1 Introduction
For a group
$\Gamma $
, a (left invariant) circular order on
$\Gamma $
is a cocycle
$c \ \colon \Gamma ^3 \to \{\pm 1, 0\}$
with certain conditions (see Section 2). Let
$CO(\Gamma )$
be the space of circular orders on
$\Gamma $
. This space
$CO(\Gamma )$
is a compact, metrizable, and totally disconnected space if
$\Gamma $
is countable. The first natural way to distinguish this space for different groups is to study its isolated points, called isolated circular orders, and this has been done, for instance, in the works [Reference Baik and SampertonBS18, Reference MatsumotoMat20, Reference Malicet, Mann, Rivas and TriestinoMMRT19, Reference Mann and RivasMR18, Reference Mann and TriestinoMT23].
Though circular orders are combinatorial objects, they can be studied in dynamical ways. In fact, if
$\Gamma $
is countable, there is a procedure to construct a faithful action of
$\Gamma $
on the circle from a circular order, which is called the dynamical realization. More strongly, it is known that a countable group
$\Gamma $
admits a circular order if and only if
$\Gamma $
acts on the circle faithfully.
The dynamical realization gives rise to a map
$$\begin{align*}R \colon CO(\Gamma) \to \operatorname{\mathrm{Hom}}(\Gamma, \operatorname{\mathrm{Homeo}}_+(S^1))/ \sim, \end{align*}$$
where
$\sim $
denotes the equivalence relation by conjugacy. Mann and Rivas proved in [Reference Mann and RivasMR18] that this map R is continuous. Employing this and ping-pong arguments, they proved that the free groups of even rank admit infinitely many isolated circular orders which are not conjugate to the others. Contrastingly, Malicet et al. proved in [Reference Malicet, Mann, Rivas and TriestinoMMRT19] that the free groups of odd rank do not admit isolated circular orders. In [Reference MatsumotoMat20], Matsumoto also used the continuity of R and ping-pong arguments to prove that
$\operatorname {\mathrm {PSL}}(2, \mathbb {Z}) \cong \mathbb {Z}_2 \ast \mathbb {Z}_3$
admits infinitely many isolated circular orders which are not the automorphic images of the others. By generalizing Mann–Rivas’ and Matsumoto’s arguments, we prove the following.
Theorem 1.1 Let
$n \kern1.4pt{\in}\kern1.4pt \mathbb {Z}_{\geq 0}, k \kern1.4pt{\in}\kern1.4pt \mathbb {Z}_{\geq 1}$
and
$m_1, \ldots , m_k \kern1.4pt{\in}\kern1.4pt \mathbb {Z}_{\geq 2}$
such that
$(\kern-1pt n,k,m_1, \ldots , m_k \kern-1pt) \neq (0,1,m_1), (0,2,2,2)$
. Then, the free product
$G = F_{2n} \ast \mathbb {Z}_{m_1} \ast \cdots \ast \mathbb {Z}_{m_k}$
admits countably many isolated circular orders which are not the automorphic images of the others.
The groups excluded in Theorem 1.1 are finite cyclic groups and the free product
$\mathbb {Z}_2 \ast \mathbb {Z}_2$
. The finite cyclic groups admit only finitely many circular orders. The group
$\mathbb {Z}_2 \ast \mathbb {Z}_2$
admits just four circular orders [Reference Clay, Mann and RivasCMR18]. In particular, these circular orders are all isolated.
Alonso et al. [Reference Alonso, Alvarez, Malicet, Cotón and TriestinoAAM+24] established the quite involved notion of ping-pong partition, which might be used to prove Theorem 1.1. However, we provide the proof by generalizing Mann–Rivas’ and Matsumoto’s argument to make this article as direct and elementary as possible.
A left order on a group is a total order which is invariant under the left-multiplication. As is the case of
$CO(\Gamma )$
, the space
$LO(\Gamma )$
of left orders on
$\Gamma $
is compact, metrizable, and totally disconnected if
$\Gamma $
is countable. Isolated left orders on several groups have been studied. For example, the following groups admit no isolated left orders: free abelian groups of rank greater than one [Reference SikoraSik04], free groups of rank greater than one [Reference McClearyMcC85], free products of two left-orderable groups [Reference RivasRiv12], virtually solvable groups [Reference Rivas and TesseraRT16], and surface groups of genus greater than one and certain amalgamated products of free groups [Reference Alonso, Brum and RivasABR17]. Isolated left orders on the following group have been constructed: the braid groups
$B_n$
[Reference Dubrovina and DubrovinDD01, Reference Dehornoy, Dynnikov, Rolfsen and WiestDDRW08], the fundamental groups of the complement of torus knots [Reference ItoIto13, Reference NavasNav11], a certain family of groups including torus knot groups and some of their amalgamated products [Reference DehornoyDeh14], amalgamated free products over
$\mathbb {Z}$
of groups with isolated left orders [Reference ItoIto16], and direct products
$F_{2n} \times \mathbb {Z}$
[Reference Mann and RivasMR18].
Let
$\widehat {\Gamma }$
be a central
$\mathbb {Z}$
-extension of
$\Gamma $
. Mann and Rivas [Reference Mann and RivasMR18] provided a criterion for a certain left order on
$\widehat {\Gamma }$
to be isolated in terms of the circular orders on
$\Gamma $
. They used this to prove that the direct product
$F_{2n} \times \mathbb {Z}$
admits countably many isolated left orders which are not conjugate to the others. Similarly, Matsumoto [Reference MatsumotoMat20] proved that the braid group
$B_3$
of
$3$
-strands, which is a central extension of
$\operatorname {\mathrm {PSL}}(2,\mathbb {Z})$
, admits countably many isolated left orders which are not the automorphic images of the others.
Applying the criterion by Mann–Rivas to Theorem 1.1, we prove the following.
Theorem 1.2 Let
$n,k,m_1, \ldots , m_k$
be as in Theorem 1.1. Let
$\widehat {G}$
be the group given as
$$\begin{align*}\widehat{G} = \langle e_1, \ldots, e_k, h_1, \ldots, h_{2n}, z \mid e_1^{m_1} = \cdots = e_k^{m_k} = z, [h_1, z] = \cdots = [h_{2n}, z] = 1 \rangle. \end{align*}$$
Then,
$\widehat {G}$
admits countably many isolated left orders which are not the automorphic images of the others.
2 Preliminaries
2.1 Circular orders
In this section, we review some facts about circular orders. We refer the reader to [Reference CalegariCal04] and [Reference Baik and SampertonBS18] for further details.
Definition 2.1 Let
$\Gamma $
be a group. A map
$c \ \colon \Gamma ^3 \to \{0, \pm 1\}$
is called a circular order on
$\Gamma $
if
-
(1)
$c(g_1 , g_2 , g_3) = 0$
if
$g_i = g_j$
for some
$i\neq j$
, -
(2)
$c(g_2, g_3, g_4)-c(g_1, g_3, g_4)+c(g_1, g_2, g_4)-c(g_1, g_2, g_3) = 0$
for every
$g_1, g_2, g_3, g_4 \in \Gamma $
, and -
(3)
$c(gg_1, gg_2, gg_3) = c(g_1, g_2, g_3)$
for every
$g, g_1, g_2, g_3 \in \Gamma $
.
For an automorphism
$\phi \ \colon \Gamma \to \Gamma $
, we define
$c_{\phi } \ \colon \Gamma ^3 \to \{ 0, \pm 1 \}$
by
$$\begin{align*}c_{\phi}(g_1, g_2, g_3) = c(\phi(g_1), \phi(g_2), \phi(g_3)) \end{align*}$$
for
$g_1, g_2, g_3 \in \Gamma $
. Then,
$c_{\phi }$
is a circular order on
$\Gamma $
, called an automorphic image of c.
Let
$CO(\Gamma )$
be the set of circular orders on
$\Gamma $
. The product topology of
$\{0, \pm 1\}^{\Gamma} $
induces the topology of
$CO(\Gamma )$
. Then, the space
$CO(\Gamma )$
is known to be compact and totally disconnected, and it is also metrizable if
$\Gamma $
is countable [Reference SikoraSik04].
Definition 2.2 A circular order c is said to be isolated if c is an isolated point in
$CO(\Gamma )$
.
For a circular order c and a finite set S of
$\Gamma $
, we set
$U_{c,S} = \{ c' \in CO(\Gamma ) \ \colon c'|_{S^3} = c|_{S^3} \}$
. Then, the neighborhood basis of c is given by
$\{ U_{c,S} \ \colon S \text { is a finite subset} \text{of } \Gamma \}$
. Hence, a circular order c is isolated if and only if there exists a finite subset S of
$\Gamma $
such that
$U_{c,S} = \{ c \}$
.
Every circular order on a countable group
$\Gamma $
is realized as an orbit of some action of
$\Gamma $
on the circle. Indeed, let c be a circular order on
$\Gamma $
and
$x_0$
a point in
$S^1$
. Here, we regard
$S^1$
as the quotient
$\mathbb {R}/\mathbb {Z}$
. We fix an enumeration
$\{ g_i \}_{i \geq 0}$
with
$g_0 = 1$
and define an embedding
$\iota \ \colon \Gamma \to S^1$
as follows. First, we set
$\iota (g_0) = x_0$
and
$\iota (g_1) = x_0 + 1/2$
. If
$\iota (g_i)$
is defined for all
$i \leq n$
, we define
$\iota (g_{n+1})$
as the midpoint of the unique connected component of
$S^1 \setminus \{ \iota (g_0), \dots , \iota (g_n) \}$
such that
$$ \begin{align*} c(g_i, g_j, g_k) = \mathrm{ord}(\iota(g_i), \iota(g_j), \iota(g_k)). \end{align*} $$
Here, the map
$\mathrm {ord} \ \colon (S^1)^3 \to \{ 0, \pm 1 \}$
is defined by
$$ \begin{align*} \mathrm{ord}(x,y,z) = \begin{cases} 1 & \text{ if } x,y,z \text{ are distinct and arranged counterclockwise}, \\ -1 & \text{ if } x,y,z \text{ are distinct and arranged clockwise}, \\ 0 & \text{ otherwise.} \end{cases} \end{align*} $$
Then, the self left-action of
$\Gamma $
induces a continuous order-preserving action on
$\iota (\Gamma ) \subset S^1$
. This action extends to a continuous action on the closure of
$\iota (\Gamma )$
and also extends to a continuous action on
$S^1$
by setting the action on the complements of the closure of
$\iota (\Gamma )$
to be affine. Now, we obtain an action
$\rho _{c} \ \colon \Gamma \to \mathrm {Homeo}_+(S^1)$
, which is called the dynamical realization of c based at
$x_0$
. By construction, the orbit
$\rho _c(\Gamma )\cdot x_0$
recovers the circular order c.
It is known that the conjugacy class of the dynamical realization
$\rho _{c}$
does not depend on the choices of the enumeration
$\{ g_i \}_{i\geq 0}$
or the basepoint
$x_0$
. Hence, the dynamical realization gives rise to a map
$$ \begin{align*} R \ \colon CO(\Gamma) \to \mathrm{Hom}(\Gamma, \mathrm{Homeo}_+(S^1))/\sim, \end{align*} $$
where
$\mathrm {Hom}(\Gamma , \mathrm {Homeo}_+(S^1))/\sim $
denotes the quotient of
$\mathrm {Hom}(\Gamma , \mathrm {Homeo}_+(S^1))$
by the equivalence relation of conjugacy. This map R is called the realization map. Note that the set
$\mathrm {Hom}(\Gamma , \mathrm {Homeo}_+(S^1))/\sim $
is equipped with the quotient topology of the pointwise convergence topology on
$\mathrm {Hom}(\Gamma , \mathrm {Homeo}_+(S^1))$
.
We will use the following properties of the realization map R.
Theorem 2.1 [Reference Mann and RivasMR18, Proposition 3.3]
The realization map R is continuous.
