Group orderings, dynamics, and rigidity

arXiv:1607.00054v3 [math.DS] 12 Jul 2017

Kathryn Mann and Cristóbal Rivas∗

Abstract
Let G be a countable group. We show there is a topological relationship between
the space CO(G) of circular orders on G and the moduli space of actions of G on the
circle; and an analogous relationship for spaces of left orders and actions on the line.
In particular, we give a complete characterization of isolated left and circular orders
in terms of strong rigidity of their induced actions of G on S 1 and R.
As an application of our techniques, we give an explicit construction of infinitely
many nonconjugate isolated points in the spaces CO(F2n ) of circular orders on free
groups, disproving a conjecture from [1], and infinitely many nonconjugate isolated
points in the space of left orders on the pure braid group P3 , answering a question of
Navas. We also give a detailed analysis of circular orders on free groups, characterizing
isolated orders.

1

Introduction

Let G be a group. A left order on G is a total order invariant under left multiplication,
i.e. such that a < b implies ga < gb for all a, b, g ∈ G. It is well known that a countable
group is left-orderable if and only if it embeds into the group of orientation-preserving
homeomorphisms of R, and each left order on a group defines a canonical embedding up
to conjugacy, called the dynamical realization. Similarly, a circular order on a group G is
defined by a cyclic orientation cocycle c : G3 → {±1, 0} satisfying certain conditions (see
§2), and for countable groups this is equivalent to the group embedding into Homeo+ (S 1 ).
Analogous to the left order case, each circular order gives a canonical, up to conjugacy,
dynamical realization G → Homeo+ (S 1 ). This correspondence is the starting point for
a rich relationship between the algebraic constraints on G imposed by orders, and the
dynamical constraints of G–actions on S 1 or R. The correspondence has already proved
fruitful in many contexts; one good example is the relationship between orderability of
fundamental groups of 3-manifolds, and the existence of certain codimension one foliations
or laminations as shown in [3].
For fixed G, we let LO(G) denote the set of all left orders on G, and CO(G) the set of
circular orders. These spaces have a natural topology; that on CO(G) comes from its identification with a subset of the infinite product {±1, 0}G×G×G , and LO(G) can be viewed
as a further subset of this (see §2 for details). While CO(G) and LO(G) have previously
∗

K. Mann was partially supported by NSF grant DMS-1606254. C. Rivas acknowledges the support of
CONICYT via FONDECYT 1150691 and via PIA 79130017.

1

been studied with the aim of understanding the structures of orders on groups, our aim
here is to relate the spaces LO(G) and CO(G) to the moduli spaces Hom(G, Homeo+ (R))
and Hom(G, Homeo+ (S 1 )) of actions of G on the line or circle.
In many cases these moduli spaces are very poorly understood. An important case is
when G is the fundamental group of a surface of genus at least 2. Here Hom(G, Homeo+ (S 1 ))
has a topological interpretation (as the space of flat circle bundles over the surface), yet
it remains an open question whether Hom(G, Homeo+ (S 1 )) has finitely or infinitely many
connected components. Our work here shows that, for any group G, the combinatorial
object CO(G) is a viable tool for understanding the space of actions of G on S 1 .
In other cases, actions of G on the circle or line are easier to describe than circular
or left orders, and thus the dynamics of actions can serve as a means for understanding
the topology of LO(G) and CO(G). A good example to keep in mind is G = F2 , since
Hom(F2 , Homeo+ (S 1 )) ∼
= Homeo+ (S 1 ) × Homeo+ (S 1 ), but the topology of LO(F2 ) and
CO(F2 ) are not so obvious.
For any group G, the space LO(G) is compact, totally disconnected and, when G is
countable, also metrizable [17]. The same result holds by the same argument for CO(G).
Consequently the most basic question is whether LO(G) or CO(G) has any isolated points
– if not, it is homeomorphic to a Cantor set. That LO(F2 ) has no isolated points was
proved by McCleary [14] (see also [15]) and generalized recently to LO(G) where G is a
free product of groups in [19]. That CO(F2 ) has no isolated points either was conjectured
in [1]. Our techniques give a (perhaps surprising) easy disproof of this conjecture using
the dynamics of actions of F2 on S 1 .

Statement of results.
Given that CO(G) is totally disconnected and Hom(G, Homeo+ (S 1 )) typically has large
connected components, one might expect little correlation between the two spaces. Our
aim is to demonstrate that there is a strong, though somewhat subtle, relationship. A
first step, and key tool is continuity:
Proposition 1.1 (Continuity of dynamical realization). Let c be a circular order on a
countable group G, and ρ a dynamical realization of c. For any neighborhood of U of ρ in
Hom(G, Homeo+ (S 1 )), there exists a neighborhood V of c in CO(G) such that each order
in V has a dynamical realization in U .
An analogous result holds for left orders and actions on R. With this Proposition and
several other tools, we give a complete characterization of isolated left and circular orders
in terms of the dynamics (namely, rigidity) of their dynamical realization.
Theorem 1.2. Let G be a countable group. A circular order on G is isolated if and only
if its dynamical realization ρ is rigid in the following strong sense: for every action ρ′ sufficiently close to ρ in Hom(G, Homeo+ (S 1 )) there exists a continuous, degree 1 monotone
map h : S 1 → S 1 fixing the basepoint x0 of the realization, and such that ρ(g)◦h = h◦ρ′ (g)
for all g ∈ G.

2

The corresponding result for left orders is Theorem 3.11.
In the course of the proof of Theorem 1.2, we establish several other facts concerning the
relationship between Hom(G, Homeo+ (S 1 )) and CO(G). When combined with standard
facts about dynamics of groups acting on the circle, this gives new information about
spaces of circular orders. For example, we prove the following corollary, a special case of
which immediately gives a new proof of the main construction (Theorem 4.6) from [1].
Corollary 1.3. Suppose G ⊂ Homeo+ (S 1 ) is a countable group acting minimally, and
such that some point x0 has trivial stabilizer. Then the order on G induced by the orbit
of x0 is not isolated in CO(G).
In particular this shows that, if the dynamical realization of a circular order c on G is
minimal, then c is not isolated. (See Theorem 3.18 and following remarks.)
Isolated orders on free groups and braid groups. In Section 4 we undertake a
detailed study of the space of circular orders on free groups. As an application of Theorem
1.2, we show:
Theorem 1.4. The space of circular orders on the free group on 2n generators has infinitely many isolated points. In fact, there are infinitely many distinct classes of isolated
points under the natural conjugation action of F2n on CO(F2n ).
This disproves the conjecture about CO(F2 ) of [1]. An alternative, counterexample
to the F2 conjecture is given in Section 4.3, where we also give an explicit singleton
neighborhood of the isolated circular order constructed therein.
We also describe explicitly the dynamics of isolated orders on free groups. To state
this, let {a1 , a2 , ..., an } be a set of free generators for Fn .
Theorem 1.5. Suppose ρ is the dynamical realization of c ∈ CO(Fn ). The order c is
±1
isolated if and only if there exist disjoint domains D(s) ⊂ S 1 for every s ∈ {a±1
1 , ..., an },
each consisting a finite union of closed intervals, such that ρ(s)(S 1 \ D(s−1 )) ⊂ D(s) holds
for all s, and the basepoint of ρ is in the complement of the domains D(s).
Note that the conclusion of this theorem is exactly the condition in the classical pingpong lemma.
These dynamical realizations have a particularly nice description under the additional
assumption that the domains D(s) are connected sets:
Theorem 1.6. Let c ∈ CO(Fn ) have dynamical realization ρ that satisfies the hypotheses
of Theorem 1.5. If, additionally, the domains D(s) are connected, then n is even, and the
dynamical realization of c is topologically conjugate to a representation Fn → PSL(2, R) ⊂
Homeo+ (S 1 ) corresponding to a hyperbolic structure on a genus n/2 surface with one
boundary component. Moreover, each such representation Fn → PSL(2, R) arises as the
dynamical realization of an isolated circular order.

3

We note that no analog of Theorem 1.4 was previously known for any group, even
for left orders. In [15], Navas asked What can be said in general about the set of isolated
(left) orders on a group, up to conjugacy? For instance, is it always finite? A corollary of
Theorem 1.4 answers this in the negative:
Corollary 1.7. The pure braid group P3 ∼
= F2 × Z has infinitely many distinct conjugacy
classes of isolated left orders.
This is proved in § 5. We expect that the existence of isolated points is not unique to free
groups and braid groups and there should be many more examples. Further questions are
raised in Section 6.
Acknowledgements. The authors thank Adam Clay and Andres Navas for their feedback and interest in this work, and Shigenori Matsumoto for helpful comments, including
pointing out an error in a previous version.

2

Background material

In this section, we recall some standard facts about left and circular orders. A reader
familiar with orders may wish to skip this section, while the less comfortable reader may
wish to consult [2], [21], or in the case of left orders, [6] for further details.
Definition 2.1. [Cocycle definition of circular orders] Let S be a set . We say that
c : S 3 → {±1, 0} is a circular order on S if
i) c−1 (0) = △(S), where △(S) := {(a1 , a2 , a3 ) ∈ S 3 | ai = aj , for some i 6= j},
ii) c is a cocycle, that is c(a2 , a3 , a4 ) − c(a1 , a3 , a4 ) + c(a1 , a2 , a4 ) − c(a1 , a2 , a3 ) = 0 for
all a1 , a2 , a3 , a4 ∈ S.
A group G is circularly orderable if it admits a circular order c which is left-invariant in
the sense that c(u, v, w) = 1 implies c(gu, gv, gw) = 1 for all g, u, v, w ∈ G.
In other words, a circular order on G is a homogeneous 2-cocycle in the standard
complex for computing the integral Eilenberg-MacLane cohomology of G, which takes the
values 0 on degenerate triples, and ±1 otherwise.
This cocycle condition is motivated by the standard order cocycle or orientation cocycle
for points on the circle. Say an ordered triple (x, y, z) of distinct points in S 1 is positively
oriented if one can read points x, y, z in order around the circle counterclockwise, and
negatively oriented otherwise. Define the order cocycle ord : S 1 × S 1 × S 1 by

 1 if (x, y, z) is positively oriented
ord(x, y, z) =
−1 if (x, y, z) is negatively oriented

0 if any two of x, y and z agree.

It is easy to check that this satisfies the cocycle condition of Definition 2.1, and is invariant
under left-multiplication in S 1 . In fact, it is invariant under Homeo+ (S 1 ) in the sense that
ord(x, y, z) = ord(h(x), h(y), h(z)) for any orientation preserving homeomorphism h.
4

As mentioned in the introduction, the topology on the space CO(G) is that inherited
from the product topology on {±1, 0}G×G×G . A neighborhood basis of a circular order c
consists of the sets of the form {c′ ∈ CO(G) : c′ (u, v, w) = c(u, v, w) for all u, v, w ∈ S}
where S ranges over all finite subsets of G.
Left orders as “degenerate” circular orders. Recall that a left order on a group
G is a total order < invariant under left multiplication. Given (G, <), we can produce a
circular order on G by defining c< (g1 , g2 , g3 ) to be the sign of the (unique) permutation σ
of {g1 , g2 , g3 } such that σ(g1 ) < σ(g2 ) < σ(g3 ).
Observe that the left order c< above is a coboundary. Indeed, if c′ (x, y) equals 1
(respectively −1 or 0) when x < y (respectively y < x or x = y), then
c< (g1 , g2 , g3 ) = c′ (g2 , g3 ) − c′ (g1 , g3 ) + c′ (g1 , g2 ).
Conversely, if a circular order c on a group G is the coboundary of a left-invariant function
c′ : G2 → {±1, 0}, i.e. we have c(u, v, w) = c′ (v, w) − c′ (u, v) + c′ (u, v) for all u, v, w ∈ G,
then one can check that c′ defines a left order on G by x ≤ y if and only if c′ (x, y) ≥ 0.
Yet another characterization of the circular order c< obtained from a left order can be
found in [1, Proposition 2.17].
In this sense, we can view LO(G) as a subset of CO(G), and give it the subset topology.
It is not hard to see that this agrees with the original topology given in [17], full details of
this are written in [1]. Because of this, throughout this paper we frequently take circular
orders as a starting point, and treat left orders as a special case.
Dynamical realization. We now describe a procedure for realizing a circular order on
a group G as an order induced from an action of G on S 1 . This starts with the following
construction.
Construction 2.2 (Order embedding). Let G be a countable group with circular order
c, and let {gi } be an enumeration of elements of G. Define an embedding ι : G → S 1
inductively as follows. Let ι(g1 ) and ι(g2 ) be arbitrary distinct points. Then, having
embedded g1 , ..., gn−1 , send gn to the midpoint of the unique connected component of
S 1 \ {ι(g1 ), ...ι(gn−1 )} such that
c(gi , gj , gk ) = ord(ι(gi ), ι(gj ), ι(gk ))
holds for all i, j, k ≤ n.
Given an embedding ι : G → S 1 , as above, the left-action of G on itself now gives a
continuous, order preserving action by homeomorphism on ι(G) ⊂ S 1 , which extends to a
homeomorphism of the closure of ι(G) in S 1 . This can be extended to an action by homeomorphisms of S 1 , for instance by permuting the complimentary interval to the closure
of the image by linear maps. See [2, 6, 7] for more details. We denote the homomorphic
embedding just constructed by φι : G → Homeo+ (S 1 ).

