in G_{g}$ which displaces any point $z\in Z$ of at
most $\varepsilon/2$ along its leaf. We have
$h(\pi_{Z}(D_{ij}\times\\{t_{ij}\\}))\subset Z_{D_{ij},L_{ij}}$ for any
distinct $i,j\in\mathcal{I}$. Since Thompson’s group action on dyadic
rationals is transitive, we can find an element $k$ which is a product of
elements from the commuting subgroups
$\\{G_{D_{ij},L_{ij}}\\}_{i,j\in\mathcal{I},i\neq j}$ (in particular, $k\in
K$), such that $kh$ fixes

$\bigcup_{i,j\in\mathcal{I},i\neq
j}\pi_{Z}(D_{ij}\times\\{t_{ij}\\})=\bigcup_{i\in\mathcal{I}}\partial\overline{Z_{C_{i},I_{i}}}.$

Hence $kh$ preserves each chart $Z_{C_{i},I_{i}}$, and therefore it can be
decomposed as a product of elements from the commuting subgroups
$\\{G_{C_{i},I_{i}}\\}_{i\in\mathcal{I}}$. In particular, we have that $kh\in
K$, and therefore $h\in K$.