Theorem 2.2 [Reference Mann and RivasMR18, Lemma 3.12]
Let
$c \in CO(\Gamma )$
and let
$\rho \ \colon \Gamma \to \operatorname {\mathrm {Homeo}}_+(S^1)$
be a dynamical realization of c based at
$x_0$
. Let U be any neighborhood of
$\rho $
in
$\operatorname {\mathrm {Hom}}(\Gamma , \operatorname {\mathrm {Homeo}}_+(S^1))$
. Then, there exists a neighborhood V of c in
$CO(\Gamma )$
such that each order in V has a dynamical realization based at
$x_0$
contained in U.
Theorem 2.3 [Reference Mann and RivasMR18, Corollary 3.24]
Suppose that
$\rho \ \colon \Gamma \to \operatorname {\mathrm {Homeo}}_+(S^1)$
is an action with an exceptional minimal set K, that
$\rho (\Gamma )$
acts transitively on the set of connected components of
$S^1 \setminus K$
, and that, for some component I, the stabilizer of I is nontrivial and acts on I as the dynamical realization of a left order with basepoint
$x_0 \in I$
. Then,
$\rho $
is the dynamical realization of a circular order with basepoint
$x_0$
.
2.2 Left orders
In this section, we recall the definition and properties of left orders. We refer the reader to [Reference Clay and RolfsenCR16, Reference Deroin, Navas and RivasDNR25] for further details.
Definition 2.3 Let
$\Gamma $
be a group. A total order
$<$
on
$\Gamma $
is called a left order on
$\Gamma $
if for every
$g \in \Gamma $
and
$g_1, g_2 \in \Gamma $
satisfying
$g_1 < g_2$
, the inequality
$gg_1 < gg_2$
holds.
For a left order
$<$
on
$\Gamma $
and an automorphism
$\phi $
of
$\Gamma $
, we define another left order
$<_{\phi }$
on
$\Gamma $
as follows:
$g_1 <_{\phi } g_2$
if and only if
$\phi (g_1) < \phi (g_2)$
.
Let
$LO(\Gamma )$
be the set of left orders on
$\Gamma $
. For a left order
$<$
on
$\Gamma $
, define
$\lambda \ \colon \Gamma \setminus \{ 1 \} \to \{ \pm 1 \}$
as follows:
$\lambda (g) = 1$
if and only if
$1 < g$
. Hence,
$LO(\Gamma )$
can be seen as a subspace of
$\{ \pm 1 \}^{\Gamma \setminus \{ 1 \}}$
.
A circular order c on
$\Gamma $
is called a coboundary if there exists a left invariant function
$c' \ \colon \Gamma ^2 \to \{ \pm 1, 0 \}$
such that
$c(f,g,h) = c'(g,h) - c'(f,h) + c'(f,g)$
for every
$f,g,h \in \Gamma $
. A coboundary c induces a left order on
$\Gamma $
by
$g\leq h$
if and only if
$c'(g,h)\geq 0$
.
Definition 2.4 [Reference Mann and RivasMR18, Definition 3.14]
Let
$\Gamma $
be a group and c a circular order on
$\Gamma $
. A subgroup H of
$\Gamma $
is said to be convex if the restriction of c to H is a left order (i.e., a coboundary), and if whenever one has
$c(h_1, g, h_2) = +1$
and
$c(h_1, 1, h_2) = +1$
with
$h_1, h_2$
in H and
$g \in \Gamma $
, then
$g \in H$
.
As shown in [Reference Mann and RivasMR18, Lemma 3.15], there is a unique maximal convex subgroup for any
$c\in CO(\Gamma )$
, which we call the linear part of c.
The linear part is characterized in terms of dynamics as follows.
Theorem 2.4 [Reference Mann and RivasMR18, Part of Proposition 3.17]
Let
$\rho $
be the dynamical realization of a circular order c based at
$x_0$
. If
$\rho $
admits an exceptional minimal set K, then the linear part is the stabilizer of the connected component of
$S^1\setminus K$
that contains
$x_0$
.
2.3 Known results on isolated circular orders on free products of cyclic groups
Let n be an integer grater than
$1$
. Since
$\mathbb {Z}_n$
is finite,
$CO(\mathbb {Z}_n)$
is a finite set. Clay et al. [Reference Clay, Mann and RivasCMR18] proved that
$\mathbb {Z}_2* \mathbb {Z}_2$
admits just four circular orders. Hence, circular orders on the groups above are all isolated.
Mann and Rivas [Reference Mann and RivasMR18] proved that for every
$n \geq 1$
, the non-abelian free group
$F_{2n}$
of rank
$2n$
admits countably many isolated circular orders which are not conjugate to each other. After that, Malicet et al. [Reference Malicet, Mann, Rivas and TriestinoMMRT19] proved that the non-abelian free group
$F_{2n-1}$
of rank
$2n-1$
admits no isolated circular orders for every
$n \geq 1$
. Matsumoto [Reference MatsumotoMat20] proved that
$\operatorname {\mathrm {PSL}}(2,\mathbb {Z}) \cong \mathbb {Z}_2 \ast \mathbb {Z}_3$
admits countably many isolated circular orders which are not the automorphic images of the others. Applying the argument of Matsumoto [Reference MatsumotoMat20, Proof of Theorem 9.2] to the isolated circular orders on
$F_{2n}$
obtained by Mann and Rivas [Reference Mann and RivasMR18], we can show that these isolated circular orders are not the automorphic images of the others as well.
By lifting these isolated orders to certain central extensions, they obtain isolated left orders. In fact, Mann and Rivas [Reference Mann and RivasMR18] proved that the direct product
$F_{2n} \times \mathbb {Z}$
admits countably many isolated left orders, and Matsumoto [Reference MatsumotoMat20] proved that the braid group
$B_3$
of
$3$
-strand admits countably many isolated left orders, which are not the automorphic images of the others.
2.4 Dynamical backgrounds
In this section, we recall dynamical backgrounds, such as hyperbolic geometry and groups acting on the circle (see [Reference GhysGhy01, Reference KatokKat92]).
Let
$\mathbb {H}^2$
be the upper half-plane, which we view via the Poincaré model or the unit disk model. Note that its ideal boundary
$\partial \mathbb {H}^2$
is naturally identified with the circle. Let
$\textrm {Isom}_+(\mathbb {H}^2)$
be the group of orientation-preserving isometries of
$\mathbb {H}^2$
, which is isomorphic to
$\operatorname {\mathrm {PSL}}(2,\mathbb {R})$
.
Definition 2.5 Let
$\Gamma $
be a group. An injective homomorphism
$\sigma :\Gamma \hookrightarrow \textrm {Isom}_+(\mathbb {H}^2)$
is called a Fuchsian representation if the induced topology on
$\sigma (\Gamma )$
is discrete. The image of a Fuchsian representation is called a Fuchsian group.
Fuchsian groups have several nice properties; for example, they admit fundamental domains. We focus on a specific set on the ideal boundary that characterizes the dynamics of Fuchsian groups.
It is known that for a Fuchsian representation
$\sigma :\Gamma \hookrightarrow \textrm {Isom}_+(\mathbb {H}^2)$
, the set of all accumulation points of a
$\sigma $
-orbit is independent of the choice of the base point. Furthermore, it is known that
$L_\sigma \subset \partial \mathbb {H}^2$
(see [Reference KatokKat92, Corollary 2.2.7]). This set of accumulation points is called the limit set of
$\sigma $
, and denoted by
$L_{\sigma }$
. A Fuchsian group is said to be elementary if the limit set contains at most two points.
Theorem 2.5 [Reference KatokKat92, Theorem 3.4.5]
For a Fuchsian representation
$\sigma :\Gamma \hookrightarrow \textrm {Isom}_+(\mathbb {H}^2)$
, the limit set
$L_\sigma $
is
$\sigma (\Gamma )$
-invariant.
The following classification of group actions on the circle is well-known.
Proposition Let
$\sigma :\Gamma \to \textrm {Homeo}_+(S^1)$
be a homomorphism. Then, there are three mutually exclusive possibilities:
-
(1) There is a finite orbit.
-
(2) All orbits are dense.
-
(3) There is a compact
$\sigma (\Gamma )$
-invariant subset
$\mathcal {M}\subset S^1$
which is infinite and different from
$S^1$
and such that the orbits of points in
$\mathcal {M}$
are dense in
$\mathcal {M}$
. This set
$\mathcal {M}$
is unique, contained in the closure of any orbit and is homeomorphic to a Cantor set.
The set
$\mathcal {M}$
in Proposition 2.6 iii) is called the exceptional minimal set of
$\sigma $
.
Theorem 2.7 [Reference KatokKat92, Theorem 3.4.6]
For a Fuchsian representation
$\sigma :\Gamma \hookrightarrow \textrm {Isom}_+(\mathbb {H}^2)$
, if its limit set
$L_\sigma $
contains more than two points, then either
-
(1)
$L_\sigma =\partial \mathbb {H}^2$
, or -
(2)
$L_\sigma $
is homeomorphic to a Cantor set.
Theorem 2.8 [Reference KatokKat92, Theorem 2.4.3]
Any elementary Fuchsian group is either cyclic or is conjugate in
$\operatorname {\mathrm {Isom}}_+(\mathbb {H}^2)$
to the subgroup generated by
${\gamma }_1, {\gamma }_2 \in \operatorname {\mathrm {Isom}}_+(\mathbb {H}^2)$
defined by
$\gamma _1(z)=kz$
and
$\gamma _2(z)=-1/z$
for some
$k\in \mathbb {R}_{>1}$
.
The latter group in Theorem 2.8 is isomorphic to
$\mathbb {Z}_2*\mathbb {Z}_2$
. Therefore, from the above theorems, for a Fuchsian representation
$\sigma :\Gamma \hookrightarrow \textrm {Isom}_+(\mathbb {H}^2)$
, if
$\Gamma $
is neither cyclic nor isomorphic to
$\mathbb {Z}_2*\mathbb {Z}_2$
, and its limit set
$L_\sigma $
is a proper subset of
$\partial \mathbb {H}^2$
, then
$L_\sigma =\mathcal {M}$
.
3 Isolated circular orders
3.1 Construction of circular orders
We fix
$n \in \mathbb {Z}_{\geq 0}, k \in \mathbb {Z}_{\geq 1}$
, and
$m_1, \ldots , m_k \in \mathbb {Z}_{\geq 2}$
and set
$$\begin{align*}G = \langle e_1, \dots, e_k, h_1, \dots h_{2n} \mid e_1^{m_1} = \cdots = e_k^{m_k} = 1 \rangle. \end{align*}$$
That is, the group G is the free product
$F_{2n} * \mathbb {Z}_{m_1} * \cdots * \mathbb {Z}_{m_k}$
of cyclic groups. Assume that G is neither cyclic nor isomorphic to
$\mathbb {Z}_2 * \mathbb {Z}_2$
.
Embedding of G into
$\mathrm{Isom}_+(\mathbb {H}^2)$
.
First, we embed the group G into
$\operatorname {\mathrm {Isom}}_+(\mathbb {H}^2) \cong \operatorname {\mathrm {PSL}}(2,\mathbb {R})$
. Roughly, the embedding is given as Figure 1. To make this precisely, we will use the following notation and lemmas.
Notation For distinct points
$a,b \in S^1$
, let
$(a,b) \subset S^1$
be the open interval from a to b according to the counterclockwise orientation. For a open interval
$J = (a,b)$
in
$S^1$
, we set
$\partial _-(J) = a$
and
$\partial _+(J) = b$
. The closed interval
$[a,b]$
is defined in the same way.
Lemma 3.2 Let a and b be distinct points in
$S^1 = \partial \mathbb {H}^2$
. For an integer n greater than one, there exists an elliptic element
$\mathbf {e} \in \operatorname {\mathrm {Isom}}_+(\mathbb {H}^2)$
of order n such that
$\mathbf {e}^{n-1}(a) = b$
and
$\{ \mathbf {e}(a), \ldots , \mathbf {e}^{n-2}(a) \} \subset (a,b)$
.
Proof Let
$R_n$
be the rotation matrix of angle
$\pi /n$
, which defines an element of
$\operatorname {\mathrm {Isom}}_+(\mathbb {H}^2)$
of order n. Take a parabolic element
$P \in \operatorname {\mathrm {Isom}}_+(\mathbb {H}^2)$
whose fixed point is a and which sends
$R_n^{-1}(a)$
to b. Then,
$\mathbf {e} = P\cdot R_n \cdot P^{-1}$
is a desired element.