5

Remark 2.3. One can check that taking a different enumeration of the elements of G, or
a different choice of ι(g1 ) and ι(g2 ) gives embeddings that are conjugate by an orientationpreserving homeomorphism of S 1 . The same is true for the corresponding homomorphic
embedding φι . Thus it is natural to make the following definition.
Definition 2.4 (Dynamical realization). Let G be a countable group with circular order
c. A dynamical realization of c is an embedding φ : G → Homeo+ (S 1 ) defined as φ =
h ◦ φι ◦ h−1 , where φι is an embedding constructed as above and h is any homeomorphism
of S 1 .
The basepoint of the dynamical realization φ is the point h(ι(id)), where id denotes
the identity element of G. Observe that φ(g)(h(ι(id))) = h(ι(g)).
By definition, dynamical realizations are all conjugate. Thus when discussing properties invariant under topological conjugacy, we often speak of the dynamical realization.
Note that a dynamical realization of a circular order always produces an action of G
such that any point in ι(G) has trivial stabilizer. Conversely, if G ⊂ Homeo+ (S 1 ) is such
that some point x0 has trivial stabilizer, then
c(g1 , g2 , g3 ) := ord(g1 (x0 ), g2 (x0 ), g3 (x0 ))
defines a circular order on G. This is simply the pullback of the order cocycle on S 1 under
the embedding of G via the orbit of x0 . We say that this is the order induced by the orbit
of x0 . In general, if G does not admit a free orbit, then the circular order on G/StabG (x0 )
can be extended to a circular order on G, for instance, by means of Lemma 3.13 below.
For a countable left-ordered group (G, <), there is an analogous construction of a dynamical realization G → Homeo+ (R). One first enumerates G, then defines an embedding
ι : G → R inductively by

 max{ι(gk ) : k < n} + 1 if gn > gk for all k < n
ι(gn ) =
min{ι(gk ) : k < n} − 1 if gn < gk for all k < n

the midpoint of [gi , gj ] if gi < gn < gj are successive among i, j < n
Alternatively, one can check that the construction given above for circular orders produces
an action on S 1 with a global fixed point whenever c = c< is a left order (i.e. degenerate)
cocycle. Identifying S 1 \ {∗} with R gives the dynamical realization of (G, <).
Conjugation. We conclude this section by defining the conjugation action of G on its
space of orders. This is important in the study of LO(G) and CO(G); indeed, a standard
technique to show that an order is not isolated is to approximate it by its conjugates
[15, 20].
Definition 2.5 (conjugate orders). Let c be a circular order on an arbitrary group G,
and let g ∈ G. The g-conjugate order is the order cg defined by cg (x, y, z) = c(xg, yg, zg).
An order is called conjugate to c if it is of the form cg for some g ∈ G.

6

Since orders are assumed to be left-invariant, we may equivalently define cg (x, y, z) =
c(g−1 xg, g−1 yg, g−1 zg). This gives an action of G on CO(G) by conjugation, and it is easy
to check that this is an action by homeomorphisms of CO(G). Note that a conjugate of a
left order is also a left order, so conjugation also gives an action of G on LO(G) by homeomorphisms. It follows directly from the definition that, given a dynamical realization of
c with basepoint x0 , a dynamical realization of cg is given by the same action of G on S 1 ,
but with basepoint g(x0 ).

3

A dynamical portrait of left and circular orders

We turn now to our main goal of studying the relationship between spaces of actions and
spaces of orders.
Let Hom(G, Homeo+ (S 1 )) denote the set of actions of a group G on S 1 by orientationpreserving homeomorphisms, i.e. the set of group homomorphisms G → Homeo+ (S 1 ).
This space has a natural topology; a neighborhood basis of an action ρ0 is given by the
sets
O(F,ǫ) (ρ0 ) := {ρ ∈ Hom(G, Homeo+ (S 1 )) | d(ρ(g)(x), ρ0 (g)(x)) < ǫ for all x ∈ S 1 , g ∈ F }
where F ranges over all finite subsets of G, and d is the standard length metric on S 1 . If G is
finitely generated, fixing a generating set S gives an identification of Hom(G, Homeo+ (S 1 ))
with a subset of Homeo+ (S 1 )|S| via the images of the generators, and the subset topology
on Hom(G, Homeo+ (S 1 )) agrees with the topology defined above.
Fixing some point p ∈ S 1 , the space Hom(G, Homeo+ (R)) of actions of G on R can be
identified with the closed subset
{ρ ∈ Hom(G, Homeo+ (S 1 ) : ρ(g)(p) = p for all g}
and its usual (compact–open) topology is just the subset topology. Given this, our primary
focus will be on the larger space Hom(G, Homeo+ (S 1 )) and its relationship with CO(G);
as the Hom(G, Homeo+ (R)) ↔ LO(G) relationship essentially follows by restricting to
subsets.
As mentioned in the introduction, due to the relationship between circular orders on
G and actions of G on S 1 given by dynamical realization, it is natural to ask about the
relationship between the two spaces CO(G) and Hom(G, Homeo+ (S 1 )), hoping that the
study of one may inform the other. For instance, one might (naively) propose the following.
Naive conjecture 3.1. Let G be a countable group. The space CO(G) has no isolated
points if (or perhaps if and only if) Hom(G, Homeo+ (S 1 )) is connected.
A supportive example is the case G = Z2 . It is not difficult to show both that
Hom(Z2 , Homeo+ (S 1 )) is connected and that CO(Z2 ) has no isolated points. This kind
of reasoning may have motivated the conjecture that CO(F2 ) has no isolated points, since
Hom(F2 , Homeo+ (S 1 )) is connected. However, our disproof of this conjecture for F2 (Theorem 1.4) shows that the naive reasoning is false.
7

Since dynamical realizations are faithful, one might try to improve the Conjecture
3.1 by restricting to the subspace of faithful actions of a group on S 1 . However, the
subset of faithful representations in Hom(F2 , Homeo+ (S 1 )) is also connected! To see this,
one can first show that the subset of faithful actions in Hom(F2 , Diff + (S 1 )) is connected,
open, and dense in Hom(F2 , Diff + (S 1 )) using a transversality argument, as remarked in
[16]. Since Hom(F2 , Diff + (S 1 )) is dense in Hom(F2 , Homeo+ (S 1 )), this implies that any
faithful action by homeomorphisms can be approximated by one by diffeomorphisms, and
hence the space of faithful actions in Hom(F2 , Homeo+ (S 1 )) is also connected.
As these examples show, the relationship between Hom(G, Homeo+ (S 1 )) and CO(G)
is actually rather subtle. Our next goal is to clarify this relationship.
Convention 3.2. For the remainder of this paper G will always denote a countable group.

3.1

The relationship between Hom(G, Homeo+ (S 1 )) and CO(G)

This section provides the groundwork for our dynamical characterization of isolated circular orders, starting with some basic observations. Let Hom(G, Homeo+ (S 1 ))/ ∼ denote the
quotient of Hom(G, Homeo+ (S 1 )) by the equivalence relation of conjugacy, and equipped
with the quotient topology. Recall that the dynamical realization of a circular order on a
countable group is well-defined up to conjugacy in Homeo+ (S 1 ). This defines a natural
“realization” map R : CO(G) → Hom(G, Homeo+ (S 1 ))/ ∼. Our first proposition is a
weaker form of Proposition 1.1 from the introduction.
Proposition 3.3 (Dynamical realization is continuous). The realization map R : CO(G) →
Hom(G, Homeo+ (S 1 ))/ ∼ is continuous.
Proof. Let c ∈ CO(G), and let ρ be a dynamical realization of c. Given a neighborhood
O(F,ǫ) of ρ in Hom(G, Homeo+ (S 1 )), we need to produce a neighborhood U of c in CO(G)
such that every circular order c′ ∈ U has a conjugacy representative of its dynamical
realization in the O(F,ǫ) –neighborhood of a conjugate of ρ.
Let S ⊂ G be a finite symmetric set with F ⊂ S and |S| > 1/ǫ. After conjugacy of ρ,
we may assume that ρ(S)(x0 ) partitions S 1 into intervals of equal length, each of length
less than ǫ. We now show that every circular order c′ that agrees with c on the finite set
S · S has a conjugate of a dynamical realization in the O(S,ǫ) ⊂ O(F,ǫ) – neighborhood of ρ.
Given such a circular order c′ , let ρ′ be a dynamical realization of c′ such that
ρ(g)(x0 ) = ρ′ (g)(x0 ) for all g ∈ S · S. Let s ∈ S. By construction ρ(s) and ρ′ (s) agree on
every point of ρ(S)(x0 ). We now compare ρ(s) and ρ′ (s) at other points. Let I be any
connected component of S 1 \S(x0 ). Note that ρ(s)(I) = ρ′ (s)(I). Let y ∈ I. If ρ(s)(I) has
length at most ǫ, then as ρ′ (s)(y) and ρ(s)(y) both lie in ρ(s)(I), they differ by a distance
of at most ǫ. If ρ(s)(I) has length greater than ǫ, consider instead the partition of ρ(s)(I)
by ρ(S)(x0 ) ∩ ρ(s)(I), this is a partition into intervals of length less than ǫ. As s−1 ∈ S
and ρ(s−1 ) and ρ′ (s−1 ) agree on ρ(S)(x0 ), considering preimages shows that ρ(s)(y) and
ρ′ (s)(y) must lie in the same subinterval of the partition, and hence differ by distance at
most ǫ.

8

Remark 3.4. Note that the same argument shows that dynamical realization is continuous as a map from LO(G) to the quotient of Hom(G, Homeo+ (R)) by conjugacy in
Homeo+ (R).
The next propositions discuss the partial “inverse” to the dynamical realization map
obtained by fixing a basepoint.
Notation 3.5. Let G be a countable group, and x0 ∈ S 1 . Let H(x0 ) ⊂ Hom(G, Homeo+ (S 1 ))
denote the subset of homomorphisms G → Homeo+ (S 1 ) such that x0 has trivial stabilizer.
Each ρ ∈ H(x0 ) induces a circular order on G using the orbit of x0 . This gives a
well-defined “orbit map” o : H(x0 ) → CO(G).
Proposition 3.6. The orbit map o : H(x0 ) → CO(G) is continuous and surjective.
Proof. To show continuity, given a finite set S ⊂ G and ρ ∈ H(x0 ), we need to find a
neighborhood U of ρ in H(x0 ) such that the cyclic order of ρ′ (S)(x0 ) agrees with that of
ρ(S)(x0 ) for all ρ′ ∈ U . But the existence of such a neighborhood follows immediately
from the definition of the topology on Hom(G, Homeo+ (S 1 )). Surjectivity of the orbit
map follows from the existence of a dynamical realization with basepoint x0 , as given in
Definition 2.4 and following remarks).
To describe the fibers of the orbit map, we will generalize the following statement
about left orders from [15, Lemma 2.8].
Lemma 3.7 (Navas, [15]). Let < be a left order on G, and ρ1 its dynamical realization
with basepoint 0 ∈ R. Let ρ2 ∈ Hom(G, Homeo+ (R)) be an action with no global fixed
point, and such that 0 has trivial stabilizer. The order induced by the ρ2 –orbit of 0 agrees
with < if and only if there is a non-decreasing, surjective map f : R → R, with f (0) = 0,
such that ρ1 (g)f = f ρ2 (g) for all g ∈ G.
The assumption that f (0) = 0 is omitted from the statement in [15], but it is necessary
and used in the proof.
For circular orders, we replace the non-decreasing, surjective map f above with a
continuous degree 1 monotone map of S 1 . This is defined as follows: A continuous map f :
S 1 → S 1 is degree 1 monotone if it is surjective and weakly order-preserving, meaning that
for all triples x, y, z, we have ord(f (x), f (y), f (z)) = ord(x, y, z) whenever f (x), f (y) and
f (z) are distinct points. One can check that this condition is equivalent to the existence of
a non-decreasing, surjective map f˜ : R → R that commutes with the translation x 7→ x+1,
and descends to the map f on the quotient R/Z = S 1 → S 1 .
Proposition 3.8. Let c1 be a circular order on G, and ρ1 its dynamical realization with
basepoint x0 . Let ρ2 ∈ Hom(G, Homeo+ (S 1 )) be such that x0 has trivial stabilizer. The
circular order induced by the ρ2 orbit of x0 agrees with c1 if and only if there is a continuous,
degree 1 monotone map f : S 1 → S 1 such that f (x0 ) = x0 and ρ1 (g) ◦ f = f ◦ ρ2 (g) for
all g ∈ G.

9

The degree 1 monotone map f is an example of a semi-conjugacy of S 1 . Proposition
3.8 says that the elements of H(x0 ) corresponding to the same circular order all differ by
such a semi-conjugacy. However, it is important to note that the relationship between ρ1
and ρ2 given in the proposition is not symmetric. Loosely speaking, ρ1 (the dynamical
realization) can be obtained from ρ2 by collapsing some intervals to points, but not viceversa. This gives a characterization of dynamical realizations as the “most minimal” (i.e.
those with “densest orbits”) actions among all actions with a given cyclic structure on an
orbit.
Proof of Proposition 3.8. Let ρ1 be a dynamical realization of c with basepoint x0 , and
let ρ2 be an action of G on S 1 . Suppose first that f is a degree 1 monotone map fixing
x0 , and such that ρ1 (g) ◦ f = f ◦ ρ2 (g) for all g ∈ G. Then the cyclic order of the orbits
of x0 under ρ1 (G) and ρ2 (G) agree. (Here we do not even need f to be continuous.)
For the converse, suppose that ρ2 defines the same circular order as ρ1 . Define a
map f : ρ2 (G)(x0 ) → ρ1 (G)(x0 ) by ρ2 (g)(x0 ) 7→ ρ1 (g)(x0 ). Following the strategy of
[15, Lemma 2.8], we show that f first extends continuously to the closure of the orbit
ρ2 (G)(x0 ), and can then be further extended to a continuous degree 1 monotone map.
Suppose that x is in the closure of the orbit of x0 under ρ2 (G). Then there exist gi in G
such that ρ2 (gi )(x0 ) → x; moreover we can choose these such that, for each i, the triples
ρ2 (g1 )(x0 ), ρ2 (gi )(x0 ), ρ2 (gi+1 )(x0 ) all have the same (positive or negative) orientation.
Assume for concreteness that these triples are positively oriented.
Since the orbit of x0 under ρ1 has the same cyclic order as that of ρ2 , the triples
ρ1 (g1 )(x0 ), ρ1 (gi )(x0 ), ρ1 (gi+1 )(x0 ) are also all positively oriented, so the sequence ρ1 (gi )(x0 )
is monotone (increasing) in the closed interval obtained by cutting S 1 at ρ1 (g1 )(x0 ). Thus
the sequence ρ1 (gi )(x0 ) converges to some point, say y. Define f (x) = y. This is well
defined, since if ρ2 (hi )(x0 ) is another sequence converging to x from the same side, we
can find subsequences gk and hk such that the triples ρ1 (g1 )(x0 ), ρ1 (hk )(x0 ), ρ1 (gk+1 )(x0 )
are positively oriented, so the monotone sequences ρ1 (gk )(x0 ) and ρ1 (gk )(x0 ) converge
to the same point. If instead ρ2 (hi )(x0 ) approaches from the opposite side, i.e. the
triples ρ2 (h1 )(x0 ), ρ2 (hi )(x0 ), ρ2 (hi+1 )(x0 ) are negatively oriented, then the midpoint construction from the definition of dynamical realization implies that the distance between
ρ1 (gi )(x0 ) and ρ1 (hi )(x0 ) approaches zero, so they converge to the same point. With this
definition, the function f is now weakly order preserving on triples, whenever it is defined.
If ρ2 (G)(x0 ) is dense in S 1 , this completes the definition of f . If not, for each interval I
that is a connected component of the complement of the closure of ρ2 (G)(x0 ), extend f over
I by defining it to be the unique affine map from I = (a, b) to the interval (f (a), f (b)) in
the complement of the closure of ρ1 (G)(x0 ). This gives a well defined continuous extension
that preserves the relation ρ1 (g) ◦ f = f ◦ ρ2 (g) and preserves the weak order preserving
property of f on ρ2 (G)(x0 ).
We conclude these preliminaries by showing the necessity of fixing the basepoint in
Propositions 3.6 and 3.8. Note that, if c is an order induced from an action of G on
S 1 with basepoint x0 , then the order induced from the basepoint g(x0 ) – which also has
trivial stabilizer – is precisely the conjugate order cg from Definition 2.5. However, one
10

may also change basepoint to a point outside the orbit of x0 . The following proposition
gives a general description of orders under change of basepoint, it will be used in the next
section. We use the notation stab(x) to denote the stabilizer of a point x.
Proposition 3.9 (Change of basepoint). Let G ⊂ Homeo+ (S 1 ) be a countable group.
Let x ∈ S 1 , and let {xi } be a sequence of points approaching x such that stab(xi ) = {id}
for all i.
i) If x has trivial stabilizer, then the circular orders from basepoints xi approach the
order from basepoint x. In particular, if xi are in the orbit of x under G, then the
order induced from the orbit of x can be approximated by its conjugates.
ii) If an interval I ⊂ S 1 satisfies stab(x) = {id} for all x ∈ I, then all choices of
basepoint in I give the same circular order.
Proof. i) Assume x has trivial stabilizer. Let S ⊂ G be any finite set. If xj is sufficiently
close to x, then the finite set {g(xj ) | g ∈ S} will have the same circular order as that of
{g(x) | g ∈ S}.
ii) Assume now that stab(x) = {id} for all x in a connected interval I. By part i),
under this assumption the map I → CO(G) given by sending a point x to the circular
order induced from x is continuous. Since I is connected and CO(G) totally disconnected,
this map is constant.
Remark 3.10. Note that if x has nontrivial stabilizer, then the set of accumulation points
of circular orders induced from such a sequence of basepoints need not be a singleton. For
an easy example, consider an action of Z on the circle such that some point x is a repelling
fixed point of the generator f of Z. Then for any points y and z close to x and separated
by x, we will have a positive cyclic order on one and only one of the triples (y, f (y), f 2 (y))
and (z, f (z), f 2 (z)).
Although such an action will not arise as the dynamical realization of any order on Z,
this behavior does occur for Z-subgroups of the dynamical realization of orders on many
groups, including F2 .

3.2

Proof of Theorem 1.2

We now prove the characterization theorem that was stated in the introduction:
Theorem 1.2. Let G be a countable group. A circular order on G is isolated if and only
if its dynamical realization ρ is rigid in the following strong sense: for every action ρ′ sufficiently close to ρ in Hom(G, Homeo+ (S 1 )) there exists a continuous, degree 1 monotone
map h : S 1 → S 1 fixing the basepoint x0 and such that ρ(g) ◦ h = h ◦ ρ′ (g) for all g ∈ G.
In particular, this implies that the dynamical realization of an isolated circular order
is rigid in the more standard sense that it has a neighborhood in Hom(G, Homeo+ (S 1 ))
consisting of a single semi-conjugacy class (see e.g. Definition 3.5 in [10]). We remark,
however, that this weaker form of rigidity does not entail that the ordering is isolated. For
instance, this weak rigidity holds for some non-isolated circular orders on fundamental
11

groups of hyperbolic closed surfaces and also on (some) solvable groups, see Example 3.25
and Example 3.27 below.
By minor modifications of the proof, we obtain the same result for left orders.
Theorem 3.11. Let G be a countable group. A left order on G is isolated if and only if its
dynamical realization ρ is rigid in the following strong sense: for every action ρ′ sufficiently
close to ρ in Hom(G, Homeo+ (R)) there exists a continuous, surjective monotone map
h : R → R fixing the basepoint x0 and such that ρ(g) ◦ h = h ◦ ρ′ (g) for all g ∈ G.
We prove Theorem 1.2 first, then give the modifications for the left order case. The
first step in the proof is a stronger version of the continuity of dynamical realization given
in Proposition 3.3.
Lemma 3.12 (Continuity of dynamical realization, II). Let c ∈ CO(G) and let ρ : G →
Homeo+ (S 1 ) be a dynamical realization of c based at x0 . Let U be any neighborhood of
ρ in Hom(G, Homeo+ (S 1 )). Then, there exists a neighborhood V of c in CO(G) such that
each order in V has a dynamical realization based at x0 contained in U .
Proof. We run a modification of the original argument in Proposition 3.3, which proved
a weaker result about the conjugacy class of ρ. Fix an enumeration ν : N → G such that
ν(0) = id, and let ρ : G → Homeo+ (S 1 ) be the dynamical realization of c build with it. Let
U be a neighborhood of ρ in Hom(G, Homeo+ (S 1 )). We may assume that U = O(F,ǫ) (ρ)
for some ǫ. Suppose first, that ρ(G)(x0 ) is dense in S 1 . Take S ⊂ G a finite set such that
ρ(g)ρ(S)(x0 ) is ǫ/2-dense in S 1 for any g ∈ F ,

(1)

and let n be such that T = ν({0, . . . , n}) contains F · S. In this way, if c′ is a circular
order of G coinciding with c over T , we can build a dynamical realization ρ′ of c′ using
enumeration ν that has the same base point as ρ, and moreover, has ρ′ (g) = ρ(g) on the
set ρ(S)(x0 ). Further, by (1) above, ρ′ belongs to the O(F,ǫ) -neighborhood of ρ.
In the case where ρ(G)(x0 ) is not dense, we claim that x0 (and hence all points in the
orbit of x0 ) are isolated points. To see this, suppose that x0 were not isolated. Then every
point of ρ(G)(x0 ) would be an accumulation point of the orbit ρ(G)(x0 ), and hence the
closure of the orbit is a Cantor set. In this case the “minimality” of dynamical realizations
given by Proposition 3.8 implies that ρ(G)(x0 ) is in fact dense. More concretely, if there is
some interval I in the complement of the closure of ρ(G)x0 , then collapsing each interval
ρ(g)(I) to a point produces a new action of G on S 1 that is not semi-conjugate to ρ by any
degree one continuous map f , contradicting Proposition 3.8. (A more detailed version of
this kind of argument is given in Lemma 3.21 and Corollary 3.24.)
Given that x0 is isolated, let t ∈ G be such that the oriented interval I := (x0 , ρ(t)(x0 ))
contains no other points from ρ(G)(x0 ). Note that, for each pair of distinct elements
g, h ∈ G, we have ρ(g)(I) ∩ ρ(h)(I) = ∅. This is because
ρ(g)(I) ∩ ρ(h)(I) 6= ∅ ⇔ I ∩ ρ(g −1 h)(I) 6= ∅
and since by definition I contains no points in the orbit of x0 , we must have ρ(g−1 h)(I) ⊃ I,
hence ρ(g−1 h)−1 (I) ⊂ I, contradicting the definition of I.
12

We use this observation to modify the construction from the proof of Proposition 3.6
as follows. Given ǫ and a finite set F ⊂ G, let S ⊂ G be a finite, symmetric set containing
F ∪ {id} and such that each interval J in the complement of the set
[
g(I)
g∈S·S

has length less than ǫ. Let c′ be a circular order that agrees with c on S · S. Then, as in
the case where ρ(G)(x0 ) is dense, we may build ρ′ a dynamical realization of c′ that agrees
with c on S · S(x0 ). In particular, this means that ρ(g)(I) = ρ′ (g)(I) for all g ∈ S · S and
that these intervals are pairwise disjoint.
For each g ∈ S ·S, let hg be the restriction of ρ′ (g)ρ(g)−1 to ρ(g)(I). Note that this is a
homeomorphism of ρ(g)(I) fixing each endpoint. Let h : S 1 → S 1 be the homeomorphism
defined by

hg (x) if x ∈ ρ(g)(I) for some g ∈ S · S
h(x) =
x
otherwise
Then hρ′ h−1 is also a dynamical realization of c′ (since it is the conjugate of a dynamical
realization by a homeomorphismn of S 1 ). We now show that hρ′ h−1 is in O(S,ǫ) (ρ).
Let s ∈ S. By construction, ρ(s) and hρ′ (s)h−1 agree on S(x0 ). Moreover, if x lies in
some interval ρ(t)(I) where t ∈ S, we have
hρ′ (s)h−1 (x) = ρ(st)ρ′ (st)−1 ρ′ (s)ρ′ (t)ρ(t)−1 (x) = ρ(s)(x)
so again ρ(s) and hρ′ (s)h−1 agree here. Finally, if x is not in any suchSinterval, then
g(I). Since
ρ(s)(x), (and hence hρ′ (s)h−1 (x) also) lies in the complement of the set
g∈S·S

both images ρ(s)(x) and hρ′ (s)h−1 (x) lie in the same complementary interval, which by
construction has length less than ǫ, they differ by a distance less than ǫ.
We are now in position to finish the proof of the Theorem.
End of proof of Theorem 1.2. Let c ∈ CO(G) be isolated, and let ρ be its dynamical
realization with basepoint x0 . Since c is isolated, there exists a finite set F ⊂ G such that
any order that agrees with c on F is equal to c. Because the orbit of x0 under ρ(F )(x0 )
is a finite set, there exists a neighborhood U of ρ in Hom(G, Homeo+ (S 1 )) such that, for
any ρ′ ∈ U , the cyclic order of the set ρ′ (F )(x0 ) agrees with that of ρ(F )(x0 ). Let ρ′ ∈ U .
If x0 has trivial stabilizer under ρ′ , then the cyclic order on G induced by the orbit of
x0 under ρ′ (G) agrees with c on F , so is equal to c. By Proposition 3.8, this gives the
existence of a map h as in the theorem.
If x0 instead has non-trivial stabilizer, say K ⊂ G, then the orbit of x0 under ρ′ (G)
gives a circular order on the set of cosets of K. Lemma 3.13 below implies that this can
be “extended” or completed to an order on G in at least two different ways, both of which
agree with c on F . In particular, one of the order completions is not equal to c. This
shows that c is not an isolated point, contradicting our initial assumption. This completes
the forward direction of the proof.
13

For the converse, assume that c is a circular order whose dynamical realization ρ
satisfies the rigidity condition in the theorem. Let U be a neighborhood of ρ so that
each ρ′ in U is semi-conjugate to ρ by a continuous degree one monotone map h as in
the statement of the theorem. Lemma 3.12 provides a neighborhood V of c such that
any c′ ∈ V has a dynamical realization in U . Proposition 3.8 now implies that the
neighborhood V consists of a single circular order, so c is an isolated point.
Lemma 3.13 (Order completions, see Theorem 2.2.14 in [2]). Let G ⊂ Homeo(S 1 ) and
let x be a point with stabilizer K ⊂ G. Any left order on K can be extended to circular
order on G that agrees with the circular order on cosets of K induced by the orbit of x.
We include a proof sketch for completeness.
Proof sketch. Suppose that <K is a left order on K. Define c(g1 , g2 , g3 ) = 1 whenever
ord(g1 (x), g2 (x), g3 (x)) = 1. When (g1 , g2 , g3 ) is a non-degenerate triple but g1 (x) =
g2 (x) 6= g3 (x), then one can declare c(g1 , g2 , g3 ) = 1 whenever g2−1 g1 <K . This determines
also the other cases where exactly two of the points gi (x) coincide. The remaining case
is when g1 (x) = g2 (x) = g3 (x), in which case we declare c(g1 , g2 , g3 ) to be the sign of the
permutation σ of {id, g1−1 g2 , g1−1 g3 } such that σ(id) < σ(g1−1 g2 ) < σ(g1−1 g3 ). Checking
that this gives a well-defined left-invariant circular order is easy and left to the reader.
We end this section with the modifications necessary for the left order version of this
theorem.
Proof of Theorem 3.11. Since any linear order is in particular a circular order, the argument from Lemma 3.12 can also be used to show the following
Let G be a countable group, and < a left order on G, with dynamical realization
ρ. Let U be a neighborhood of ρ in Hom(G, Homeo+ (R)). Then there exists
a neighborhood V of < in LO(G) such that each order in V has a dynamical
realization in U .
Combining this with Lemma 3.7 (in place of Proposition 3.8, which was used in the circular
order case) now shows that any order < on G with dynamical realization ρ satisfying the
rigidity assumption is an isolated left order.
For the other direction of the proof, assuming < is an isolated left order, one runs the
beginning of the proof of Theorem 1.2: since < is isolated, there exists a finite set F ⊂ G
such that any order that agrees with < on F is equal to <. Because the orbit of x0 under
ρ(F )(x0 ) is a finite set, there exists a neighborhood U of ρ in Hom(G, Homeo+ (R)) such
that, for any ρ′ ∈ U , the linear order of the set ρ′ (F )(x0 ) agrees with that of ρ(F )(x0 ).
Let ρ′ ∈ U . Order completion applied here again allows us to assume that x0 has trivial
stabilizer under ρ′ (applying order completion in this case will give a left rather than
circular order), and so the cyclic order on G induced by the orbit of x0 under ρ′ (G) agrees
with < on F , so is equal to <. By Lemma 3.7, this gives the existence of a map h as in
the theorem.