Lemma 3.3 For disjoint closed intervals
$H^+$
and
$H^-$
in
$S^1$
, there exists a hyperbolic element
$\mathbf {h} \in \operatorname {\mathrm {Isom}}_+(\mathbb {H}^2)$
such that
$S^1\setminus H^+ \subset \mathbf {h}(H^-)$
.
Proof Let x (resp. y) be a point in
$\mathrm {Int} (H^-)$
(resp.
$\mathrm {Int} (H^+)$
). Let g be a hyperbolic element of
$\operatorname {\mathrm {Isom}}_+(\mathbb {H}^2)$
whose repelling (resp. attracting) fixed point is x (resp. y). Then, there exists a positive integer n such that
$S^1\setminus H^+ \subset g^n(H^-)$
. Hence,
$\mathbf {h} = g^n$
is a desired element.
To realize Figure 1, we fix
$k + 4n$
closed intervals in
$S^1$
that are mutually disjoint. We label the intervals as
$$\begin{align*}X_1, \ldots, X_k, Y_1^+, Y_2^-, Y_1^-, Y_2^+, \ldots, Y_{2n-1}^+, Y_{2n}^-, Y_{2n-1}^-, Y_{2n}^+ \end{align*}$$
in counterclockwise order. Let
$\mathbf {e}_i$
be an element obtained by Lemma 3.2 applied to
$X_i$
and
$m_i$
, and let
$\mathbf {h}_j$
be an element obtained by Lemma 3.3 applied to
$H^{\pm } = Y_j^{\pm }$
. Resetting
$Y_j^+$
as
$S^1\setminus \mathrm {Int}(\mathbf {h}_j(Y_j^-))$
, we obtain the elements
$\mathbf {e}_1, \ldots , \mathbf {e}_k, \mathbf {h}_1, \ldots , \mathbf {h}_{2n}$
and the intervals
$X_1, \ldots , X_k, Y_1^{\pm }, \ldots , Y_{2n}^{\pm }$
that fit into Figure 1.
Setting
$\rho (e_i) = \mathbf {e}_i$
and
$\rho (h_j) = \mathbf {h}_j$
, we obtain a homomorphism
$\rho \ \colon G \to \operatorname {\mathrm {Isom}}_+(\mathbb {H}^2)$
, and the standard ping-pong argument shows the following.
Lemma 3.4 The homomorphism
$\rho \ \colon G \to \operatorname {\mathrm {Isom}}_+(\mathbb {H}^2)$
is injective.
For the reader’s convenience, we provide the proof.
Proof Let
$Y_j$
be the union of
$Y_j^+$
and
$Y_j^-$
, and let
$Z \subset S^1$
be the union of
$X_i$
’s and
$Y_j$
’s. First, note that
$\mathbf {e}_i^{l} (Z \setminus X_i) \subset X_i$
and
$\mathbf {h}_j^{l} (Z \setminus Y_j) \subset Y_j$
unless
$\mathbf {e}_i^{l} = \operatorname {\mathrm {id}}$
and
$\mathbf {h}_j^{l} = \operatorname {\mathrm {id}}$
, respectively. Let
$w = w_1 \dots w_m$
be a nontrivial reduced word of
$\{ e_1, \ldots , e_k, h_1, \ldots , h_{2n} \}$
. We will prove that
$\rho (w)$
is not the unit element of
$\operatorname {\mathrm {Isom}}_+(\mathbb {H}^2)$
. If
$w_1$
and
$w_m$
are
$e_i$
and
$e_j$
for some i and j, respectively, then
$\rho (w)(X\setminus X_j)$
is a proper subset of
$X_i$
. Hence,
$\rho (w)$
is not the unit element of
$\operatorname {\mathrm {Isom}}_+(\mathbb {H}^2)$
. The other cases are proved in the same way.
As
$\operatorname {\mathrm {Isom}}_+(\mathbb {H}^2)$
is a subgroup of
$\operatorname {\mathrm {Homeo}}_+(S^1)$
through the action on the ideal boundary, the inclusion
$G \to \operatorname {\mathrm {Isom}}_+(\mathbb {H}^2)$
given in Lemma 3.4 gives rise to a homomorphism
$$\begin{align*}\rho \ \colon G \to \operatorname{\mathrm{Homeo}}_+(S^1). \end{align*}$$
Under
$\rho $
, we regard G as a subgroup of
$\operatorname {\mathrm {Homeo}}_+(S^1)$
. In particular, we abbreviate
$\rho (g)(x)$
as
$g(x)$
for
$g \in G$
and
$x \in S^1$
.
Recall from the proof of Lemma 3.4 that
$Z \subset S^1$
is the union of
$X_i$
’s and
$Y_j$
’s. We label each connected component of the complement
$S^1 \setminus \mathrm {Int}(Z)$
as
$J_1, \ldots , J_k, K_1^{\pm }, \ldots , K_{2n}^{\pm }$
so that
$$\begin{align*}J_1, J_2, \ldots, J_k, K_1^-, K_2^+, K_1^+, K_2^-, \ldots, K_{2n-1}^-, K_{2n}^+, K_{2n-1}^+, K_{2n}^- \end{align*}$$
are arranged counterclockwise and that
$\partial _-(J_1) = \partial _+(X_1)$
(see Figure 2).
Arrangement of
$J_1, J_2, \ldots, J_k, K_1^-, K_2^+, K_1^+, K_2^-, \ldots, K_{2n-1}^-, K_{2n}^+, K_{2n-1}^+, K_{2n}^-$
.
We set
$x_{e_1} = \partial _-(X_1)$
. Because of the remark at the end of Section 2.4, the limit set
$L_{\rho }$
of
$\rho $
is an exceptional minimal set.
Let P be the polygon depicted in Figure 1, that is, the union of the intervals
$J_i$
’s,
$K_i^{\pm }$
’s of
$\partial \mathbb {H}^2$
and a fundamental domain of
$\rho (G)$
in
$\mathbb {H}^2$
. Here, we regard P as a closed subset of
$\mathbb {H}^2 \cup \partial \mathbb {H}^2$
.
Lemma 3.5
-
(1) The point
$x_{e_1}$
is not contained in the limit set
$L_{\rho }$
. -
(2) Let
$I_o$
be the connected component of
$S^1\setminus L_{\rho }$
containing
$x_{e_1}$
. Then, for every connected component J of
$P \cap \partial \mathbb {H}^2$
, there exists
$g \in G$
such that J is contained in
$g(I_o)$
.
Proof First, note that the set of connected components of
$P \cap \partial \mathbb {H}^2$
is
$\{ J_1, \ldots , J_k, K_1^{\pm }, \ldots , K_{2n}^{\pm } \}$
. We set
$J_0 = K_{2n}^{-}$
and
$K_0 = J_k$
. Then, for every
$i \in \{ 1, \ldots , k \}$
, we have
$\partial _+(J_{i-1}) = \partial _-(e_i(J_{i}))$
. Moreover, we have
$\partial _+(K_j^{\pm }) = \partial _-(h_{j+1}^{\pm 1}(K_{j+1}^{\pm }))$
for every odd integer j in
$\{ 1, \ldots , 2n \}$
, and
$\partial _+(K_j^{+}) = \partial _-(h_{j-1}^{-1}(K_{j-1}^{-}))$
and
$\partial _+(K_{j-1}^{-}) = \partial _-(h_{j}(K_{j}^+))$
for every even integer j in
$\{ 1, \ldots , 2n \}$
(see Figure 2). Since the point
$x_{e_1}$
is the boundary point of the intervals
$K_{2n}^{-}$
and
$e_1(J_1)$
, this point is not in
$L_{\rho }$
. Hence, i) follows.
Let
$I_{o}$
be the connected component of
$S^1\setminus L_{\rho }$
containing
$x_{e_1}$
. Then,
$I_o$
is depicted as in Figure 3, where we set
$\beta = e_1 \ldots e_k$
,
${\gamma }_i = [h_{2i-1}, h_{2i}],$
and
${\gamma }=\prod _{j=1}^{n} {\gamma }_j$
for
${i = 1, \ldots , n}$
. This implies ii).
The tiling on
$I_o$
.
Lemma 3.6 For every connected component J of
$\partial \mathbb {H}^2 \setminus L_{\rho }$
, there exists
$g \in G$
such that
$I_o = g(J)$
. That is,
$\rho (G)$
acts transitively on the set of connected components of
$\partial \mathbb {H}^2 \setminus L_{\rho }$
.
Proof Let J be a connected component of
$\partial \mathbb {H}^2 \setminus L_{\rho }$
. Since P tessellates
$\overline {\mathbb {H}^2} \setminus L_{\rho }$
, there exists
$h \in G$
and
$I \in \{ J_1, \ldots , J_k, K_1^{\pm }, \ldots , K_{2n}^{\pm } \}$
such that
$I \subset h(J)$
. In particular, together with Lemma 3.5, there exists
$g \in G$
such that
$I_o \cap g(J) \neq \emptyset $
. Since
$g(J)$
is also a connected component of
$\partial \mathbb {H}^2 \setminus L_{\rho }$
and
$I_o \cap g(J) \neq \emptyset $
, either
$I_o \cap g(J) \subset \partial I_o$
or
$I_o = g(J)$
holds. However, the former statement contradicts to the fact that
$L_{\rho }$
is homeomorphic to the Cantor set. Hence, we have
$I_o = g(J)$
.
Write
$\alpha = \beta \gamma = e_1\dots e_k[h_1, h_2]\dots [h_{2n-1},h_{2n}]$
. Since any stabilizer of a gap in a finitely generated subgroup of
$\operatorname {\mathrm {Isom}}_+(\mathbb {H}^2) \cong \operatorname {\mathrm {PSL}}(2,\mathbb {R})$
is cyclic, Figure 3 implies the following.
Lemma 3.7 The stabilizer of
$I_o$
is the infinite cyclic subgroup of G generated by
$\alpha $
, and acts on
$I_o$
as the dynamical realization of a left order with basepoint
$x_{e_1} \in I_o$
.
Proof Since the interval
$[x_{e_1},\alpha (x_{e_1})]$
is tiled as shown in Figure 3, all intervals
$[\alpha ^l(x_{e_1}),\alpha ^{l+1}(x_{e_1})]$
are tiled recursively for all integers l. Recall that
$I_o$
is the connected component of
$S^1 \setminus L_{\rho }$
containing
$x_{e_1}$
. Hence,
$\bigcup _{l\in \mathbb {Z}}[\alpha ^l(x_{e_1}),\alpha ^{l+1}(x_{e_1})] \subset I_o$
. Moreover,
$\bigcup _{l\in \mathbb {Z}}[\alpha ^l(x_{e_1}),\alpha ^{l+1}(x_{e_1})] = I_o$
holds. Indeed, the limit points
$\displaystyle \lim _{l\to \infty } \alpha ^{l} (x_{e_1})$
and
$\displaystyle \lim _{l\to \infty } \alpha ^{-l} (x_{e_1})$
belong to the limit set
$L_{\rho }$
, and
$I_o$
is disjoint from the limit set, while its endpoints
$\partial _-(I_o)$
and
$\partial _+(I_o)$
are contained in it.
Since the open interval
$(x_{e_1},\alpha (x_{e_1}))$
does not contain any point of the orbits of
$x_{e_1}$
, the intersection of
$I_o$
and the orbit of
$x_{e_1}$
is
$\{ \alpha ^l(x_{e_1}) \mid l \in \mathbb {Z} \}$
. Hence, the stabilizer of
$I_o$
is the infinite cyclic subgroup generated by
$\alpha $
. This orbit clearly coincides with that of the dynamical realization of a left order of the infinite cyclic group.
Lemmas 3.6 and 3.7, together with Theorems 2.3 and 2.4, imply the following.
Proposition The action
$\rho $
is a dynamical realization of a circular order
$c \in CO(G)$
with basepoint
$x_{e_1}$
, and the linear part is the subgroup of G generated by
$\alpha $
.
3.2 Free subgroup of G
In this section, we find a free subgroup F of G of finite index, the restriction of
$\rho $
to which is a ping-pong action.