14

3.3

Convex subgroups and dynamics

Using our work above, we give some additional properties of isolated circular orders. For
this, we need to describe certain subgroups associated to a circular order. Recall that a
subgroup H of a left-ordered group is called convex if, whenever one has h1 < g < h2
with h1 , h2 in H and g ∈ G, then g ∈ H. We extend this to circularly ordered groups as
follows. (A similar definition appears in [8].)
Definition 3.14. A subgroup H of a circularly ordered group is convex if the restriction
of c to H is a left order, and if whenever one has c(h1 , g, h2 ) = +1 and c(h1 , id, h2 ) = +1
with h1 , h2 in H and g ∈ G, then g ∈ H.
In particular {id} is (trivially) a convex subgroup.
Lemma 3.15 (The linear part of an action). Let c be a circular order on G. Then there
is a unique maximal convex subgroup H ⊂ G. We call H the linear part of c.
Proof. Let ρ be a dynamical realization of a circular order c on G, with basepoint x0 . Note
that the definition of convex is easily seen to be equivalent to the following condition:
(∗) ρ(H) acts on S 1 with a fixed point, and if g(x0 ) lies in the connected component of
S 1 \ fix(ρ(H)) containing x0 , then g ∈ H.
In other words, if IH denotes the connected components of S 1 \ fix(ρ(H)) containing x0 ,
then H = {g ∈ G : g(x0 ) ∈ IH }. Identifying IH with the line, the induced linear order on
H agrees with the order on the orbit of x0 under ρ(H).
Now, assume that H and K are two proper convex subgroups. Then, since H ∪ K 6= G,
and g(x0 ) ∈
/ IH ∪ IK for g ∈ G \ (H ∪ K), we have that IH ∪ IK 6= S 1 . Therefore, the cyclic
order in G induces a linear order on H ∪ K that we denote <. Assume h ∈ H \K, then (up
to replacing h with its inverse) we have h(x0 ) > k(x0 ) for all k ∈ K, and h−1 (x0 ) < k(x0 )
for all k ∈ K. Thus, H ⊃ K. It follows that the union of convex subgroups is convex, and
so the union of all convex subgroups is the (unique) maximal element.
To give a more complete dynamical description of the linear part of a circular order,
we use the following general fact about groups acting on the circle.
Lemma 3.16. Let G be any group, and ρ : G → Homeo+ (S 1 ) any action on the circle.
Then there are three mutually exclusive possibilities.
i) There is a finite orbit. In this case, all finite orbits have the same cardinality.
ii) The action is minimal, i.e. all orbits are dense.
iii) There is a (unique) compact G–invariant subset K ⊂ S 1 , homeomorphic to a Cantor
set, and contained in the closure of any orbit. This set is called the exceptional
minimal set.
A proof can be found in [7, Sect. 5].
Proposition 3.17 (Dynamical description of the linear part). Let ρ be the dynamical
realization of a circular order c. According to the trichotomy above, the linear part of c
has the following description.
15

i) (finite orbit case) the linear part is the finite index subgroup of G that fixes any
finite orbit pointwise.
ii) (minimal case) the linear part is trivial.
iii) (exceptional case) the linear part is the stabilizer of the connected component of
S 1 \ K that contains x0 .
Proof. We use the alternative characterization (∗) of the linear part of an order given in
the proof of Lemma 3.15. Suppose first that ρ(G) has a finite orbit, say O, and let H be
the finite index subgroup fixing this orbit pointwise. Then H is a left-ordered subgroup
since it acts on S 1 with a fixed point, and H is convex with IH equal to the connected
component of S 1 \ O containing x0 . Moreover, for any g ∈
/ H, the action of ρ(g) has no
fixed point (it cyclically permutes the connected components of S 1 \ O and so g is not in
any convex subgroup. This shows that H is the maximal convex subgroup of G.
More generally, if H is a convex subgroup of any circular order on a group G, and
g∈
/ H, then ρ(g)(IH ) ∩ IH is always empty. To see this, note that the endpoints of IH are
accumulation points of ρ(H)(x0 ), so if ρ(g)(IH ) ∩ IH 6= ∅, then there exists some element
of the form ρ(gh)(x0 ) ∈ IH . Since H is convex, this means that g ∈ H.
This argument shows that IH is always a wandering interval for the action of ρ(G),
and in particular, that the action of G cannot be minimal. Moreover, when the maximal
convex subgroup His non-trivial, IH is the maximal (with respect to inclusion) wandering
interval containing x0 , namely a connected component of the complement of the exceptional minimal set. This completes the proof.
We now state and prove the main result in this section.
Theorem 3.18. If c is an isolated circular order on an infinite group G, then c has
nontrivial linear part, and the induced left order on the linear part is an isolated left
order.
In particular, using Proposition 3.17, this implies that the dynamical realization of
any isolated circular order on any group G cannot be minimal. This was Corollary 1.3
stated in the Introduction, and it immediately gives a new proof of the main construction
(Theorem 4.6) of [1]. The fact that the linear part of an isolated circular order has an
isolated linear order implies also that it has a nontrivial Conradian soul, as defined in [15],
and that the soul admits only finitely many left orders.
For the proof, we will use a consequence of the following theorem of Margulis.
Proposition 3.19 (Margulis, [11]). Let G be any group, and ρ : G → Homeo+ (S 1 ) any
action on the circle. Either ρ(G) preserves a probability measure on S 1 , or ρ(G) contains
a nonabelian free subgroup.
The construction of the nonabelian subgroup comes from the existence of contracting
intervals, meaning intervals I such that there exists a sequence gn in G such that the
diameter of gn (I) approaches 0. An exposition of the proof can be found in [7]. In the
case where the action of G is minimal, one can find contracting intervals containing any
point. Put more precisely, we have the following corollary of the proof given in [7].
16

Proposition 3.20. [Ghys [7]] Let ρ : G → Homeo+ (S 1 ) be a minimal action on the circle.
Either ρ(G) is conjugate to a group of rotations, or the following condition holds: For any
point x ∈ S 1 , there is a neighborhood I of x such that for any y ∈ S 1 , there is a sequence
of elements gy,n ∈ G such that gy,n (I) converges to y as n tends to ∞.
We say that the neighborhood I in the above proposition is a contractible neighborhood
of x.
In order to prove Theorem 3.18, we also need to describe dynamical realizations with
exceptional minimal set. We state this as a separate lemma; it will be used again in the
next section.
Lemma 3.21 (Condition for dynamical realizations). Suppose that ρ : G → Homeo+ (S 1 )
is such that x0 has trivial stabilizer, and ρ has an exceptional minimal set K. If ρ is a
dynamical realization of a circular order with basepoint x0 , then each connected component
I of S 1 \ K contains a point of the orbit ρ(G)(x0 ) and has nontrivial stabilizer in G.
Proof. Let ρ : G → Homeo+ (S 1 ) be a dynamical realization with basepoint x0 and exceptional minimal set K. Since ρ(G) permutes the connected components of S 1 \K, it suffices
to show that each interval of S 1 \ K contains at least two points in the orbit ρ(G)(x0 ).
Suppose for contradiction that some connected component I of S 1 \ K contains one or
no points in the orbit ρ(G)(x0 ). Then, for each g ∈ G, the connected component ρ(g)(I)
of S 1 \ K also contains at most one point in the orbit of x0 . Collapsing each interval
ρ(g)(I) to a point gives a new circle, on which G acts by homeomorphisms. Let ρ2 denote
this new action, and note that ρ2 (G)(x0 ) has the same cyclic order as ρ(G)(x0 ). More
precisely, let h : S 1 → S 1 be a map such that, for
S each g ∈ G, the image h(ρ(g)(I)) is
a singleton, h is injective on the complement of g ρ(g)(I) and h fixes x0 . Define ρ2 by
h ◦ ρ(g) = ρ2 (g) ◦ h.
Since ρ is a dynamical realization, Proposition 3.8 gives a continuous degree one monotone f : S 1 → S 1 fixing x0 and such that f ◦ ρ2 (g) = ρ(g) ◦ f . In other words, f is an
inverse for h on the orbit of x0 . However, since the endpoints of I lie in the exceptional
minimal set K, they are also in the closure of the orbit ρ(G)(x0 ). But h(I) is a point, so
f cannot be continuous at this point. This gives the desired contradiction.
Proof of Theorem 3.18. Suppose that c is an isolated circular order on G. We use proposition 3.16 to describe its dynamical realization ρ. Let x0 be the basepoint of ρ. First, we
will show that ρ cannot be minimal. Assume for contradiction that ρ is minimal. Then
Proposition 3.20 implies that either G is abelian and ρ conjugate to an infinite group of
rotations, or we have an interval I containing x0 and, for each y ∈ S 1 a sequence contracting I to {y}. The rotations case gives an order which is not isolated, this is shown
in Lemma 3.22 below. (While it is relatively easy to see that such an order on an abelian
group is not isolated if, say, G has rank at least one, it is more difficult for groups like the
group of rotations by nπ/2k , n, k ∈ Z and deserves a separate lemma.)
Thus, we now have only to deal with the second case of a contractible interval. By
the proof of Proposition 3.9, there is an interval I0 = [a, b] containing x0 in its interior,
such that the circular order induced from any point y ∈ I with trivial stabilizer coincides
17

with c. By taking a smaller neighborhood of x0 if necessary, we may assume that I is
a contractible neighborhood of x0 . Let y1 , y2 and z in I be such that y1 , z, y2 , x0 are in
counterclockwise order. By minimality, we may assume that all these points have trivial
stabilizer. Let (gi,n )n (i = 1, 2) be a sequence of elements in G such that gi,n (I) converges
to yi . Let n ≥ 0 be such that
g1,n (I), {z}, g2,n (I), {x0 }
are in counterclockwise order. Then, c(id, g1,n , g2,n ) = 1, but if cz is the circular order
induce from z, then cz (id, g1,n , g2,n ) = −1, contradicting our assumption on I that any
order induce form a point in it must coincide with c. Thus the dynamical realization of
an isolated circular order cannot be minimal.
Now we describe the case when ρ is not minimal. By Lemma 3.16, it has either a finite
orbit or invariant Cantor set. In the first case, since G is infinite, x0 is not an element of
a finite orbit, and so c has a nontrivial linear part. In the second case, Proposition 3.21
implies that x0 lies in the complement of the exceptional minimal set K, and that the
stabilizer of the connected component of S 1 \ K containing x0 is nontrivial. Hence, c has
nontrivial linear part.
Now let H ⊂ G denote the linear part of c. If the left order on H were not isolated
in LO(H), then order completion from Lemma 3.13 allows us to approximate c in CO(G)
using left orders on H approaching the restriction of c to H. Thus, the left order on H
must be isolated.
It remains only to prove the lemma on abelian groups.
Lemma 3.22 (Infinite groups of rotations are not isolated). Let G be an infinite group,
and suppose ρ(G) ⊂ SO(2) ∼
= R/Z is the dynamical realization of a circular order c. Then
c is not isolated.
Proof. Identify G with its image in the additive group R/Z. Note that the order on
G ⊂ R/Z ∼
= S 1 agrees with the usual cyclic order on points of S 1 .
Case i. G is not a torsion group. Let Ĝ ⊂ R be the set of lifts of elements to R, so we have
a short exact sequence 0 → Z → Ĝ → G → 0, and consider the vector space V over Q
generated by Ĝ ⊂ R. By assumption, G is not torsion, so V 6⊂ Q. Let λ ∈ V \ Q. Choose
λ′ ∈ R linearly independent over Q from V , and define φ : V → R by λq 7→ λ′ q for q ∈ Q,
and α 7→ α for any α in the complement of the span of λ. Then φ(Ĝ) ∼
= Ĝ as additive
groups, and φ descends to an embedding of G in R/Z with a different cyclic order. This
order can be made arbitrarily close to the original one by taking λ′ as close as we like to
λ.
Case ii. G ⊂ Q/Z.
In this case G is an infinite abelian torsion group and we can
decompose it into a direct sum of groups G = ⊕Gp , where Gp is the group of all elements
who have order a power of p, for each prime p. These are simply the elements a/pk ∈ R/Z.
We use the following basic fact.
Fact 3.23. Let Ap = {a/pk : k ∈ N} ⊂ R/Z. Then, for any k, the function x 7→ x + pk x
is an automorphism of Ap .
18

P∞
i
To see this, one checks easily that any map x 7→
i=0 ai p x gives a well defined
endomorphism.
(For fixed x ∈ Ap , all but finitely many terms in the formal power series
P
i
ai p x vanish mod Z). Since 1 + pk is invertible in the p-adic integers, x 7→ x + pk x has
an inverse, so is an automorphism.
Given any finite subset S of G, we can find some Gp ⊂ G and N > 0 such that
Gp ∩ {a/pk : k > N } is nonempty and does not contain any element of S. Let k be the
smallest integer greater than N such that Gp contains an element of the form a/pk . Using
the fact above, define a homomorphism G → R/Z to be the identity on Gq for q 6= p,
and to be x 7→ x + pk−1 x on Gp . Note that this is well defined, injective, restricts to the
identity on S, and changes the cyclic order of elements of Gp .
We conclude this section with a converse to Lemma 3.21 and corollary to the proof of
Proposition 3.8. It will also be useful in the next section.
Corollary 3.24. Suppose that ρ : G → Homeo+ (S 1 ) is an action with exceptional minimal set K, that ρ(G) acts transitively on the set of connected components of S 1 \ K, and
that, for some component I, the stabilizer of I is nontrivial and acts on I as the dynamical
realization of a linear order with basepoint x0 ∈ I. Then ρ is the dynamical realization of
a circular order with basepoint x0 .
Proof. Let ρ be as in the proposition and let x0 ∈ I. Then x0 has trivial stabilizer.
Suppose for contradiction that ρ is not a dynamical realization, and let ρ2 be the dynamical
realization of the circular order on G given by the orbit of x0 under ρ(G). Proposition
3.8 then gives the existence of a continuous degree one monotone map f : S 1 → S 1 such
that f (x0 ) = x0 and ρ2 (g) ◦ f = f ◦ ρ(g). If f is a homeomorphism, then ρ is a dynamical
realization. Thus, f must not be injective.
Given that f is not injective, the construction of f in the proof of Proposition 3.8 implies that there is a connected component J of the complement of the closure of ρ2 (G)(x0 )
such that f (J) is a singleton, say y. Since the exceptional minimal set K is contained in
the closure of ρ2 (G)(x0 ), it follows that J lies inside a connected component of S 1 \ K.
After conjugacy, we may find such an interval J such that J ⊂ I. But then, since by
assumption ρ restricts to a dynamical realization of the stabilizer stabG (I), Lemma 3.7
provides a continuous surjective map fˆ : I → I such that ρ(g) ◦ fˆ = fˆ ◦ ρ2 (g) for every g
fixing I. But fˆ is local inverse of f , so it cannot be continuous. This provides the desired
contradiction.

3.4

Illustrative non-examples

We give some examples to show that Theorem 1.2 does not hold when ρ is assumed to
have a related or slightly weaker form of rigidity.
Example 3.25. Let G be the fundamental group of a closed surface Σg of genus g ≥ 2,
and ρ : G → PSL(2, R) ⊂ Homeo+ (S 1 ) the homomorphism arising from a hyperbolic
structure on Σg . This is an embedding of G into PSL(2, R) as a cocompact lattice. Let
x0 ∈ S 1 be a point with trivial stabilizer. Such a point exists, as there are only countably
19

many points with nontrivial stabilizer, each one an isolated fixed point of an infinite cyclic
subgroup of G. Then the orbit of x0 induces a circular order, say c0 , on G, and since ρ is
minimal Proposition 3.8 implies that ρ is a dynamical realization of c0 with basepoint x0 .
By Corollary 1.3, this is not an isolated circular order. However, the action does have a
form of rigidity, which we now describe.
The main theorem of Matsumoto [12], together with minimality of ρ implies that there
exists a neighborhood U of ρ in Hom(G, Homeo+ (S 1 )) such that, for all ρ′ in U , there
exists a continuous, degree one monotone map h such that h ◦ ρ′ = ρ ◦ h. This is quite
similar to our “strong rigidity”, except that here h will not generally fix the basepoint x0
To see directly that c0 is not isolated, one can change the basepoint as in Remark 3.10
to approximate c0 by its conjugates. This corresponds to conjugating ρ by some small
homeomorphism h that does not fix x0 .
One can modify the group G in the example above to give a dynamical realization
ρ of a circular order c with nontrivial linear part (and isolated left order on the linear
part!) and so that ρ still has some form of rigidity – but where c fails again to be an
isolated circular order due to basepoint considerations. Here is a brief sketch of one such
construction.
Example 3.26. Let G = π1 (Σg ) × Z. Define ρ : G → Homeo+ (S 1 ) by starting with
the action of π1 (Σg ) defined above, then “blowing up” each point in the orbit of x0 ,
replacing it with an interval (i.e. performing the Denjoy trick), and inserting an action of
Z by translations supported on these intervals, that commutes with the action of π1 (Σg ).
Corollary 3.24 shows that these are dynamical realizations of the circular order on G
obtained from the orbit of x0 .
Moreover, much like in the case above, one can argue from Matsumoto’s result that
any nearby action ρ′ of G on S 1 is semi-conjugate to ρ, in the sense that both can be
collapsed to a common minimal action where the Z factor acts trivially. However, ρ is not
an isolated circular order; one can produce arbitrarily nearby orders by performing the
same construction, but blowing up the orbit of a nearby point instead of x0 , and choosing
the basepoint there.
Our last example concerns circular orders on solvable groups.
Example 3.27. Consider the Baumslag-Solitar groups BS(1, 2) = ha, b | aba−1 = b2 i
acting by ρ(a) : x 7→ 2x and ρ(b) : x 7→ x + 1 for x ∈ R ∪ {∞} = S 1 . As in Example 3.25,
ρ is a dynamical realization of any circular order induced from a point x0 having trivial
stabilizer. For this, one can take x0 to be any point in R \ Q.
We claim that the representation ρ is rigid, in the sense that all nearby actions of
BS(1, 2) on the circle are semi-conjugate to it. Indeed, from the invariance under conjugation of rotation number, b always has a fixed point in S 1 . Furthermore, since the fixed
points of ρ(a) are hyperbolic, and therefore stable, for any representation ρ′ close to ρ,
the element ρ′ (a) has a fixed point. By iterating a fixed point of ρ′ (b) under ρ′ (a−1 ), we
obtain a global fixed point of ρ′ . This implies that, up to semi-conjugacy, we can view ρ′
as an action of BS(1, 2) on R ∪ {∞} fixing ∞. Moreover, in this model, ρ′ (a) has a fixed
20

point, say p, on R and we have ρ′ (b)(p) > p. But now, in [18] it is shown that BS(1, 2)
has only four semi-conjugacy classes of actions on the line. Two of them giving actions
where a has no fixed points, and the other two are affine actions: one in which b is a
negative translation and the other in which b is a positive translation. This implies that ρ′
is semi-conjugate to ρ. However, the ordering on BS(1, 2) induced from x0 is not isolated,
it is approximated by its conjugates in the same way as in Example 3.25.
Similar arguments can be applied to many orderings on (not necessarily affine) solvable
groups. See [20].

4

Circular orders on free groups

In this section, we use the results of Section 3 to show that there are infinitely many
nonconjugate circular orders on free groups of even rank, and characterize the dynamical
realizations of isolated circular orders on free groups generally, proving Theorems 1.4 and
1.5.
We start with a definition related to the conditions in Theorem 1.5.
Definition 4.1. Let a1 , a2 , ..., an ∈ Homeo+ (S 1 ). We say these elements have ping-pong
dynamics if there exist pairwise disjoint closed sets D(ai ) and D(a−1
i ) such that, for each
i, we have

ρ(ai ) S 1 \ D(a−1
i ) ⊂ D(ai ).

−1
1
We call any such sets D(ai ) and D(a−1
i ) satisfying ai S \ D(ai ) ⊂ D(ai ) attracting
±1
domains for ρ(ai ) and ρ(a−1
i ) respectively, and use the notation D(ai ) to denote D(ai ) ∪
−1
D(ai ). These attracting domains need not be connected. In this case, we use the notation
D1 (s), D2 (s), ... for the connected components of D(s). Note that the definition of ping±1
pong dynamics implies that for each s ∈ {a±1
1 , ..., an }, and for each domain Dk (t) with
−1
t 6= s , there exists a unique j such that s(Dk (t)) lies in the interior of Dj (s).
The terminology is motivated by the following lemma, a version of the classical pingpong lemma.
Lemma 4.2 (Ping-pong). Let a1 , a2 , ..., an ∈ Homeo+ (S 1 ) have ping-pong dynamics.
Then a1 , ..., an generate a free group. More precisely, for any x0 not in the interior of
an attracting domain D(a±1
i ), the orbit of x0 is free and its cyclic order is completely
±1
determined by: i) the cyclic order of the sets Dj (s) and {x0 }, for s ∈ {a±1
1 , ..., an }, and
ii) the collection of containment relations
s(Dk (t)) ⊂ Dj (s),
s(x0 ) ∈ Dj (s)
Proof of Lemma 4.2. Let w1 , w2 and w3 be distinct reduced words in the letters ai and
a−1
i . We need to show that the cyclic order of the triple w1 (x0 ), w2 (x0 ), w3 (x0 ) is well
defined (i.e. x0 has trivial stabilizer) and completely determined by the cyclic order of,
and containment relations among, x0 and the connected components of the attracting
domains.
21

We proceed by induction on the maximum word length of wi . For the base case, if each
wi is either trivial or a generator, then either wi (x0 ) = x0 (trivial case), or wi (x0 ) ∈ Dj (wi ),
for some j determined by the containment relations. Since the wi are distinct, no two of
the points w1 (x0 ), w2 (x0 ) and w3 (x0 ) lie in the same domain D(wi ). Thus, all are distinct
points, and the cyclic order of the sets Dj (s) and {x0 } determines the cyclic order of the
triple.
For the inductive step, assume that for all triples of reduced words wi of length at most
k, the cyclic order of the triple w1 (x0 ), w2 (x0 ), w3 (x0 ) is determined, and assume also that
the points wi (x0 ) have wi (x0 ) ∈ Dj (wi ) for some j determined by wi . (For completeness
and consistency, consider {x0 } to be the attracting domain for the empty word). Let
w1 , w2 , w3 be reduced words of length at most k + 1.
±1
Write wi = si vi , where si ∈ {a±1
1 , ...an }, so vi is a word of length at most k. (If wi is
empty, skip this step, and set vi to be the empty word.) We have vi (x0 ) ∈ Dk (ti ) for some
k, and since wi is a reduced word, ti 6= si . By inductive hypothesis, the sets Dk (ti ) and
cyclic order of the points vi (x0 ) are known. Finally, we have wi (x0 ) = si vi (x0 ) ∈ si (Dk (ti )),
and si (Dk (ti )) ⊂ Dj (si ) for some j given by the containment relations.
If these sets Dj (si ) are distinct, then we are done since their cyclic order is known. We
may also ignore the case where all wi have the same initial first letter s, since the cyclic order of the triple s−1 w1 (x0 ), s−1 w2 (x0 ), s−1 w3 (x0 ) agrees with that of w1 (x0 ), w2 (x0 ), w3 (x0 ).
So we are left with the case where exactly two of the three domains Dj (si ) containing wi (x0 )
agree. For concreteness, assume w1 and w2 start with s and w3 does not, and w1 (x0 ) and
w2 (x0 ) lie in Dj (s).
Consider the shorter reduced words v1 = s−1 w1 , v2 = s−1 w2 and s−1 . By hypothesis,
the cyclic order of the points v1 (x0 ), v2 (x0 ), s−1 (x0 ) is determined. This order agrees
with the order of the triple w1 (x0 ), w2 (x0 ), x0 . As x0 ∈
/ Dj (s) and w3 (x0 ) ∈
/ Dj (s), this
order is the same as that of w1 (x0 ), w2 (x0 ), w3 (x0 ). Thus, the cyclic order of the triple
w1 (x0 ), w2 (x0 ), w3 (x0 ) is determined by this initial configuration.
The most basic and familiar example of ping-pong dynamics is as follows.
Example 4.3. Let D(a), D(a−1 ), D(b), and D(b−1 ) be disjoint closed intervals in S 1 . Let
a and b be orientation-preserving homeomorphisms of the circle such that
a(S 1 \ D(a−1 )) ⊂ D(a),
b(S 1 \ D(b−1 )) ⊂ D(b).

(2)

By Lemma 4.2, any point x0 in the complement of the union of these attracting domains
induces a circular order on F2 by c(w1 , w2 , w3 ) = ord(w1 (x0 ), w2 (x0 ), w3 (x0 )).
Remark 4.4. Note that there are two dynamically distinct cases in the construction
−1
above: either we can
components of
 take D(b) and D(b ) to lie in different connected

1
−1
S \ D(a) ∪ D(a ) , or to lie in the same connected component of S 1 \ D(a) ∪ D(a−1 ) .
Convention 4.5. For the remainder of this section, whenever we speak of actions with
ping-pong dynamics, we assume that each attracting domain D(s) has finitely many connected components.
22

The next Proposition shows that ping-pong dynamics come from isolated circular orders, proving one direction of Theorem 1.5.
Proposition 4.6. Let ρ : Fn → Homeo+ (S 1 ) be the dynamical realization
of a circular
S
order c, with basepoint x0 . If ρ has ping-pong dynamics and x0 ∈ S 1 \ i D(a±1
i ), then c
is isolated in CO(Fn ).
Proof. Let ρ : Fn → Homeo+ (S 1 ) be the dynamical realization of a circular order c, with
basepoint x0 , and assume that ρ has ping-pong dynamics. Following the convention above,
let D1 (s), D2 (s), ... denote the (finitely many) connected components of D(s), so for each
±1
−1 and each connected component D (t), there exists j and i
s ∈ {a±1
k
1 , ...an }, each t 6= s
such that we have containment relations of the form
◦

◦

ρ(s)(Dk (t)) ⊂ D j (s) and ρ(s)(x0 ) ∈ D i (s)

(3)

◦

Here we use the notation D i (s) for the interior of Di (s), and in this proof we will use the
fact that the image lies in the interior.
We now show these dynamics are stable under small perturbations. To make this
′
±1
precise, for each s ∈ {a±1
1 , ...an } and each connected component Dk (s), let Dk (s) be an
◦

ǫ–enlargement of Dk (s), with ǫ chosen small enough so that Dk (s) ⊂ D ′ k (s), but all the
domains Dk′ (s) remain pairwise disjoint. Now if ρ′ (s) is sufficiently close to ρ(s), then we
will have
ρ′ (s)(S 1 \ D ′ (s−1 )) ⊂ ρ′ (s)(S 1 \ D(s−1 )) ⊂ D ′ (s).
◦

◦

Moreover, as in (3), we will also have ρ′ (s)(Dk′ (t)) ⊂ D ′ j (s) and ρ′ (s)(x0 ) ∈ D ′ i (s) (with
the same indices).
Thus, there exists a neighborhood U of ρ in Hom(Γ, Homeo+ (S 1 )) such that any ρ′ ∈ U
is a ping-pong action for which the sets Dj′ (s) may be taken as the connected components
of attracting domains. Moreover, these components are in the same cyclic order as the
components Dj (s), the containments from equation (3) are still valid, and x0 is in the
same connected component of the complement of the domains. Fix such a neighborhood
U of ρ.
By Lemma 4.2, the cyclic order of the orbit of x0 depends only on the cyclic order of
′
′
the domains Dj′ (a±1
i ) and x0 , so for each ρ ∈ U , the cyclic order of ρ (Fn )(x0 ) agrees with
that of ρ(Fn )(x0 ). Now by Lemma 3.12, there exists a neighborhood V of c in CO(Fn )
such that any order c′ ∈ V has a dynamical realization ρ′ with basepoint x0 such that
ρ′ ∈ U . As we observed above, the order of the orbit ρc (F2 )(x0 ) agrees with that of
ρ(F2 )(x0 ), and so the two circular orders agree.
Although much less obvious, the converse to Proposition 4.6 is also true:
Proposition 4.7. Suppose ρ is a dynamical realization of an isolated circular order
on Fn = ha1 , a2 , ..., an i. Then there exist disjoint closed sets D(s) ⊂ S 1 for every
±1
s ∈ {a±1
1 , ..., an }, each consisting a finite union of intervals, and each disjoint from the
basepoint, such that ρ(s)(S 1 \ D(s−1 )) ⊂ D(s) holds for all s.
23