For
$l \in \mathbb {Z}_{> 0}$
, we set
$[l] = \{ 1, \ldots , l \}$
. For
$2 \leq t \leq k$
, we set
$$ \begin{align*} \Lambda_t &= \big(([m_1] \times \cdots \times [m_{t-1}])\setminus \{ (m_1,\ldots, m_{t-1}) \} \big) \times [m_t-1], \\ \Xi_t &= [m_{t+1}] \times \cdots \times [m_k], \end{align*} $$
where we regard
$\Xi _k = \{ 0 \}$
. For
$\lambda = (u_1, \ldots , u_t) \in \Lambda _t$
and
$\xi = (u_{t+1}, \ldots , u_{k}) \in \Xi _t$
, we set
$$ \begin{align*} f_{\lambda} &= [e_t^{u_t}, e_{t-1}^{u_{t-1}}\dots e_1^{u_1}], \\ g_{\xi} &= e_k^{u_k}\dots e_{t+1}^{u_{t+1}}, \end{align*} $$
where we regard
$g_{\xi } = 1$
when
$\xi \in \Xi _k$
. Let
$S_1$
and
$S_2$
be the subsets of G defined by
$$ \begin{align*} S_1 &= \left\{ {}^{g_{\xi}} f_{\lambda} \mid 2 \leq t \leq k, \lambda \in \Lambda_t, \xi \in \Xi_t \right\}, \\ S_2 &= \left\{ {}^{e_k^{i_k}\dots e_1^{i_1}} h_l \, \middle| \, l \in [2n], (i_1, \dots i_k) \in [m_1] \times \cdots \times [m_k] \right\}. \end{align*} $$
Let F be the subgroup of G generated by
$S_1 \cup S_2$
. Let
$G_1$
be the subgroup of G generated by
$\{ e_1, \ldots , e_k \}$
.
Notation Let
$\Gamma $
be a group. For elements
$g,h$
of
$\Gamma $
, we set
$[g,h] = ghg^{-1}h^{-1}$
and
${}^{g}{h} = ghg^{-1}$
.
Lemma 3.10 Let w be an element of
$G_1$
. Then, there exists an element
$\gamma $
of the subgroup
$\langle S_1 \rangle $
generated by
$S_1$
and
$i_1, \ldots , i_k$
such that
$\gamma w = e_k^{i_k} \dots e_1^{i_1}$
.
Proof Let
$\| w \|$
denote the word length of w with respect to
$\{ e_1, \ldots , e_k \}$
. The statement is trivial if
$\| w \| = 0$
. Let
$l \geq 1$
and assume that the statement holds for elements of
$G_1$
whose word lengths are equal to
$l-1$
. Let w be an element of
$G_1$
with
$\| w \| = l$
. Let
$e_j^{\pm 1}$
be the suffix of w in the reduced word expression and set
$w = ve_j^{\pm 1}$
. Then,
$\| v \| = l-1$
. By the above assumption, there exist
$\gamma \in \langle S_1 \rangle $
and
$i_1, \ldots , i_k$
such that
$\gamma v = e_k^{i_k} \dots e_1^{i_1}$
. Hence, we have
$$\begin{align*}\gamma w = e_k^{i_k} \dots e_1^{i_1} \cdot e_j^{\pm 1}. \end{align*}$$
If
$j = 1$
, the statement holds for w. Assume that
$j \neq 1$
. We set
$\gamma ' = {}^{e_k^{i_k} \dots e_j^{i_j}} [e_j^{\mp 1}, e_{j-1}^{i_{j-1}}\dots e_1^{i_1}]$
. By the equality
${}^{cd}[a,b] = {}^{c}[da,b]\cdot {}^{c}[b,d]$
, we have
$$ \begin{align*} \gamma' &= {}^{e_k^{i_k} \dots e_{j+1}^{i_{j+1}} \cdot e_j^{i_j}} [e_j^{\mp 1}, e_{j-1}^{i_{j-1}}\dots e_1^{i_1}] \\ &= {}^{e_k^{i_k} \dots e_{j+1}^{i_{j+1}}} [e_{j}^{i_j \mp 1}, e_{j-1}^{i_{j-1}}\dots e_1^{i_1}] \cdot {}^{e_k^{i_k} \dots e_{j+1}^{i_{j+1}}}[e_{j-1}^{i_{j-1}}\dots e_1^{i_1}, e_j^{i_j}] \\ &= {}^{e_k^{i_k} \dots e_{j+1}^{i_{j+1}}} [e_{j}^{i_j \mp 1}, e_{j-1}^{i_{j-1}}\dots e_1^{i_1}] \cdot \left({}^{e_k^{i_k} \dots e_{j+1}^{i_{j+1}}}[e_j^{i_j}, e_{j-1}^{i_{j-1}}\dots e_1^{i_1}]\right)^{-1}. \end{align*} $$
Hence,
$\gamma '$
is an element of
$\langle S_1 \rangle $
. Moreover, we have
$$ \begin{align*} \gamma' \gamma w &= {}^{e_k^{i_k} \dots e_j^{i_j}} [e_j^{\mp 1}, e_{j-1}^{i_{j-1}}\dots e_1^{i_1}] \cdot e_k^{i_k} \dots e_1^{i_1} \cdot e_j^{\pm 1} \\ &= e_k^{i_k} \dots e_{j+1}^{i_{j+1}} \cdot e_j^{i_j \mp 1} \cdot e_{j-1}^{i_{j-1}} \dots e_1^{i_1}. \end{align*} $$
This completes the proof.
Lemma 3.11 For an element g of G, there exist an element f of F and
$i_1, \ldots , i_k$
such that
$fg = e_k^{i_k}\dots e_1^{i_1}$
.
Proof If g is contained in
$G_1$
, then the statement holds by Lemma 3.10. Assume that g is not contained in
$G_1$
. Then, there exists an element of
$\{ h_1^{\pm 1}, \ldots , h_{2n}^{\pm 1} \}$
in the reduced word expression of g. Then, there exist
$w \in G_1$
,
$i \in \{ 1, \ldots , 2n \}$
and
$g' \in G$
whose prefix in the reduced word expression is not equal to
$h_i^{\mp 1}$
such that
$$\begin{align*}g = w \cdot h_i^{\pm 1} \cdot g'. \end{align*}$$
Then, by Lemma 3.10, there exist
$\gamma _1 \in \langle S_1 \rangle $
and
$i_1, \ldots , i_k$
such that
$$\begin{align*}\gamma_1 g = e_{k}^{i_k} \dots e_1^{i_1} \cdot h_i^{\pm 1} \cdot g'. \end{align*}$$
We set
$\gamma _2 = {}^{e_{k}^{i_k} \dots e_1^{i_1}} h_i^{\mp 1}$
, which is an element of
$\langle S_2 \rangle $
. Then, we have
$$\begin{align*}\gamma_2 \gamma_1 g = g'. \end{align*}$$
Note that
$\gamma _2 \gamma _1$
is an element of F. This procedure strictly decreases the number of elements of
$\{ h_1^{\pm 1}, \ldots , h_{2n}^{\pm 1} \}$
in the reduced word expression of g. Taking this procedure inductively, we obtain an element
$\gamma \in F$
such that
$\gamma g \in G_1$
. Applying Lemma 3.10 to
$\gamma g$
, we obtain the lemma.
We set
$$\begin{align*}Q = \bigcup_{i_k = 0}^{m_k-1}\dots \bigcup_{i_1 = 0}^{m_1-1} e_k^{i_k} \dots e_1^{i_1}(P). \end{align*}$$
Then,
$Q\cap \mathbb {H}^2$
turns out to be a fundamental domain of F by the following two lemmas.
Lemma 3.12 For every
$f \in F\setminus \{ 1 \}$
, the interior
$\mathrm {Int}(Q)$
is disjoint from
$f(\mathrm {Int}(Q))$
.
Proof Assume that
$\mathrm {Int}(Q)\cap f(\mathrm {Int}(Q))$
is non-empty. Since
$P\cap \mathbb {H}^2$
is a fundamental domain of
$\rho (G)$
, this implies that there exist
$i_1, \ldots , i_k, j_1, \ldots , j_k$
such that
$$\begin{align*}\\[-20pt]e_k^{i_k} \dots e_1^{i_1}(P) = fe_k^{j_k}\dots e_1^{i_1}(P), \end{align*}$$
and hence
$e_k^{i_k} \dots e_1^{i_1} = fe_k^{j_k}\dots e_1^{i_1}$
. Let
$q \ \colon G \to \mathrm {H}_1(G)$
be the abelianization of G. Note that the abelianization
$\mathrm {H}_1(G)$
is isomorphic to
$\mathbb {Z}^{2n} \times \mathbb {Z}_{m_1} \times \cdots \times \mathbb {Z}_{m_k}$
. The elements of
$S_1$
are contained in the kernel of q and the image
$q(S_2)$
is contained in the subspace
$\mathbb {Z}^{2n} \times \{ 0 \} \times \cdots \times \{ 0 \} \subset \mathrm {H}_1(G)$
. Hence,
$q(e_k^{i_k} \dots e_1^{i_1}) = q(fe_k^{j_k}\dots e_1^{i_1})$
implies that
$i_l = j_l$
modulo
$m_l$
for every
$1 \leq l \leq k$
. Therefore, we obtain
$f = 1$
.
Lemma 3.13 The equality
$\displaystyle \bigcup _{f \in F} f(Q \cap \mathbb {H}^2) = \mathbb {H}^2$
holds.
Proof Since
$P\cap \mathbb {H}^2$
is a fundamental domain of
$\rho (G)$
, it is enough to prove that
$g(P) \subset \bigcup _{f \in F} f(Q)$
for every
$g \in G$
. By Lemma 3.11, we take an element f of F such that
$fg$
is of the form
$e_k^{i_k}\dots e_1^{i_1}$
. Then, we have
$fg(P) = e_k^{i_k} \dots e_1^{i_1}(P) \subset Q$
, which implies
$g(P) \subset \bigcup _{f \in F} f(Q)$
.
According to [Reference Mann and RivasMR18], we define the notion of ping-pong dynamics as follows.
Definition 3.1 [Reference Mann and RivasMR18, Definition 4.1]
Let
$a_1, a_2, \ldots , a_n \in \operatorname {\mathrm {Homeo}}_+(S^1)$
. We say that these elements have ping-pong dynamics if there exist pairwise disjoint closed sets
$D(a_i)$
and
$D(a_i^{-1})$
such that, for each i, we have
$$\begin{align*}a_i (S^1\setminus D(a_i^{-1})) \subset D(a_i). \end{align*}$$
We call such sets
$D(a_i)$
and
$D(a_i^{-1})$
satisfying
$a_i (S^1 \setminus D(a_i^{-1})) \subset D(a_i)$
attracting domains for
$a_i$
and
$a_i^{-1}$
, respectively, and use the notation
$D(a_i^{\pm 1})$
to denote
$D(a_i) \cup D(a_i^{-1})$
. These attracting domains need not be connected. In this case, we use the notation
$D_1(s), D_2(s), \ldots $
for the connected components of
$D(s)$
. Note that the definition of ping-pong dynamics implies that for each
$s \in \{ a_1^{\pm 1}, \ldots , a_n^{\pm 1}\}$
, and for each domain
$D_k(t)$
with
$t \neq s^{-1}$
, there exists a unique j such that
$s(D_k(t))$
lies in the interior of
$D_j(s)$
.
In the following, we will prove that the elements of
$S_1 \cup S_2$
have ping-pong dynamics. For
$\lambda = (u_1, \ldots , u_t) \in \Lambda _t$
, we set
$$ \begin{align} J_{\lambda}^{-} &= [e_{t-1}^{u_{t-1}} \dots e_1^{u_1}e_t^{u_t-1}(x_{e_t}), e_{t-1}^{u_{t-1}} \dots e_1^{u_1}e_t^{u_t}(x_{e_t})], \nonumber \\ J_{\lambda}^+ &= [e_t^{u_t}e_{t-1}^{u_{t-1}} \dots e_1^{u_1}(x_{e_t}), e_t^{u_t}e_{t-1}^{u_{t-1}} \dots e_1^{u_1}e_t^{-1}(x_{e_t})]. \end{align} $$
For every
$l \in [2n]$
, we set
$$\begin{align*}J_{h_l}^- = [x_l^-, x_l^+], \ J_{h_l}^+ = [h_l(x_l^+), h_l(x_l^-)]. \end{align*}$$
The relationship between these intervals is as follows.
Lemma 3.14 Let
$2 \leq t, t' \leq k, \lambda \in \Lambda _t, \xi \in \Xi _t, \lambda ' \in \Lambda _{t'}$
and
$\xi ' \in \Xi _{t'}$
. Let
$l, l' \in [2n]$
and
$(i_1, \ldots i_k), (i^{\prime }_1, \ldots , i^{\prime }_k) \in [m_1]\times \cdots \times [m_k]$
. Then, the following hold:
-
(1)
$g_{\xi }(J_{\lambda }^+) \cap g_{\xi '}(J_{\lambda '}^{+}) \neq \emptyset $
if and only if
$\lambda = \lambda '$
and
$\xi = \xi '$
, -
(2)
$\mathrm {Int}(g_{\xi }(J_{\lambda }^-)) \cap \mathrm {Int}(g_{\xi '}(J_{\lambda '}^-)) \neq \emptyset $
if and only if
$\lambda = \lambda '$
and
$\xi = \xi '$
, -
(3)
$g_{\xi }(J_{\lambda }^+) \cap g_{\xi '}(J_{\lambda '}^{-}) = \emptyset $
, -
(4)
$g_{\xi }(J_{\lambda }^{+}) \cap e_k^{i_k} \dots e_1^{i_1}(J_{h_l}^{\pm }) = \emptyset $
, -
(5)
$g_{\xi }(J_{\lambda }^{-}) \cap e_k^{i_k} \dots e_1^{i_1}(J_{h_l}^{\pm }) = \emptyset $
, -
(6)
$e_k^{i_k} \dots e_1^{i_1}(J_{h_l}^{+}) \cap e_k^{i^{\prime }_k} \dots e_1^{i^{\prime }_1}(J_{h_{l'}}^{-}) = \emptyset $
, and -
(7)
$e_k^{i_k} \dots e_1^{i_1}(J_{h_l}^{\pm }) \cap e_k^{i^{\prime }_k} \dots e_1^{i^{\prime }_1}(J_{h_{l'}}^{\pm }) = \emptyset $
if and only if
$(i_1, \ldots , i_k) = (i^{\prime }_1, \ldots , i^{\prime }_k)$
and
$l = l'$
.
Proof We set
$(\lambda , \xi ) = (u_1, \ldots , u_t, u_{t+1}, \ldots , u_k) \in [m_1] \times \cdots \times [m_k]$
. Then, we have
$$ \begin{align*} g_{\xi}(J_{\lambda}^+) &= e_k^{u_k} \dots e_1^{u_1}\big([x_{e_t}, e_t^{-1}(x_{e_t})]\big), \\ g_{\xi}(J_{\lambda}^-) &= e_k^{u_k} \dots e_{t+1}^{u_{t+1}} e_{t-1}^{u_{t-1}} \dots e_1^{u_1}\big([e_t^{u_t -1}(x_{e_t}), e_t^{u_t}(x_{e_t})]\big). \end{align*} $$
We set
$(\lambda ', \xi ') = (u^{\prime }_1, \ldots , u^{\prime }_{t'}, u^{\prime }_{t'+1}, \ldots , u^{\prime }_k) \in [m_1] \times \cdots \times [m_k]$
.
We first prove i). The “if” part is clear. To prove the “only if” part, we assume that
$g_{\xi }(J_{\lambda }^+) \cap g_{\xi '}(J_{\lambda '}^{+}) \neq \emptyset $
. Note that
$$ \begin{align*} g_{\xi}(J_{\lambda}^{+}) &\subset [e_k^{u_{k}-1}(x_{e_k}), e_k^{u_{k}}(x_{e_k})], \\ g_{\xi'}(J_{\lambda'}^{+}) & \subset [e_k^{u^{\prime}_{k}-1}(x_{e_k}), e_k^{u^{\prime}_{k}}(x_{e_k})]. \end{align*} $$
If
$u_k \neq u^{\prime }_{k}$
, then either of the following holds:
-
(1)
$g_{\xi }(J_{\lambda }^+) \cap g_{\xi '}(J_{\lambda '}^{+}) = \{ e_k^{u_k}(x_{e_k}) \}$
; -
(2)
$g_{\xi }(J_{\lambda }^+) \cap g_{\xi '}(J_{\lambda '}^{+}) = \{ e_k^{u^{\prime }_k}(x_{e_k}) \}$
.
If (1) holds, then
$e_k^{u_k}(x_{e_k}) \in g_{\xi }(J_{\lambda }^+)$
, and hence
$x_{e_k} \in e_{k-1}^{u_{k-1}} \dots e_1^{u_1}\big ( [x_{e_t}, e_{t}^{-1}(x_{e_t})] \big )$
. This implies that
$u_j = m_j$
for every
$j \in [k-1]$
and
$t = k$
. This contradicts
$\lambda \in \Lambda _k$
. In the same way, (2) also contradicts to
$\lambda ' \in \Lambda _k$
. Hence, we obtain
$u_k = u^{\prime }_k$
. Taking this procedure inductively, we obtain
$u_j = u^{\prime }_j$
for every
$1 \leq j \leq k$
. Hence, we have
$[x_{e_t}, e_t^{-1}(x_{e_t})] \cap [x_{e_{t'}}, e_{t'}^{-1}(x_{e_{t'}})] \neq \emptyset $
, which implies
$t = t'$
. This completes the proof of i).
For ii), we assume that
$\mathrm {Int}(g_{\xi }(J_{\lambda }^-)) \cap \mathrm {Int}(g_{\xi '}(J_{\lambda '}^-)) \neq \emptyset $
. Since
$$ \begin{align*} &\mathrm{Int}(g_{\xi}(J_{\lambda}^{-})) \subset (e_k^{u_{k}-1}(x_{e_k}), e_k^{u_{k}}(x_{e_k})) \text{ and} \\ &\mathrm{Int}(g_{\xi'}(J_{\lambda'}^{-})) \subset (e_k^{u^{\prime}_{k}-1}(x_{e_k}), e_k^{u^{\prime}_{k}}(x_{e_k})), \end{align*} $$
we have
$u_k = u^{\prime }_{k'}$
. Taking this procedure inductively, we have
$u_j = u^{\prime }_j$
for
$1 \leq j \leq k$
except for
$j = t, t'$
, and we obtain
$u_t = m_t$
and
$u^{\prime }_{t'} = m_{t'}$
. This implies that
$$\begin{align*}(e_t^{u_t -1}(x_{e_t}), e_t^{u_t}(x_{e_t})) \cap (e_{t'}^{u^{\prime}_{t'} -1}(x_{e_{t'}}), e_{t'}^{u^{\prime}_{t'}}(x_{e_{t'}})) \neq \emptyset. \end{align*}$$
Hence, we obtain
$t = t'$
. This completes the proof of ii).
For iii), we assume that
$g_{\xi }(J_{\lambda }^+) \cap g_{\xi '}(J_{\lambda '}^{-}) \neq \emptyset $
. By the arguments same as above, we have
$u_j = u^{\prime }_{j}$
for every
$1 \leq j \leq k$
except for
$j = t'$
and
$u_{t'} = m_{t'}$
. However, this contradicts to
$\lambda ' \in \Lambda _{t'}$
.
For iv), we assume that
$g_{\xi }(J_{\lambda }^{+}) \cap e_k^{i_k} \dots e_1^{i_1}(J_{h_l}^{\pm }) \neq \emptyset $
. By the arguments similar to the above, we have
$u_j = i_j$
for every
$1 \leq j \leq k$
and
$J_{\lambda ^+} \cap J_{h_l}^{\pm } \neq \emptyset $
, which is not the case.
The proofs of v) and vi) are the same as one of iv). The proof of vii) is the same as one of i).
Lemma 3.15 Let
$2 \leq t \leq k, \lambda \in \Lambda _t$
, and
$\xi \in \Xi _t$
. Let
$l \in [2n]$
and
$(i_1, \dots i_k), (j_1, \dots , j_k) \in [m_1]\times \cdots \times [m_k]$
. Then, the following hold:
-
(1)
$e_{k}^{j_k} \dots e_1^{j_1} (x_{e_1})$
is not contained in
$g_{\xi }(J_{\lambda }^{\pm })$
; -
(2)
$e_{k}^{j_k} \dots e_1^{j_1} (x_{e_1})$
is not contained in
$e_k^{i_k} \dots e_1^{i_1}(J_{h_{l}}^{\pm })$
.
We omit the proof since it is similar to one of Lemma 3.14.
Proposition There exist attracting domains
$D(\rho (s))$
for
$s \in S_1 \cup S_2 \cup S_1^{-1} \cup S_2^{-1}$
such that the elements
$$\begin{align*}&\{ D(\rho(s)) \mid s \in S_1 \cup S_2 \cup S_1^{-1} \cup S_2^{-1} \} \\&\quad\cup \{ e_k^{j_k} \dots e_1^{j_1}(x_{e_1}) \mid (j_1, \ldots j_k) \in [m_1] \times \cdots \times [m_k] \} \end{align*}$$
are pairwise disjoint. In particular, the elements of
$\rho (S_1 \cup S_2)$
have ping-pong dynamics.
Proof By the definitions of
$f_{\lambda }, g_{\xi }, J_{\lambda }^{\pm }$
, and
$J_{h_l}^{\pm }$
, we have
$$ \begin{align*} &{}^{g_{\xi}}f_{\lambda}(S^1 \setminus g_{\xi}(J_{\lambda}^-)) \subset g_{\xi}(J_{\lambda}^+), \\ &{}^{e_k^{i_k}\dots e_1^{i_1}} h_l(S^1\setminus e_k^{i_k}\dots e_1^{i_1} (J_{h_l}^-)) \subset e_k^{i_k}\dots e_1^{i_1}(J_{h_l}^+). \end{align*} $$
For a closed interval
$I = [a,b] \subset S^1$
and a sufficiently small
$\epsilon> 0$
, we set
${I_{\epsilon } = [a-\epsilon , b+\epsilon ]}$
. By Lemmas 3.14 and 3.15, there exists
$\epsilon> 0$
such that the elements of the union
$$ \begin{align*} &\{ (g_{\xi}(J_{\lambda}^{+}))_{\epsilon} \mid 2 \leq t \leq k, \lambda \in \Lambda_t, \xi \in \Xi_t \} \\ &\cup \{ \mathrm{Cl}(({}^{g_{\xi}}f_{\lambda})^{-1}(S^1 \setminus (g_{\xi}(J_{\lambda}^+))_{\epsilon})) \mid 2 \leq t \leq k, \lambda \in \Lambda_t, \xi \in \Xi_t \} \\ &\cup \{ e_k^{i_k}\dots e_1^{i_1}(J_{h_l}^{\pm}) \mid (i_1, \ldots, i_k) \in [m_1]\times \cdots \times [m_k], l \in [2n] \} \\ &\cup \{ e_k^{i_k} \dots e_1^{i_1}(x_{e_1}) \mid (i_1, \ldots, i_k) \in [m_1]\times \cdots \times [m_k] \} \end{align*} $$
are pairwise disjoint. Here,
$\mathrm {Cl}(X)$
denotes the closure of X. Setting
$$ \begin{align*} &D\big(\rho({}^{g_{\xi}}f_{\lambda})\big) = (g_{\xi}(J_{\lambda}^{+}))_{\epsilon}, \\ &D\big(\rho(({}^{g_{\xi}}f_{\lambda})^{-1})\big) = \mathrm{Cl}(({}^{g_{\xi}}f_{\lambda})^{-1}(S^1 \setminus (g_{\xi}(J_{\lambda}^+))_{\epsilon})), \\ &D\big(\rho({}^{e_{k}^{i_k}\dots e_{1}^{i_1}}h_l)\big) = e_k^{i_k}\dots e_1^{i_1}(J_{h_l}^{+}), \\ &D\big(\rho(({}^{e_{k}^{i_k}\dots e_{1}^{i_1}}h_l)^{-1})\big) = e_k^{i_k}\dots e_1^{i_1}(J_{h_l}^{-}), \end{align*} $$
we obtain the proposition.