This proposition gives the other direction of Theorem 1.5.
Proof of Proposition 4.7. Assume that ρ is the dynamical realization of an isolated circular
order on Fn , and let x0 be the basepoint for ρ. We start by proving the following claim.
To simplify notation, here Fn (x0 ) denotes the orbit of x0 under ρ(Fn ), and Fn (x0 ) denotes
its closure.
Claim. For any y ∈ Fn (x0 ) \ {x0 }, there exists a neighborhood U of y in S 1 , and s ∈
±1
{a±1
1 , ..., an } such that, for each point ρ(g)(x0 ) ∈ U ∩ Fn (x0 ), the (reduced) word g in
±1
±1
{a1 , ..., an } has the same initial letter s.
We prove this claim by contradiction. If the claim is not true, then there is some set
of the form
±1
{ρ(sw)(x0 ) : sw is a reduced word in a±1
1 , ..., an }
±1
(where s ∈ {a±1
1 , ..., an } is fixed), with points of the form ρ(tv)(x0 ) t 6= s, arbitrarily close
to its closure. In other words, given any ǫ > 0, we can find two points ρ(sw)(x0 ) and
ρ(tv)(x0 ), with s 6= t, that are distance at most ǫ/2 apart. Write w = wk wk−1 ...w1 as a
±1
reduced word in the letters a±1
1 , ..., an , and similarly write v = vl vl−1 ...v1 , and consider
all the images of x0 under initial strings of these words, i.e. the points ρ(wk′ ...w1 )(x0 )
and ρ(vl′ ...v1 )(x0 ) for k′ ≤ k and l′ ≤ l. We may assume that none of these points lie in
the shorter than ǫ/2-length interval between ρ(sw)(x0 ) and ρ(tv)(x0 ) – otherwise, we may
replace sw and tv with two of these initial strings, say u and u′ , that still have different
initial letters, so that ρ(u)(x0 ) and ρ(u′ )(x0 ) have distance less than ǫ/2 apart, and so that
the images of x0 under initial strings of u and u′ do not lie in the small interval between
ρ(u)(x0 ) and ρ(u′ )(x0 ).
Now we modify ρ by replacing ρ(s) with hρ(s), where h is a homeomorphism supported
on a small neighborhood of the interval between ρ(sw)(x0 ) and ρ(tv)(x0 ). Choose h such
that the triple hρ(s)ρ(w)(x0 ), ρ(tv)(x0 ), x0 has the opposite orientation from the triple
ρ(sw)(x0 ), ρ(tv)(x0 ), x0 . Additionally, we may take the interval where h is supported to
be small enough to not contain any point of the form ρ(wk′ ...w1 )(x0 ) or ρ(vl′ ...v1 )(x0 ). We
leave the images of the other generators unchanged. Call this new action ρ′ .
Even though ρ(s) 6= ρ′ (s) and s may appear as a letter in v or w, we claim that
the triple ρ′ (sw)(x0 ), ρ′ (tv)(x0 ), x0 does indeed have the opposite orientation from the
triple ρ(sw)(x0 ), ρ(tv)(x0 ), x0 . This is because the support of h is disjoint from all points
ρ(wk′ ...w1 )(x0 ) and ρ(vl′ ...v1 )(x0 ), so one can see inductively that in fact ρ′ (wk′ ...w1 )(x0 ) =
ρ(wk′ ...w1 )(x0 ) and ρ′ (vl′ ...v1 )(x0 ) = ρ(vl′ ...v1 )(x0 ) for all k′ ≤ k and l′ ≤ l. It follows that
ρ′ (sw)(x0 ) = hρ(sw)(x0 ), and ρ′ (tv)(x0 ) = ρ(tv)(x0 ). (It is easy to check that this even
works if t = s−1 .) Thus, by definition of h, the triples have opposite orientations.
We have just shown that, for any ǫ > 0, there exists an action of Fn on S 1 such that
the image of any point under any generator of Fn is at most distance ǫ from the original
action ρ, and yet the cyclic order of the orbit of x0 under the new action is different. Given
any finite subset of Fn we can choose ǫ small enough so that the new order on the orbit
of x0 will agree with the previous one on this finite set. Order completion now gives an

24

arbitrarily close circular order to the original, hence could not have been isolated. This
completes the proof of the claim.
Now we finish the proof of the proposition. Assume that ρ is the dynamical realization
of an isolated circular order with basepoint x0 . We construct the domains D(s) in three
steps.
Step 1. For each s, declare that D(s) contains every point of the form ρ(sw)(x0 ), and
every point y in the closure of {ρ(sw)(x0 ) : sw a reduced word in a±1
i }. The claim we just
proved implies that accumulation points of {ρ(sw)(x0 ) : sw a reduced word in a±1
i } are
disjoint from those of {ρ(tw)(x0 ) : tw a reduced word in a±1
}
for
s
=
6
t.
i
Step 2. If I is a connected component of the complement of Fn (x0 ), and the endpoints
of I are both in the already constructed D(s), then declare I ⊂ D(s). Note that the sets
D(s) are pairwise disjoint, and so far we have ρ(s)(D(t)) ⊂ D(s) for every t 6= s−1 .
Step 3. As defined so far, the sets D(s) cover all of S 1 except for some intervals complementary to Fn (x0 ), precisely, those intervals with boundary consisting of one point in
D(s) and the other point in D(t) for some s 6= t. (There is also an exceptional case where
one of the endpoints is x0 , i.e. allowing for the empty word.) Our first claim, i.e. that
every point in Fn (x0 ) has a neighborhood containing only points of {ρ(sw)(x0 ) : sw} for
some fixed s, implies, since S 1 is compact, that there are only finitely many such complementary intervals. Also, since ρ is a dynamical realization, it is not possible to have a
complementary interval I with both endpoints in Fn (x0 ) \ Fn (x0 ) (see Construction 2.2).
Further, form Theorem 3.18, it follows that the isolated circular order associated to ρ has
a non-trivial linear part which is isomorphic to Z (since higher rank free groups has no
isolated left orders, see [15]), thus there is an element hmin ∈ Fn , such that the (small)
interval Imin = (x0 , hmin (x0 )) has no point of Fn (x0 ) in its interior. The same holds for
g(Imin ) for any g ∈ Fn . This implies that any complementary interval of the sets D(s)
must have each endpoint inside Fn (x0 ).
Suppose that I is such a complementary interval, so I is of the form (s1 s2 ...sk (x0 ), t1 t2 , ...tl (x0 ))
±1
where s1 s2 ...sk and t1 t2 ...tl are reduced words in the generators a±1
1 , ..., an and s1 6= t1 .
−1
−1
This implies that s1 (I) = (s2 ...sk (x0 ), s1 t1 t2 ...tl (x0 )) is also a complementary interval
−1
−1
since s1 ...sk is a reduced word i.e. s2 6= s−1
1 . Similarly, t1 (I) = (t1 s1 s2 ...sk (x0 ), t2 ...tl (x0 ))
−1
is also a complementary interval. Note that, for any u ∈
/ {s−1
1 , t1 } the endpoints of u(I)
are (reduced) words beginning in u, so u(I) ⊂ D(u).
Iterating this argument, it follows that, for any proper initial string s1 ...sp with p < k,
the interval (s1 ...sp )−1 (I) is complementary, as are its images under s−1
p+1 and sp . The
−1
−1
interval (s1 ...sk ) (I) = (x0 , (s1 ...sk ) t1 ...tl (x0 )) is also complementary, and also its image under sk , but its images under every other generator are contained in some already
defined D(s). An analogous statement holds for images of I under initial strings of t1 ...tl .
In particular, this reasoning shows that hmin = (s1 ...sk )−1 t1 ...tl , and that any complementary interval is equal to the image of (x0 , hmin (x0 )) under the inverse of a prefix of hmin .
For convenience, we switch to some simpler notation, writing hmin = u1 u2 ...un , so that
the complementary intervals are
−1
−1
(x0 , u1 . . . un (x0 )), (u−1
1 (x0 ), u2 . . . un (x0 )), . . . , (un . . . u1 (x0 ), x0 ).

25

We now show how to enlarge the domains D(s) to contain parts of these complementary
intervals, in order to have them satisfy the “ping-pong condition” ρ(s)(S 1 \D(s−1 )) ⊂ D(s),
for each of the generators s. We remind the reader that, for every other interval J in (an
already defined portion of) a set D(s), and for every t 6= s−1 , we have t(J) ⊂ D(t).
Thus, after extending domains we need only check what happens on the complementary
intervals.
−1
First, choose p1 ∈ (x0 , u1 . . . un (x0 )) and add [p1 , u1 . . . un (x0 )] to D(u1 ) and [u−1
1 (x0 ), u1 (p1 )]
−1
to D(u1 ). Note that this is coherent with Step 1. Note also that in this way we have
−1
−1
−1
that u−1
1 (Imin \ D(u1 )) ⊂ D(u1 ) and that u1 (u1 (Imin ) \ D(u1 )) ⊂ D(u1 ) (so in par−1
−1
ticular s(Imin \ D(u1 )) ⊂ D(s) and s(u1 (Imin \ D(u1 )) ⊂ D(s) for every generator
s). Then, we choose p2 ∈ (u−1
1 (p1 ), u2 . . . un (x0 )) and add [p2 , u2 . . . un (x0 )] to D(u2 ) and
−
−1
−1
[u−1
u
1(x
),
u
(p
)]
to
D(u
0
2
2
1
2
2 ).
We repeat this procedure iteratively; in the j th step we choose pj ∈ (u−1
j−1 (pj−1 ), uj . . . un (x0 ))
−1
−1
−1
and add [pj , uj . . . un (x0 )] to D(uj ) and [uj . . . u1 (x0 ), uj (pj )] to D(u−1
j ). This proce−1
dure ends with the choice of pn ∈ (un−1 (pn−1 ), un (x0 )) and the addition of [pn , un (x0 )] to
−1
−1
−1
D(un ) and the addition of [u−1
n . . . u1 (x0 ), un (pn )] to D(un ). It is easy to check that
the domains D(s)’s remain disjoint, and that now the ping pong condition is satisfied
globally.
Fixing n, it is now relatively easy to produce infinitely many non-conjugate actions
of Fn on S 1 by varying the number of connected components and cyclic orientation of
domains D(s) for each generator s. Moreover, these can be chosen such that taking the
orbit of a point produces infinitely many nonconjugate circular orders on Fn (the reader
may try this as an exercise, otherwise we will see some explicit examples shortly). However,
the existence of such actions is not enough to prove that CO(Fn ) has infinitely many
nonconjugate isolated points. This is because not every ping-pong action arises as the
dynamical realization of a circular order; therefore Proposition 4.6 does not automatically
apply.
As a concrete example, only one of the two cases in Remark 4.4 is the dynamical
realization of a circular order. We will soon see that a circular order on F2 produced by
Construction 4.3 when D(b) and D(b−1 ) lie in different connected components of S 1 \
D(a) ∪ D(a−1 ) is isolated, but one produced when D(b) and D(b−1 ) lie in the same
connected component is not isolated.
To illustrate this point, and as a warm up to the proof of Theorem 1.4, we now prove
exactly which ping-pong actions come from dynamical realizations in the simple case where
the domains D(s) are all connected.

4.1

Schottky groups and simple ping-pong dynamics

Definition 4.8. Say that an action of Fn on S 1 has simple ping-pong dynamics if the generators satisfy the requirements of a ping-pong action, and there exist connected attracting
domains D(s) for all generators and inverses.
This short section gives a complete characterization of dynamical realizations with
26

simple ping-pong dynamics. To do this, we will use some elementary hyperbolic geometry
and results on classical Schottky subgroups of PSL(2, R). Although our exposition aims
to be self-contained, a reader looking for more background can refer to [4, Ch. 1 and 2]
for a good introduction.
On ping-pong in PSL(2, R). There is a natural action of PSL(2, R) on S 1 = R∪{∞} by
Möbius transformations. A finitely generated subgroup G ⊂ PSL(2, R) is called Schottky
exactly when it has ping-pong dynamics. The benefit of working in PSL(2, R) rather than
Homeo+ (S 1 ) is that Möbius transformations of the circle extend canonically to the interior
of the disc. Considering the interior of the disc as the Poincaré model of the hyperbolic
plane, Schottky groups act properly discontinuously by isometries. Thus, it makes sense to
describe the hyperbolic surface obtained by quotient the disc by such a ping-pong action.
We will prove the following.
Theorem 4.9. Let c ∈ CO(Fn ). If the dynamical realization ρc of c has simple pingpong dynamics, then n is even, and ρc is topologically conjugate to a representation
ρ : Fn → PSL(2, R) ⊂ Homeo+ (S 1 ) corresponding to a hyperbolic structure on a genus
n/2 surface with one boundary component.
Note that the conclusion of the theorem is (as it should be) independent of any choice
of generating set for Fn , even though the definition of ping-pong dynamics is phrased in
terms of a specific set of generators.
As a special case, Theorem 4.9 immediately gives many concrete examples of actions
of free groups on S 1 that do not arise as dynamical realizations of any circular order (c.f.
Lemma 4.11) and justifies the remarks at the end of the previous subsection.
The main idea of the proof of Theorem 4.9 can be summarized as follows: For PSL(2, R)
actions, the condition that the quotient has a single boundary component exactly captures
the condition that Fn acts transitively on the connected components of the complement
of the exceptional minimal set, i.e. the condition of Corollary 3.24. For general ping-pong
actions in Homeo+ (S 1 ), we use the “minimality” property of being a dynamical realization
(Proposition 3.8) to produce a conjugacy into PSL(2, R), then cite the PSL(2, R) case.
As motivation and as a first step in the proof, we start with an example and a nonexample.
Lemma 4.10 (Example of dynamical realization). Let a1 , b1 , a2 , b2 , ..., an , bn denote generators of F2n . Let ρ : F2n → PSL(2, R) ⊂ Homeo+ (S 1 ) have simple ping-pong dynamics.
Suppose that x0 and the attracting domains are in the cyclic (counterclockwise) order
−1
−1
−1
x0 , D(a1 ), D(b1 ), D(a−1
1 ), D(b1 ), D(a2 ), D(b2 ), ..., D(an ), D(bn ).

Then ρ is the dynamical realization of a circular order with basepoint x0 .
Note that the n = 1 case is one of the cases from Remark 4.4.

27

D(a−1
1 )
D(b1 )

b1
D(b−1
1 )
D(a1 )
γ̃

γ

a1

x0

Figure 1: The open disc as universal cover Σ̃. The shaded area is a fundamental domain
Proof. By the ping-pong lemma, x0 has trivial stabilizer under ρ(F2n ), so its orbit defines
a circular order c on F2n . We claim that the dynamical realization of this circular order
is ρ. To see this, we first describe the action of ρ more concretely using some elementary
hyperbolic geometry. Having done this, we will be able to apply Corollary 3.24 to the
action.
The arrangement of the attracting domains specified in the lemma implies that the
quotient of H2 by ρ(F2n ) is, topologically, the interior of a surface Σ of genus n with one
boundary component. Geometrically, it is a surface of infinite volume with a singe end,
as illustrated in Figure 1 for the case n = 1. (This is elementary; see Proposition I.2.17
and discussion on page 51 of [4] for more details.)
Let γ ⊂ Σ be a simple geodesic curve isotopic to the boundary. The complement of
γ in Σ has two connected components, a compact genus n surface with boundary γ (call
this surface Σ′ ), and an infinite volume annulus, call this A. Fixing a basepoint on Σ and
considering γ as a based curve lets us think of it as an element of π1 (Σ) = F2n , and its
image ρ(γ) ∈ PSL(2, R) is a hyperbolic element translating along an axis, γ̃ ⊂ H2 , which
projects to the curve γ ⊂ Σ.
We now describe the exceptional minimal set for ρ in terms of the geometry of this
surface. It is a standard fact that the exceptional minimal set is precisely the limit points
of the universal cover of the compact part Σ̃′ (see [4, Sec. 3.1, Prop 3.6]). This can also
be described by looking at images of γ̃ under the action of ρ(F2n ). These images are
disjoint curves, each bounding on one side a fundamental domain for the compact surface
Σ′ , and on the other a half-plane that contains a fundamental domain for the annulus A.
The intersection of the half-planes that cover A with the S 1 boundary of H2 make up the
complement of the exceptional minimal set for ρ(F2n ).
What is important to take from this discussion is that there is a single orbit for the
permutation action of ρ(F2n ) on the complementary intervals to its exceptional minimal
set, and this corresponds to the fact that Σ has a single boundary component, in our case
bounded by γ. As the exceptional minimal set is contained in the interior of the union of
28

±1
the attracting domains D(a±1
i ) and D(bi ), the basepoint x0 for the dynamical realization
lies in a complementary interval I. In our case, taking γ to be the based curve represented
by the commutator [a, b],1 our specification of the cyclic order of x0 and the attracting
domains were chosen so that x0 lies in the interval bounded by the lift γ̃ that is the axis of
this commutator. The configuration is summarized in Figure 1. Moreover, ρ(γ) preserves
I, acts on it by translations, and generates the stabilizer of I in ρ(F2n ).
In summary, S 1 is the union of an exceptional minimal set contained in the closure of
ρ(G)(x0 ) and the orbit of the open interval I. As the action of hγi on I is the dynamical
realization of a left order, Corollary 3.24 implies that ρ is the dynamical realization of
c.

By contrast, geometric representations from surfaces with more boundary components
do not arise as dynamical realizations.
Lemma 4.11 (Non-example). Let ρ : Fn → PSL(2, R) ⊂ Homeo+ (S 1 )) be such that
H2 /ρ(Fn ) is a surface with more than one infinite volume (i.e. non-cusped) end. Then ρ
is not the dynamical realization of a circular order on Fn .
A particular case of the lemma is the ping-pong action of F2 = ha, bi with attracting domains in cyclic order D(a), D(a−1 ), D(b), D(b−1 ). There the quotient H2 /ρ(F2 ) is
homeomorphic to a sphere minus three closed discs.
Proof. Let ρ be as in the Lemma, and let Σ = H2 /ρ(Fn ). Let γ1 , γ2 , ... be geodesic simple
closed curves homotopic to the boundary components of Σ. As in the proof of Lemma 4.10,
ρ(Fn ) has an exceptional minimal set, and the connected components of the complement
of this set are the boundaries of disjoint half-planes, bounded by lifts of the curves γi to
geodesics in H2 . For fixed i, the union of these half-planes bounded by lifts of γi is a
ρ(Fn )-invariant set.
Since we are assuming that there is more than one boundary component, for any
candidate for a basepoint x0 , there is some i such that x0 is not contained in an interval
bounded by a lift of γi . Equivalently, there is a connected component of the complement
of the exceptional minimal set that does not contain a point in the orbit of x0 . By
Lemma 3.21, this implies that ρ cannot be a dynamical realization of a circular order with
basepoint x0 .
Remark 4.12. Given ρ as in Lemma 4.11, and a point x ∈ S 1 with trivial stabilizer,
we get a circular order on Fn from the orbit of x. The lemma simply says that ρ is not
its dynamical realization. But it is not hard to give a positive description of what the
dynamical realization actually is: it is the action obtained from ρ by collapsing every
interval in the complement of the exceptional minimal set to a point, except those that
contain a point in the orbit of x. The resulting action is conjugate to a representation
ρ′ : Fn → PSL(2, R) such that H2 /ρ′ (Fn ) is a surface with either all (or all but one) infinitevolume end of the surface H2 /ρ(Fn ) replaced by a finite-volume cusp Put otherwise, either
1

here we use the convention for the action that [a, b](x) = b−1 a−1 ba(x)

29

for all i or all but one i, we have that ρ′ (γi ) is parabolic rather than hyperbolic. Both
cases do in fact occur; for instance, the cusp-only case arises by taking the orbit of a point
with trivial stabilizer under ρ′ , since such an action ρ′ is already minimal Proposition
3.8 implies that it is the dynamical realization of this order. We leave the details as an
exercise for the reader.
One can also show that such an action ρ′ is not rigid, by finding an arbitrarily small
deformation of ρ′ so that a parabolic element ρ′ (γi ) becomes an infinite order elliptic. This
new action is no longer semi-conjugate to the original – in fact, even the circular order of
the orbit of x under the subgroup generated by γi will change.
Now we can prove Theorem 4.9; the goal in the proof is to reduce an arbitrary action
to the examples from the PSL(2, R) case considered in the previous two lemmas.
Proof of Theorem 4.9. Let ρ : Fn → Homeo+ (S 1 ) be the dynamical realization of a circular order, with basepoint x0 . Assume that ρ has simple ping-pong dynamics. Let D(a±1
i )
−1
be the attracting domains for the generators ai and inverses ai .
Define a representation ρ′ : Fn → PSL(2, R) ⊂ Homeo+ (S 1 ) by setting ρ′ (ai ) to be the
unique hyperbolic element such that ρ′ (ai )(D(ai )) = ρ(ai )(D(ai )). Then ρ′ is a simple
±1
′
ping-pong action, and we can take D ′ (a±1
i ) = D(ai ) to be the attracting domains for ρ .
We will now show that
i) there exists f ∈ Homeo+ (S 1 ) such that f ◦ ρ′ (g) ◦ f −1 = ρ(g) for all g ∈ Fn , and
ii) the quotient of H2 by ρ′ is a genus n/2 surface with one boundary component, in
particular, n is even.
This will suffice to prove the theorem.
By Lemma 4.2, the cyclic order of ρ(Fn )(x0 ) and ρ′ (Fn )(x0 ) agree, so the map f :
ρ(Fn )(x0 ) → ρ′ (Fn )(x0 ) given by f (ρ(g)(x0 )) = ρ′ (g)(x0 ) is cyclic order preserving. We
claim that, similarly to Proposition 3.8, f extends continuously to the closure of ρ(Fn )(x0 ).
To see this, we look at successive images of attracting domains, following [4, Prop. 1.11].
±1
If w = s1 s2 ...sk is a reduced word with si ∈ {a±1
1 , ...an } define the set D(s1 , s2 , ..., sk )
by D(s1 , ..., sk ) := ρ(s1 ...sk−1 )D(sk ). Make the analogous definition for D ′ (s1 , ..., sk−1 ) :=
◦

ρ′ (s1 , ..., sk−1 )D ′ (sk ). Note that D(s1 , s2 , ..., sk ) ⊂ D(s1 , s2 , ...sk−1 ).
Let gk ∈ Fn be a sequence of distinct elements such that lim ρ(gk )(x0 ) = y. Write
i→∞

each gk as a reduced word in the generators and their inverses. Since the set of generators
and inverses is finite, after passing to a subsequence gki , we may assume that there exist
and such that gki = s1 s2 ...si . Thus,
s1 , s2 , ... with si+1 6= s−1
i
ρ(gki )(x0 ) ∈ D(s1 , s2 , ..., si ) and
y∈

∞
\

D(s1 , s2 , ..., si ).

i=1
◦

Returning to our original sequence, as D(s1 , s2 , ..., si ) ⊂ D(s1 , s2 , ..., si−1 ), and ρ(gk )(x0 ) →
y, for any fixed i, we will have ρ(gk )(x0 ) ∈ D(s1 , s2 , ..., si ) for all sufficiently large k.
30

Since the containment relations among the domains ρ(sj )(D(si )) and ρ′ (sj )(D ′ (si ))
agree, it follows that ρ′ (gk )(x0 ) ∈ D ′ (s1 , s2 , ..., si ) whenever ρ(gk )(x0 ) ∈ D(s1 , s2 , ..., si ).
Since ρ′ is a ping-pong action with image in PSL(2,
T R), it follows from hyperbolic geometry
(see Lemma 1.10 in [4]) that the intersection i D ′ (s1 , s2 , ..., si ) is a single point, say
y ′ . Thus, ρ′ (gk )(x0 ) converges to y ′ , and we may define f (y) = y ′ , giving a continuous
extension of f .
Since ρ is a dynamical realization, as in the proof of Lemma 3.21, Proposition 3.8
implies that f has a continuous inverse and so must be a homeomorphism onto its image.
We may then extend f over each complementary interval to the orbit of x0 , to produce a
homeomorphism of S 1 such that f ρ′ (g)f −1 = ρ(g) for all g ∈ Fn .
Topologically, the quotient of H2 by ρ′ (Fn ) is the interior of an orientable surface Σ
with π1 (Σ) = Fn , and therefore genus g and b = n−2g +1 boundary components, for some
g. Because ρ′ has simple ping-pong dynamics, it has at least one boundary component.
We showed in Lemma 4.11 that, if ρ′ is a dynamical realization, then there is at most
one boundary component. Since we are assuming that ρ, and hence its conjugate ρ′ is a
dynamical realization, there is exactly one boundary component, and so n = 2g is even.
This proves assertion ii.

4.2

Infinitely many nonconjugate circular orders

Finally, we give the proof that CO(F2n ) has infinitely many nonconjugate isolated points.
These will come from ping-pong actions with disconnected domains that are “lifts” of the
Schottky actions described in the previous section.
Proof of Theorem 1.4. Let G = F2n , and let ρ : G → PSL(2, R) be a representation such
that H2 /ρ(G) is an infinite volume genus n surface with one end. Fix a generating set
a1 , b1 , ..., an , bn for G.
Definition 4.13. Fix k > 1. The standard k-lift ρ̂ of ρ is a representation defined as
follows: for each generator s let ρ̂(s) be the unique lift of ρ(s) to the k-fold cyclic cover of
S 1 such that ρ̂(s) has fixed points. This determines a homomorphism ρ̂ : G → Homeo+ (S 1 )
whose image commutes with the order k rigid rotation.
±1
Let D(a±1
i ), D(bi ) be the attracting domains for the generators and their inverses
with respect to the representation ρ. For each generator or inverse s, let D̂(s) be the
pre-image of an attracting D(s) under the (k-fold) covering map π : S 1 → S 1 . Then D̂(s)
is a (disconnected) attracting domain for ρ̂(s), and ρ̂ has ping-pong dynamics. Let x̂0 be
a lift of x0 . The next lemma shows that, for infinitely many choices of k, this lift ρ̂ is the
dynamical realization of a circular order with basepoint x̂0 . Having done this, Proposition
4.6 implies that this circular order is isolated in CO(F2n ).

Lemma 4.14. If k and 2n − 1 are relatively prime, then the standard k-lift ρ̂ constructed
above is the dynamical realization of a circular order.