The following holds from Lemmas 3.12 and 3.13.
Corollary The group
$F = \langle S_1 \cup S_2 \rangle $
is a free subgroup with
$[G:F] = m_1\ldots m_k$
.
3.3 Isolation of c
By Proposition 3.8, we take a circular order c whose dynamical realization with basepoint
$x_{e_1}$
coincides with
$\rho $
. In this section, we prove the following.
Proposition The circular order c is isolated.
The following is a slight generalization of [Reference Mann and RivasMR18, Lemma 4.2].
Lemma 3.19 Let
$\Gamma $
be a group and
$F_{r}$
a rank r free subgroup of
$\Gamma $
of finite index. Set
$m = [\Gamma :F_r]$
. Let
$\{ \gamma _1 = 1, \gamma _2, \ldots , \gamma _m \}$
be a complete set of representatives of
$F_r \backslash \Gamma $
. Let
$a_1, \ldots , a_r$
be a free generating set of
$F_r$
. Let
$\sigma \ \colon \Gamma \to \operatorname {\mathrm {Homeo}}_+(S^1)$
be a homomorphism such that the elements
$\sigma (a_1), \ldots , \sigma (a_r)$
have ping-pong dynamics. Let
$x_0 \in S^1$
. Assume that the elements of
$$\begin{align*}\{ D(\sigma(a_i)) \mid i \in [r] \} \cup \{ D(\sigma(a_i)^{-1}) \mid i \in [r] \} \cup \{ \sigma(\gamma_j)(x_0) \mid j \in [m] \} \end{align*}$$
are pairwise disjoint. Then, the action
$\sigma $
is free at
$x_0$
and the cyclic order of its orbit is completely determined by the following data:
-
(1) the cyclic order of the sets
$D_l(\rho (s))$
and
$\{ \sigma (\gamma _j)(x_0) \}$
for
$s \in \{ a_1^{\pm 1}, \ldots , a_r^{\pm 1} \}$
; -
(2) the collection of containment relations
$$ \begin{align*} &\sigma(s)(D_{l'}(t)) \subset D_{l}(s), \\ &\sigma(s)(\sigma(\gamma_j)(x_0)) \in D_{l}(s). \end{align*} $$
We omit the proof since it is essentially the same as that of [Reference Mann and RivasMR18, Lemma 4.2].
Lemma 3.20 There exists a neighborhood U of
$\rho $
in
$\operatorname {\mathrm {Hom}}(G, \operatorname {\mathrm {Homeo}}_+(S^1))$
such that for every
$\sigma \in U$
, the following hold:
-
(1)
$\sigma $
acts on
$x_{e_1}$
freely; -
(2) the circular order of
$\{ \sigma (g)(x_{e_1}) \}_{g \in G}$
coincides with that of
$\{ \rho (g)(x_{e_1}) \}_{g \in G}$
.
Proof For
$A \subset S^1$
and for
$\epsilon>0$
, let
$N(A, \epsilon )$
be the open
$\epsilon $
-neighborhood of A. We take attracting domains
$D(\rho (s))$
as in Proposition 3.16, where
$s \in S_1\cup S_2 \cup S_1^{-1} \cup S_2^{-1}$
. Then, there exists
$\epsilon> 0$
such that the elements of the union
$$ \begin{align*} &\{ N(D(\rho(s)), \epsilon) \mid s \in S_1 \cup S_2 \cup S_1^{-1} \cup S_2^{-1} \} \\ &\cup \{ N(e_k^{i_k} \dots e_1^{i_1}(x_{e_1}), \epsilon) \mid (i_1, \ldots, i_k) \in [m_1]\times \cdots \times [m_k] \} \end{align*} $$
are pairwise disjoint. We set
$$\begin{align*}S = S_1 \cup S_2 \cup S_1^{-1} \cup S_2^{-1} \cup \{ e_{k}^{i_k}\dots e_{1}^{i_1} \mid (i_1, \ldots, i_k) \in [m_1] \times \cdots \times [m_k] \} \end{align*}$$
and
$$\begin{align*}U = \left\{ \sigma \in \operatorname{\mathrm{Hom}}(G, \operatorname{\mathrm{Homeo}}_+(S^1)) \, \middle| \, \sup_{x \in S^1} \sup_{g \in S} d(\rho(g)(x), \sigma(g)(x)) < \epsilon \right\}, \end{align*}$$
where d denotes the standard metric on
$S^1$
. Obviously, this U is an open neighborhood of
$\rho $
.
Take
$\sigma \in U$
. Then, by the definition of
$\epsilon $
, we can take attracting domains of
$\sigma (s)$
for
$s \in S_1 \cup S_2 \cup S_1^{-1} \cup S_2^{-1}$
satisfying the assumption of Lemma 3.19. Moreover, the data (1) and (2) in Lemma 3.19 for this
$\sigma $
coincide with those for
$\rho $
. Hence, Lemma 3.19 implies that
$\sigma $
acts on
$x_{e_1}$
freely and that the circular order of
$\{ \sigma (g)(x_{e_1}) \}_{g \in G}$
coincides with one of
$\{ \rho (g)(x_{e_1}) \}_{g \in G}$
.
Proof of Proposition 3.18
By Proposition 3.8, the action
$\rho $
is a dynamical realization of a circular order c based at
$x_{e_1}$
. We take an open neighborhood U of
$\rho $
as in Lemma 3.20. Then, Theorem 2.2 implies that there exists a neighborhood V of c in
$CO(G)$
such that each order in V has a dynamical realization based at
$x_{e_1}$
contained in U. Hence, by Corollary 3.17 and (2) of Lemma 3.19, every order in V coincides with c, which implies the isolation of c.
3.4 Infinitely many isolated orders on G
In this section, we construct infinitely many isolated circular orders by lifting the action
$\rho $
to actions on finite covering spaces of
$S^1$
.
For an integer
$d \geq 1$
, let
$\tau _d \ \colon S^1 \to S^1$
be the
$(1/d)$
-rotation and set
$$\begin{align*}\operatorname{\mathrm{Homeo}}_{+}^{(d)}(S^1) = \{ f \in \operatorname{\mathrm{Homeo}}_+(S^1) \mid f \circ \tau_d = \tau_d \circ f \}. \end{align*}$$
Note that the d-fold covering
$p_d \ \colon S^1 \to S^1$
induces the exact sequence
$$\begin{align*}0 \to \mathbb{Z}_d \to \operatorname{\mathrm{Homeo}}_+^{(d)}(S^1) \xrightarrow{\pi} \operatorname{\mathrm{Homeo}}_+(S^1) \to 1. \end{align*}$$
For an integer
$m \geq 2$
, the homomorphism
$\phi \ \colon \mathbb {Z}_m \to \operatorname {\mathrm {Homeo}}_+(S^1)$
defined by
$\phi (1) = \tau _m$
lifts to a homomorphism
$\widetilde {\phi } \colon \mathbb {Z}_m \to \operatorname {\mathrm {Homeo}}_+^{(d)}(S^1)$
if and only if there exists
$i \in \mathbb {Z}_{\geq 0}$
such that d divides
$mi+1$
. Note that if d divides
$mi + 1$
, then the lift
$\widetilde {\phi }$
is given by setting
$\widetilde {\phi }(1)$
as the lift of
$\phi (1)$
with
$\operatorname {\mathrm {rot}}(\widetilde {\phi }(1)) = (mi+1)/dm$
. In particular, the homomorphism
$\rho \ \colon G \to \operatorname {\mathrm {Homeo}}_+(S^1)$
lifts to a homomorphism
$\widetilde {\rho } \ \colon G \to \operatorname {\mathrm {Homeo}}_+^{(d)}(S^1)$
if and only if for every
$l \in [k]$
, there exists
$i_l \in \mathbb {Z}_{\geq 0}$
such that d divides
$m_l i_l + 1$
.
We take a specific lift
$\rho ^{(d)} \ \colon G \to \operatorname {\mathrm {Homeo}}_+^{(d)}(S^1)$
as follows. Assume that for every
$l \in [k]$
, there exists
$i_l \in \mathbb {Z}_{\geq 0}$
such that d divides
$m_l i_l + 1$
. For every
$i \in [2n]$
, let
$\rho ^{(d)}(h_i)$
be the lift of
$\rho (h_i)$
with
$\operatorname {\mathrm {rot}}(\rho ^{(d)}(h_i)) = 0$
. For every
$l \in [k]$
, let
$\rho ^{(d)}(e_l)$
be the lift of
$\rho (e_l)$
with
$\operatorname {\mathrm {rot}}(\rho ^{(d)}(e_l)) = (m_l i_l + 1)/dm_l$
. This provides a well-defined lift
$\rho ^{(d)} \ \colon G \to \operatorname {\mathrm {Homeo}}_+^{(d)}(S^1)$
of
$\rho $
.
Let
$\widetilde {x_{e_1}}$
be a point of
$p_d^{-1}(x_{e_1})$
. Let
$I_1, \ldots , I_d$
be the connected components of
$p_d^{-1}(I_o)$
, where
$I_1$
is the interval containing
$\widetilde {x_{e_1}}$
and they are aligned in this order with respect to the orientation of
$S^1$
.
Lemma 3.21 There exist infinitely many
$a \in \mathbb {Z}_{\geq 0}$
such that
$d = m_1 \ldots m_k \cdot a + 1$
satisfies the following:
-
(1) for every
$l \in [k]$
, there exists
$i_l \in \mathbb {Z}_{\geq 0}$
such that d divides
$m_l i_l + 1$
; -
(2) for every
$i \in [d]$
, there exists a unique
$j \in [d]$
such that
$I_i = \rho ^{(d)}(\alpha ^j)(I_1)$
, where
$\alpha = e_1 \dots e_k [h_1, h_2] \dots [h_{2n-1}, h_{2n}]$
.
In the proof of Lemma 3.21, we use the following lemma.
Lemma 3.22 Let
$p_0, p_1$
be elements of
$\mathbb {Z}_{\geq 1}$
and
$N \in \mathbb {Z}_{\geq 0}$
. Assume that
$Np_0 \neq p_1$
. Then, there exist infinitely many
$a \in \mathbb {Z}_{\geq 0}$
such that
$p_0a+1$
and
$p_1a+N$
are relatively prime.
Proof If
$p_0> p_1$
, then for sufficiently large a, we have
$p_0 a + 1> p_1 a + N$
. Note that there exist infinitely many
$a \in \mathbb {Z}_{\geq 0}$
such that
$p_0 a + 1$
is prime by Dirichlet’s theorem on arithmetic progressions. Hence, there exist infinitely many
$a \in \mathbb {Z}_{\geq 0}$
such that
$p_0a+1$
and
$p_1a+N$
are relatively prime.
Assume that
$p_0 \leq p_1$
. We set
$p_1 = p_0 q + r$
, where
$q \in \mathbb {Z}_{\geq 1}$
and
$0 \leq r < p_0$
. Then,
$\mathrm {gcd}(p_0 a + 1, p_1 a + N) = \mathrm {gcd}(p_0 a + 1, r a + N-q)$
. Note that
$(q,r) \neq (N,0)$
by the assumption on
$p_0, p_1,$
and N. Hence, by the argument similar to the case of
$p_0> p_1$
, we obtain infinitely many
$a \in \mathbb {Z}_{\geq 0}$
such that
$p_0a+1$
and
$p_1a+N$
are relatively prime.
Proof of Lemma 3.21
First note that for every
$a \in \mathbb {Z}_{\geq 0}$
, item i) holds.
Let a be an element of
$\mathbb {Z}_{\geq 0}$
and set
$d = m_1 \ldots m_k \cdot a + 1$
. Let
$\rho ^{(d)} \ \colon G \to \operatorname {\mathrm {Homeo}}_+^{(d)}(S^1)$
be the lift defined above. Recall that
$\alpha = e_1 \ldots e_k [h_1, h_2]\dots [h_{2n-1}, h_{2n}]$
is a stabilizer of
$I_o$
. By tracking the orbit of a point of
$\partial I_1$
, it is verified that
$$\begin{align*}\operatorname{\mathrm{rot}}(\rho^{(d)}(\alpha)) = \left( 2n-1+k + \displaystyle\sum_{l = 1}^{k}i_l \right) \bigg/ d. \end{align*}$$
Hence,
$\rho ^{(d)}(\alpha )(I_i) = I_j$
if and only if
$$\begin{align*}j-i = 2n-1+k+\sum_{l = 1}^{k}i_l \ \ \text{ mod } d. \end{align*}$$
Therefore, it suffices to prove that there exist infinitely many such d such that d and
$2n-1+k+\sum _{l = 1}^{k}i_l$
are relatively prime.
Set
$N = k+2n-1, p_0 = m_1 \ldots m_k$
, and
$p_1 = \sum _{l= 1}^{k} m_1 \dots \widehat {m_l} \dots m_k$
, where the hat means that the term under the hat is omitted. Here, we regard
$p_1$
as
$1$
when
$k = 1$
. Then, we have
$$ \begin{align} &p_0 a + 1 = d, \nonumber \\ &p_1 a + N = 2n-1+k+\sum_{l = 1}^{k}i_l. \end{align} $$
If
$Np_0 \neq p_1$
, then applying Lemma 3.22 to (3.2), we obtain the lemma. Hence, it suffices to show that
$Np_0 \neq p_1$
.
If
$k = 1$
, then
$Np_0 = 2nm_1$
and
$p_1 = 1$
. Hence,
$Np_0 \neq p_1$
.
If
$k = 2$
, then
$$\begin{align*}Np_0 - p_1 = (2n+1)m_1m_2 - (m_1 + m_2) = 2nm_1m_2 + (m_1-1)(m_2-1)-1. \end{align*}$$
Since
$2nm_1m_2 \geq 0$
and
$(m_1-1)(m_2-1) \geq 1$
, the equality
$Np_0 = p_1$
holds if and only if
$n = 0$
and
$m_1 = m_2 = 2$
. Hence, by the assumption on G (i.e., the group G is not isomorphic to
$\mathbb {Z}_2 \ast \mathbb {Z}_2$
), we have
$Np_0 \neq p_1$
.
If
$k \geq 3$
, then
$$ \begin{align*} Np_0 - p_1 &\geq (k-1)p_0 - p_1 = (k-1)m_1 \dots m_k - \sum_{l=1}^{k}m_1 \dots \widehat{m_l} \dots m_k \\ &\geq (k-1)m_1 \dots m_k - \frac{k}{2}m_1 \dots m_k = \frac{k-2}{2}m_1 \dots m_k> 0. \end{align*} $$
Hence, we obtain
$Np_0 \neq p_1$
. This completes the proof.
The following proposition is an analog of Proposition 3.8.
Proposition Let a be an element of
$\mathbb {Z}_{\geq 0}$
satisfying items i) and ii) in Lemma 3.21. Then, the lift
$\rho ^{(d)} \ \colon G \to \operatorname {\mathrm {Homeo}}_+^{(d)}(S^1)$
of
$\rho $
is a dynamical realization of a circular order
$c^{(d)} \in CO(G)$
with basepoint
$\widetilde {x_{e_1}}$
, and the linear part is the subgroup of G generated by
$\alpha ^d$
.
Proof By ii) of Lemma 3.21, the stabilizer of
$I_1$
is the cyclic subgroup of G generated by
$\alpha ^d$
. Note that the inverse image
$p_d^{-1}(L_{\rho })$
of the exceptional minimal set
$L_{\rho }$
of
$\rho $
gives rise to an exceptional minimal set of the lift
$\rho ^{(d)}$
.
To use Theorem 2.3, we prove that for every connected component J of
$S^1 \backslash p_d^{-1}(L_{\rho })$
, there exists
$g \in G$
such that
$I_1 = \rho ^{(d)}(g)(J)$
. Let J be a connected component of
$S^1 \backslash p_d^{-1}(L_{\rho })$
. Then,
$p_d(J)$
is a connected component of
$S^1 \backslash L_{\rho }$
. Hence, by Lemma 3.6, there exists
$g' \in G$
such that
$I_o = \rho (g')(p_d(J))$
, which implies that there exists
$i \in [d]$
such that
$I_i = \rho ^{(d)}(g')(J)$
. By Lemma 3.22, there exists
$j \in [d]$
such that
$I_i = \rho ^{(d)}(\alpha ^j)(I_1)$
. Hence, we have
$I_1 = \rho ^{(d)}(\alpha ^{-j})(I_i) = \rho ^{(d)}(\alpha ^{-j}g')(J)$
.
Applying Theorems 2.3 and 2.4 to
$\rho ^{(d)}$
, we obtain the proposition.
Take attracting domains
$D(\rho (s))$
for
$s \in S_1 \cup S_2 \cup S_1^{-1} \cup S_2^{-1}$
as in Proposition 3.16. Let d be an integer satisfying items i) and ii) in Lemma 3.21 and set
$D(\rho ^{(d)}(s)) = p_d^{-1}(D(\rho (s)))$
. Since
$p_d \ \colon S^1 \to S^1$
is the d-fold covering and
$\rho ^{(d)}$
is a lift of
$\rho $
, we have the following.
Lemma 3.24 The closed sets
$D(\rho ^{(d)}(s)) = p_d^{-1}(D(\rho (s)))$
are attracting domains for
$s \in S_1 \cup S_2 \cup S_1^{-1} \cup S_2^{-1}$
such that the elements
$$\begin{align*}&\{ D(\rho^{(d)}(s)) \mid s \in S_1 \cup S_2 \cup S_1^{-1} \cup S_2^{-1} \} \\&\quad\cup \{ \rho^{(d)}(e_k^{j_k} \dots e_1^{j_1})(\widetilde{x_{e_1}}) \mid (j_1, \ldots j_k) \in [m_1] \times \cdots \times [m_k] \} \end{align*}$$
are pairwise disjoint.
The following is an analog of Lemma 3.20.
Lemma 3.25 Let a be an element of
$\mathbb {Z}_{\geq 0}$
satisfying items i) and ii) in Lemma 3.21. Then, there exists a neighborhood U of
$\rho ^{(d)}$
in
$\operatorname {\mathrm {Hom}}(G, \operatorname {\mathrm {Homeo}}_+(S^1))$
such that for every
$\sigma \in U$
, the following hold:
-
(1)
$\sigma $
acts on
$\widetilde {x_{e_1}}$
freely; -
(2) the circular order of
$\{ \sigma (g)(\widetilde {x_{e_1}}) \}_{g \in G}$
coincides with one of
$\{ \rho ^{(d)}(g)(\widetilde {x_{e_1}}) \}_{g \in G}$
.
We omit the proof since it is essentially the same as that of Lemma 3.20. As a corollary, we obtain the following.
Corollary Let a be an element of
$\mathbb {Z}_{\geq 0}$
satisfying items i) and ii) in Lemma 3.21. Then, the circular order
$c^{(d)}$
is isolated.
We now prove that each isolated circular order
$c^{(d)}$
is not an automorphic image of the others, which implies Theorem 1.1. To prove this, we need the following.
Lemma 3.27 Let
$\Gamma $
be a group and c a circular order on
$\Gamma $
. Let
$\Gamma _0$
be the linear part of c. Then, for every automorphism
$\phi $
of
$\Gamma $
, the linear part of the induced order
$c_{\phi }$
is
$\phi ^{-1}(\Gamma _0)$
.
Proof We fix an enumeration
$\Gamma = \{ {\gamma }_i \}_{i \geq 0}$
with
${\gamma }_0 = 1$
. Let
$\sigma $
be the dynamical realization of c based at
$x_0$
with respect to the enumeration above. Let
$\sigma _{\phi }$
be the dynamical realization of
$c_{\phi }$
based at
$x_0$
with respect to the enumeration
$\Gamma = \{ \phi ({\gamma }_i) \}_{i \geq 0}$
. By the definition of the dynamical realization, we have
$\sigma _{\phi } = \sigma \circ \phi $
. Hence, the exceptional minimal set K of
$\sigma _{\phi }$
coincides with one of
$\sigma $
.
Let I be the connected component of
$S^1 \backslash K$
containing
$x_0$
. By Theorem 2.4, the linear part of c (resp. of
$c_{\phi }$
) is the stabilizer
$\mathrm {Stab}_{\sigma }(I)$
(resp.
$\mathrm {Stab}_{\sigma _{\phi }}(I)$
) of I with respect to
$\sigma $
(resp.
$\sigma _{\phi }$
). Since
$\mathrm {Stab}_{\sigma _{\phi }}(I) = \phi ^{-1}(\mathrm {Stab}_{\sigma }(I)) = \phi ^{-1}(\Gamma _0)$
, we obtain the lemma.
Proof of Theorem 1.1
By Lemma 3.22 and Corollary 3.26, it suffices to prove that each isolated circular order
$c^{(d)}$
is not an automorphic image of the others.
Let d and
$d'$
be integers satisfying items i) and ii) of Lemma 3.22. Then, the linear part of
$c^{(d)}$
(resp.
$c^{(d')}$
) is the cyclic subgroup of G generated by
$\alpha ^d$
(resp.
$\alpha ^{d'}$
) by Proposition 3.23. Assume that there exists an automorphism
$\phi $
of G such that
${c^{(d)} = (c^{(d')})_{\phi }}$
. Then, Lemma 3.27 implies that the cyclic subgroup
$\langle \alpha ^d \rangle $
, which is the linear part of
$c^{(d)}$
, coincides with
$\phi ^{-1} (\langle \alpha ^{d'} \rangle )$
. Hence, we have either
$\phi (\alpha )^d = \alpha ^{d'}$
or
$\phi (\alpha )^d = \alpha ^{-d'}$
.
Note that
$\alpha $
is a primitive element of G. Hence, so is
$\phi (\alpha )$
. Since G can be embedded into
$\operatorname {\mathrm {Isom}}_+(\mathbb {H}^2)$
as a discrete subgroup, every element of G of infinite order is a multiple of a unique primitive element. Hence,
$\phi (\alpha )^d = \alpha ^{\pm d'}$
implies that
$\phi (\alpha ) = \alpha ^{\pm 1}$
and
$d = d'$
. This completes the proof.
4 Isolated left orders
In this section, we prove Theorem 1.2, which states that the group
$$\begin{align*}\widehat{G} = \langle e_1, \ldots, e_k, h_1, \ldots, h_{2n}, z \mid e_1^{m_1} = \cdots =e_k^{m_k} = z, [h_1, z] = \cdots = [h_{2n}, z] = 1 \rangle \end{align*}$$
admits countably many isolated left orders which are not the automorphic images of the others.
4.1 Isolated left orders on a central
$\mathbb {Z}$
-extension
In this section, we recall Mann–Rivas’s criterion [Reference Mann and RivasMR18] for certain left orders on a central
$\mathbb {Z}$
-extension to be isolated in terms of circular orders.
Let
$<$
be a left order on a group
$\Gamma $
. A subgroup
$\Lambda $
of
$\Gamma $
is said to be cofinal for
$<$
if for every
${\gamma } \in \Gamma $
, there exist
$\lambda _1, \lambda _2 \in \Lambda $
such that
$\lambda _1 < {\gamma } < \lambda _2$
. Let
$LO_{\Lambda }(\Gamma )$
be the set of left orders on
$\Gamma $
for which the subgroup
$\Lambda $
is cofinal.
Theorem 4.1 [Reference Mann and RivasMR18, Proposition 5.1]
Let
$\Gamma $
be a group and
$0 \to \mathbb {Z} \to \widehat {\Gamma } \to \Gamma \to 1$
a central
$\mathbb {Z}$
-extension of
$\Gamma $
. Then, there is a continuous map
$\pi _{\widehat {\Gamma }}^* \ \colon LO_{\mathbb {Z}}(\widehat {\Gamma })\to CO(\Gamma )$
. Moreover, each circular order on
$\Gamma $
is in the image of one such map
$\pi _{\widehat {\Gamma }}^*$
.
The map
$\pi _{\widehat {\Gamma }}^* \ \colon LO_{\mathbb {Z}}(\widehat {\Gamma }) \to CO(\Gamma )$
is defined as follows. Let
$<$
be an element of
$LO_{\mathbb {Z}}(\widehat {\Gamma })$
. Let z be the generator of
$\mathbb {Z} \subset \widehat {\Gamma }$
satisfying
$1 < z$
. Since
$\mathbb {Z}$
is cofinal for
$<$
, for every
${\gamma } \in \Gamma $
, there exists a unique lift
$\widehat {{\gamma }} \in \widehat {\Gamma }$
of
${\gamma }$
such that
$1 \leq \widehat {{\gamma }} \leq z$
. For every distinct elements
${\gamma }_1, {\gamma }_2, {\gamma }_3 \in \Gamma $
, let
$\sigma $
be the permutation such that
$1 \leq \widehat {{\gamma }}_{\sigma (1)} < \widehat {{\gamma }}_{\sigma (2)} < \widehat {{\gamma }}_{\sigma (3)} < z$
. The map
$\pi _{\widehat {\Gamma }}^*$
is defined by
$\pi _{\widehat {\Gamma }}^*(<)({\gamma }_1, {\gamma }_2, {\gamma }_3) = \mathrm {sign}(\sigma )$
.
For a countable group
$\Gamma $
, the central extension
$\widehat {\Gamma }$
in the last statement of Theorem 4.1 is given as follows. Let c be a circular order on
$\Gamma $
. Let
$\rho $
be a dynamical realization of c based at x. Let
$T \ \colon \mathbb {R} \to \mathbb {R}$
be the translation by one and
$\mathrm {H}\widetilde {\mathrm {omeo}}_+(S^1)$
the group of homeomorphisms of
$\mathbb {R}$
which commute with T. Note that the group
$\mathrm {H}\widetilde {\mathrm {omeo}}_+(S^1)$
gives rise to a central
$\mathbb {Z}$
-extension of
$\operatorname {\mathrm {Homeo}}_+(S^1)$
. Then, the group
$\widehat {\Gamma }$
is given as the pullback of
$\mathrm {H}\widetilde {\mathrm {omeo}}_+(S^1)$
.
Take a lift
$\hat {x} \in \mathbb {R}$
of x. Then,
$\hat {x}$
has a free orbit under
$\widehat {\Gamma }\subset \operatorname {\mathrm {Homeo}}(\mathbb {R})$
, which induces a left order
$<$
on
$\widehat {\Gamma }$
for which
$\mathbb {Z}$
is cofinal. This left order
$<$
satisfies
$\pi _{\widehat {\Gamma }}^*(<) = c$
(see [Reference Mann and RivasMR18] for details).
The isolation of certain left orders on
$\widehat {\Gamma }$
can be detected by the isolation of the induced circular orders on
$\Gamma $
as follows.
Theorem 4.2 [Reference Mann and RivasMR18, Proposition 5.4]
Assume that
$\Gamma $
is finitely generated and c is an isolated circular order on G. If
$0 \to \mathbb {Z} \to \hat {\Gamma } \to \Gamma \to 1$
is a central extension and
$< \in LO_{\mathbb {Z}}(\widehat {\Gamma })$
is a left order such that
$\pi _{\widehat {\Gamma }}^*(<) = c$
, then
$<$
is isolated in
$LO(\widehat {\Gamma })$
.
Let
$\widehat {\phi } \ \colon \widehat {\Gamma } \to \widehat {\Gamma }$
be an automorphism such that
$\widehat {\phi }(\mathbb {Z}) = \mathbb {Z}$
. Then,
$\widehat {\phi }$
descends to an automorphism
$\phi \ \colon \Gamma \to \Gamma $
and forms the following commutative diagram:
Lemma 4.3 The automorphisms
$\widehat {\phi }$
and
$\phi $
induce the following commutative diagram:
Here,
$\widehat {\phi }^*$
(resp.
$\phi ^*$
) is given by
$\widehat {\phi }^*(c) = c_{\widehat {\phi }}$
(resp.
$\phi ^*(<) = <_{\phi }$
).
Proof First, we prove that for
$< \in LO_{\mathbb {Z}}(\widehat {\Gamma })$
, the induced order
$<_{\widehat {\phi }}$
is also an element of
$LO_{\mathbb {Z}}(\widehat {\Gamma })$
. Let
$\widehat {{\gamma }}$
be an element of
$\widehat {\Gamma }$
. Since
$\mathbb {Z}$
is cofinal for
$<$
, there exist
$z^k$
and
$z^l$
such that
$z^k < \widehat {\phi }(\widehat {{\gamma }}) < z^l$
. By the definition of
$<_{\widehat {\phi }}$
, we have
$\widehat {\phi }^{-1}(z^k) <_{\widehat {\phi }} \widehat {{\gamma }} <_{\widehat {\phi }} \widehat {\phi }^{-1}(z^l)$
. Since
$\widehat {\phi }^{-1}(z^k), \widehat {\phi }^{-1}(z^k) \in \mathbb {Z}$
, the subgroup
$\mathbb {Z}$
is cofinal for
$<_{\widehat {\phi }}$
. This implies that
$\widehat {\phi } \ \colon LO_{\mathbb {Z}}(\widehat {\Gamma }) \to LO_{\mathbb {Z}}(\widehat {\Gamma })$
is well defined.
Let
$<$
be an element of
$LO_{\mathbb {Z}}(\widehat {\Gamma })$
. Let z be the generator of
$\mathbb {Z}$
satisfying
$1 <_{\widehat {\phi }} z$
. Note that
$\widehat {\phi }(z)$
is the generator of
$\mathbb {Z}$
satisfying
$1 < \widehat {\phi }(z)$
. Let
${\gamma }$
be an element of
$\Gamma $
and
$\widehat {{\gamma }} \in \widehat {\Gamma }$
be the unique lift of
${\gamma }$
satisfying
$1 \leq _{\widehat {\phi }} \widehat {{\gamma }} <_{\widehat {\phi }} z$
. Then, the element
$\widehat {\phi }(\widehat {{\gamma }})$
is the unique lift of
$\phi ({\gamma })$
satisfying
$1 = \widehat {\phi }(1) \leq \widehat {\phi }(\widehat {{\gamma }}) < \widehat {\phi }(z)$
.
For distinct elements
${\gamma }_1, {\gamma }_2, {\gamma }_3$
of
$\Gamma $
, let
$\sigma $
be the permutation such that
$$ \begin{align} 1 \leq_{\widehat{\phi}} \widehat{{\gamma}}_{\sigma(1)} <_{\widehat{\phi}} \widehat{{\gamma}}_{\sigma(2)} <_{\widehat{\phi}} \widehat{{\gamma}}_{\sigma(3)} <_{\widehat{\phi}} z. \end{align} $$
This implies that
$$\begin{align*}\pi_{\widehat{\Gamma}}^* \widehat{\phi}^* (<)({\gamma}_1, {\gamma}_2, {\gamma}_3) = \pi_{\widehat{\Gamma}}^*(<_{\widehat{\phi}})({\gamma}_1, {\gamma}_2, {\gamma}_3) = \mathrm{sign}(\sigma). \end{align*}$$
By (4.1), we obtain
$$\begin{align*}1 \leq \widehat{\phi}(\widehat{{\gamma}}_{\sigma(1)}) < \widehat{\phi}(\widehat{{\gamma}}_{\sigma(2)}) < \widehat{\phi}(\widehat{{\gamma}}_{\sigma(3)}) < \widehat{\phi}(z). \end{align*}$$
This implies that
$$\begin{align*}\phi^* \pi_{\widehat{\Gamma}}^*(<)({\gamma}_1, {\gamma}_2, {\gamma}_3) = \pi_{\widehat{\Gamma}}^*(<)(\phi({\gamma}_1), \phi({\gamma}_2), \phi({\gamma}_3)) = \mathrm{sign}(\sigma). \end{align*}$$
Hence, we obtain
$\pi _{\widehat {\Gamma }}^* \widehat {\phi }^* = \phi ^* \pi _{\widehat {\Gamma }}^*$
.
4.2 Proof of Theorem 1.2
Recall that
$$\begin{align*}\widehat{G} = \langle e_1, \ldots, e_k, h_1, \ldots, h_{2n}, z \mid e_1^{m_1} = \cdots = e_k^{m_k} = z, [h_1, z] = \cdots = [h_{2n}, z] = 1 \rangle. \end{align*}$$
The subgroup
$\langle z \rangle \cong \mathbb {Z}$
coincides with the center of
$\widehat {G}$
, and the quotient
$\widehat {G} / \langle z \rangle $
is isomorphic to G. Hence, we obtain a central extension
$$\begin{align*}0 \to \mathbb{Z} \to \widehat{G} \to G \to 1. \end{align*}$$
Lemma 4.4 Let a be an element of
$\ \mathbb {Z}_{\geq 0}$
satisfying items i) and ii) in Lemma 3.21 and set
$d = m_1 \ldots m_k \cdot a + 1$
. Let
$\rho ^{(d)} \ \colon G \to \operatorname {\mathrm {Homeo}}_+^{(d)} \subset \operatorname {\mathrm {Homeo}}_+(S^1)$
be the action defined in Section 3.4. Then, the pullback
$(\rho ^{(d)})^* (\mathrm {H}\widetilde {\mathrm {omeo}}_+(S^1))$
is isomorphic to
$\widehat {G}$
.
Proof It suffices to construct a homomorphism
$\widetilde {\rho ^{(d)}} \ \colon \widehat {G} \to \mathrm {H}\widetilde {\mathrm {omeo}}_+(S^1)$
such that the following diagram commutes:
By the definitions of d and
$\rho ^{(d)}$
, we have
$$\begin{align*}\operatorname{\mathrm{rot}}(\rho^{(d)}(e_l)) = \dfrac{m_li_l+1}{dm_l} = \dfrac{1}{m_l} \end{align*}$$
and
$\operatorname {\mathrm {rot}}(\rho ^{(d)}(h_i)) = 0$
. Let
$\widetilde {\rho ^{(d)}}(e_l) \in \mathrm {H}\widetilde {\mathrm {omeo}}_+(S^1)$
(resp.
$\widetilde {\rho ^{(d)}}(h_i) \in \mathrm {H}\widetilde {\mathrm {omeo}}_+(S^1)$
) be the lift of
$\rho ^{(d)}(e_l)$
(resp.
$\rho ^{(d)}(h_i)$
) with
$\widetilde {\operatorname {\mathrm {rot}}}(\widetilde {\rho ^{(d)}}(e_l)) = 1/m_l$
(resp.
${\widetilde {\operatorname {\mathrm {rot}}}(\widetilde {\rho ^{(d)}}(h_i)) = 0}$
), where
$\widetilde {\operatorname {\mathrm {rot}}}$
denotes the Poincaré translation number. Let
$\widetilde {\rho }(z) = T$
, the translation by one. Then, this
$\widetilde {\rho ^{(d)}}$
makes the above diagram commute.
Proof of Theorem 1.2
Let a be an element of
$\mathbb {Z}_{\geq 0}$
satisfying items i) and ii) in Lemma 3.21. By Proposition 3.23, Theorem 4.1, and Lemma 4.4, we obtain a left order
$<^{(d)} \in LO_{\mathbb {Z}}(\widehat {G})$
such that
$\pi _{\widehat {G}}^*(<^{(d)}) = c^{(d)}$
. Together with Theorem 4.2, the left order
$<^{(d)}$
is isolated in
$LO(\widehat {G})$
.
Assume that there exists an automorphism
$\widehat {\phi } \ \colon \widehat {G} \to \widehat {G}$
such that
$<^{(d)}$
coincides with
$(<^{(d')})_{\widehat {\phi }}$
. Since
$\widehat {\phi }$
preserves the center
$\langle z \rangle \cong \mathbb {Z}$
of
$\widehat {G}$
, the automorphism
$\widehat {\phi }$
descends to an automorphism
$\phi \ \colon G \to G$
. By Lemma 4.3, we obtain
$$\begin{align*}c^{(d)} = \pi_{\widehat{G}}^*(<^{(d)}) = \pi_{\widehat{G}}^*((<^{(d')})_{\widehat{\phi}}) = (c^{(d')})_{\phi}. \end{align*}$$
By Theorem 1.1, we have
$d = d'$
. This completes the proof.