31

Proof. Again, we begin by describing the dynamics of ρ̂. All of the facts stated here follow
easily from the construction of ρ̂ as a lift of ρ. Further explanation and a more detailed
description of such lifts can be found in Section 2 of [9].
Let K be the exceptional minimal set for ρ. Then K̂ := π −1 (K) is the exceptional
minimal set for ρ̂, and x̂0 ∈ S 1 \ K̂. Let I denote the connected component of S 1 \ K
containing x0 , and Iˆ the connected component of S 1 \ K̂ containing x̂0 . Using Corollary
3.24, it suffices to show that ρ̂(G) acts transitively on the set of connected components of
S 1 \ K̂ and that the stabilizer of Iˆ acts on Iˆ as a dynamical realization of a linear order.
Let Iˆ = I0 , and let I1 , I2 , ..., Ik−1 denote the other connected components of π −1 (I),
in cyclic (counterclockwise) order.
Let g ∈ F2n be the product of commutators g = [a1 , b1 ]...[an , bn ], so ρ(g) is the stabilizer
of I in ρ(G). For the lifted action, we have ρ̂(g)(Ii ) = Ij , where j = i + 2n − 1 mod k.
(This follows from the fact that the rotation number of ρ̂(g) is (2n − 1)/k, see [9].) Hence,
the stabilizer of Iˆ is the infinite cyclic subgroup generated by gk . Moreover, ρ̂(g k ) acts on
Iˆ without fixed points, as it is a lift of ρ(g k ) which acts on I without fixed points. Thus,
the action of the subgroup generated by gk acts on Iˆ as the dynamical realization of a left
order on Z.
It remains only to show that ρ̂(G) acts transitively on the set of connected components
of S 1 \ K̂ so that we can apply Corollary 3.24. This is where we use the hypothesis that
k and 2n − 1 are relatively prime. Assuming that they are relatively prime, the fact that
ρ̂(g)(Ii ) = Ij , where j = i + 2n − 1 mod k will now imply that ρ̂ acts transitively on the
lifts of I. In detail, if Jˆ is any other connected component of S 1 \ K̂, with π(Jˆ) = J, then
ˆ = Jˆ for some m,
we may find f ∈ F2n such that ρ(f )(I) = J. It follows that ρ̂(g m f )(I)
so the action is transitive. This completes the proof of the lemma.
To finish the proof of the theorem, we need to show that different choices of k give
infinitely many distinct conjugacy classes of circular orders. This follows from the fact we
mentioned above that the rotation number of ρ̂(g) is (2n − 1)/k. Rotation number is a
(semi)-conjugacy invariant of homeomorphisms of S 1 . Hence, the dynamical realizations
of conjugate orders cannot assign different rotation numbers to the same element.

4.3

Explicit singleton neighborhoods of isolated orders

Recall that a neighborhood basis of an order c ∈ CO(G) is given by sets of the form
OS (c) := {c′ ∈ CO(G) : c′ (u, v, w) = c(u, v, w) for all u, v, w ∈ S}, where S ranges over all
finite subsets of G. Given an isolated circular order c, it is therefore very natural to ask:
what is the minimum cardinality of S such that OS (c) is a singleton?
In this section, we take the isolated circular order on F2 = ha, bi whose dynamical
realization has simple ping-pong dynamics (from Lemma 4.10) and give an upper bound
on the cardinality of such a set S, by exhibiting a specific set with 5 elements. (We expect
the bound |S| ≤ 5 given here is sharp, but have not pursued this point.) The proof of
the Proposition below also gives an independent and very short proof that CO(F2 ) has
isolated points; the only previous tool that we use here is the ping-pong lemma.

32

Fixing notation, let c be the isolated circular from Lemma 4.10 in the case n = 1. We
show the following:
Proposition 4.15. If c′ agrees with c on the set S 3 = {id, a, ba, a−1 ba, b−1 a−1 ba}3 , then
c′ = c.
Proof. Let ρ : F2 → Homeo+ (S 1 ) be the dynamical realization of c with basepoint x0 . We
need to show that for any action ρ′ of F2 on S 1 such that the cyclic order of ρ′ (S)(x0 )
agrees with that of ρ(S)(x0 ), the cyclic order of ρ′ (G)(x0 ) then agrees with that of ρ(G)(x0 ).
Consider an action of F2 on S 1 with the points
x0 , ab−1 a−1 b(x0 ), b(x0 ), a−1 b(x0 ), b−1 a−1 b(x0 )
in this cyclic (counterclockwise) order. Here, and in the remainder of the proof, we drop
the notation ρ′ for the action.
We begin by choosing some additional points in S 1 in a careful way, so as to arrive at
the configuration illustrated in Figure 2 below. This will let us define attracting domains
for the action.
Let p1 ∈ S 1 be a point such that the triple x0 , p1 , ab−1 a−1 b(x0 ) is positively oriented.
Choose x′0 very close to x0 , and such that x′0 , x0 , p1 is positively oriented. If x′0 is chosen
sufficiently close to x0 , then
x′0 , p1 , ab−1 a−1 b(x′0 ), b(x′0 ), a−1 b(x′0 ), b−1 a−1 b(x′0 )
will also be in positive cyclic order. Let X denote this set of 6 points.
The ordering of X implies that a−1 (p1 ) lies in the connected component of S 1 \ X
bounded by a−1 b(x′0 ) and b−1 a−1 b(x′0 ). Let p2 be a point in this component such that the
triple a−1 (p1 ), p2 , b−1 a−1 b(x′0 ) is positively oriented. Similarly, our assumption on order
implies that b(p2 ) lies in the component of S 1 \ X bounded by b(x′0 ) and a−1 b(x′0 ), and
we let p3 be a point in this component such that b(p2 ), p3 , a−1 b(x′0 ) is positively oriented.
This configuration is summarized in Figure 2.
Now let D(a), D(b), D(a−1 ) and D(b−1 ) be the disjoint intervals [p1 , a(p3 )], [b(x′0 ), b(p2 )],
[p3 , a−1 (p1 )] and [p2 , x′0 ] as shown in Figure 2. Our choice of configuration of points implies
that b(S 1 \ D(b−1 )) ⊂ D(b). Similarly, a(S 1 \ D(a−1 )) ⊂ D(a). We also have that x0 is in
the complement of the union of domains.
Since these are the same cyclic order and containment relations as c, Lemma 4.2 implies
that the cyclic order on G induced by the orbit of x0 coincides with c.
Remark 4.16. A similar strategy can be used to produce neighborhoods of the lifts of
these orders on F2 , however, many more points are needed. Compare also the framework
of [13], especially Lemma 4.8.

5

Applications to linear orders

Suppose 0 → A → B → C → 0 is a short exact sequence of groups. If A is left-ordered,
and C circularly ordered, then it is well known that there is a natural circular order on
33

b(x′0 )

a(p3 )

D(b)
b(p2 )

D(a)

ab−1 a−1 b(x0 )

p3
p1
a

−1

b(x′0 )

◦ x0
x′0
D(a−1 )
D(b−1 )
a−1 (p1 )

p2

b−1 a−1 b(x0 )

Figure 2: Configuration of points giving ping-pong domains on S 1
B such that both maps A → B and B → C are monotone. (See [2, Lemma 2.2.12] for
a proof.) Here, we discuss a different method of constructing orders on certain central
extensions.
Recall that a subgroup H ⊂ G is cofinal for a left order < on G if, for all g ∈ G, there
exist h1 , h2 ∈ H such that h1 < g < h2 . If H ⊂ G, we let LOH (G) ⊂ LO(G) denote the
subspace of LO(G) consisting of orders where H is cofinal.
π

Proposition 5.1 (see also [22]). Let G be a group, and let Z → Ĝ → G be a central
∗ : LO (Ĝ) → CO(G). Moreover, each
extension. Then, there is a continuous map π ∗ = πĜ
Z
circular order on G is in the image of one such map π ∗ .
This theorem is essentially proved in [22], although there is no comment on continuity
there.
π

Proof. Let Z → Ĝ → G be a central extension. Given < in LOZ (Ĝ) define π ∗ (<) as follows.
Let z be the generator of Z such that z > id. Since Z is cofinal, for each g ∈ G, there exists
a unique ĝ ∈ Ĝ such that id ≤ ĝ < z. Given distinct elements g1 , g2 , g3 ∈ G, let σ be the
permutation such that id ≤ ĝσ(1) < ĝσ(2) < ĝσ(3) < z. Define π ∗ (<)(g1 , g2 , g3 ) := sign(σ).
One checks that this is a well defined circular order on G.
To show continuity, given a finite set S ⊂ G, let Ŝ := {ĝ ∈ Ĝ : g ∈ S}. If <1 and <2
are two left orders that agree on Ŝ ∪ {id, z}, then the definition of π ∗ ensures that π ∗ (<1 )
and π ∗ (<2 ) agree as circular orders on S.
For the last remark, we give a proof for countable groups that highlights the relationship between the dynamical realization of c and π ∗ (c). The general case is given in
[22], and uses essentially the same idea. Let HomeoZ (R) denote the group of orientationpreserving homeomorphisms of R that commute with integer translations. This group is
the universal central extension of Homeo+ (S 1 ). Given c ∈ CO(G), let ρ be a dynamical
realization of c with basepoint x, and let x̂ ∈ R be a lift of x. Let Ĝ be the pullback of the
34

central extension 0 → Z → HomeoZ (R) → Homeo+ (S 1 ) → 1 using ρ. It is easily checked
that x̂ has a free orbit under Ĝ ⊂ Homeo(R) so induces a left order < on Ĝ, and in this
left order Z is cofinal. By construction, π ∗ (<) = c.
We note that, as remarked to the authors by S. Matsumoto, π ∗ is not necessarily
one-to-one, although the last step in the proof does give a partial inverse.
Lemma 5.2. Let G be a finitely generated group, and z ∈ G a central element. The set
of left invariant orders on G where hzi is cofinal is open in LO(G).
Proof. Suppose that < is a left order where hzi is cofinal. Then for each generator gi in
a finite generating set, there exists ki ∈ Z such that z ki ≤ gi < z ki +1 . We claim that this
finite collection of inequalities also implies that hzi is cofinal. Indeed, that z is central
implies that z ki +kj ≤ gi gj < z ki +kj +2 for all i, j, and inductively, that each word in the
generators is bounded above and below by powers of z.
Remark 5.3. This can fail when G is not finitely generated. Indeed, let G = Z[x] under
addition, and let z = 1 ∈ Z[x]. We can order Z[x] by letting 0 <π p(x) if and only if
p(π) is a non-negative real number. On the other hand any finite set S of G lies inside
a subgroup H that admits a complement, say H ⊕ H ′ = G. On H we can put again <π ,
and extend this ordering lexicographically to G using any ordering on H ′ ≃ G, so that
H is a proper convex subgroup. The resulting ordering <′ on G agrees with <π on S, so
by choosing S large it can be made arbitrarily close to <π . However, if on H ′ we put the
lexicographic ordering coming from the natural identification of Z[x] with a direct sum
of infinitely many copies of Z, then every element of G fixes a point in the dynamical
realization of <′ .
Using Lemma 5.2 and Proposition 5.1, we can produce isolated left orders from isolated
circular orders on finitely generated groups.
Proposition 5.4. Assume that G is finitely generated and c is an isolated circular order
π
on G. If Z → Ĝ → G is a central extension and <∈ LOZ (G) a left order such that
π ∗ (<) = c, then < is isolated in LO(Ĝ).
Proof. By Proposition 5.1, π ∗ : LO(Ĝ) → CO(G) is continuous. We claim that, since G
(and hence Ĝ) is finitely generated, π ∗ is also locally injective. To see this, suppose < and
<′ ∈ LO(Ĝ) have the same image under CO(G). Let ρ and ρ′ be the dynamical realizations
of < and <′ respectively; up to conjugacy, we can assume that ρ(z) = ρ′ (z) = x 7→ x + 1,
where z is the generator of the Z subgroup of Ĝ. We can also assume that the induced
actions of G on R/Z agree, since this action is the dynamical realization of π ∗ (<) = π ∗ (<′ ).
Fix a finite generating set S = {s1 , ...sn } for G, and choose generators {ŝ1 , ...ŝn , z} for Ĝ
where ŝi is the (unique) lift of si to Ĝ such that id < ŝi < z. Since π ∗ (<) = π ∗ (<′ ), for
each s ∈ S we have ρ(ŝi ) = ρ′ (si )z ni for some ni ∈ Z. In particular, if < and <′ are close
enough so that they agree on the elements ŝi and z, we will necessarily have ρ(ŝi ) = ρ′ (ŝi ),
thus the dynamical realizations, and hence the orders < and <′ agree.
Now let < be an order with π ∗ (<) isolated, as in the statement of the proposition.
Continuity and local injectivity of π ∗ combined with the fact that π ∗ (<) is isolated implies
35

that < must have also been isolated in LOZ (Ĝ). Lemma 5.2 says that LOZ (Ĝ) is an open
neighborhood of < in LO(Ĝ), so < is isolated in LO(Ĝ).
Proposition 5.1, 5.4, and Theorem 1.4 together imply the following.
Corollary 5.5. The space of left orders of F2 ×Z has infinitely many nonconjugate isolated
points.
Proof. The existence of infinitely many isolated points is an immediate consequence of
5.1, 5.4 and Theorem 1.4, together with the fact that all central extensions of F2 by Z
are trivial, i.e. direct products. The argument that the examples obtained by pulling
back the nonconjugate “standard k-lift” orders on F2 to F2 × Z are nonconjugate can be
done similarly to the argument that the original orders on F2 were non-conjugate. Since
these dynamical realizations have image in HomeoZ (R), elements have a well defined,
conjugation-invariant, translation number, and the translation number of the commutator
[a, b] in the k-fold lift is 1/k (again, this is standard and more background can be found in
[9]). Thus, varying k gives non-conjugate dynamical realizations, and hence nonconjugate
orders.
This example is interesting for three reasons. First, F2 × Z is isomorphic to P3 , the
pure braid group on three strands. It is known that the braid group Bn admits isolated
orderings. Could they always come from isolated ordering on Pn ? Related work in an
extensive study of orderings on braid groups can be found in [5]. Secondly, F2 × Z is
an example of a right-angled Artin group. Perhaps it is possible to give a complete
characterization of which RAAGs have isolated left orders. Finally, this example also
shows that direct products behave quite differently than free products – in [19], it is
shown that the free product of any two left orderable groups has no isolated left orders.

6

Further questions

Since we have found infinitely many isolated points in the compact space CO(F2n ), they
must accumulate somewhere. It is not hard to show the following, we leave the proof as
an exercise.
Proposition 6.1 (Accumulation points of lifts in CO(F )). Let A ⊂ CO(F2 ) denote the set
of all finite lifts of the Fuchsian circular order. Then the accumulation points of A are left
orders on F2 . These left orders have dynamical realizations conjugate into the universal
covering group of PSL(2, R), which acts by homeomorphisms on the line commuting with
integer translations.
Since LO(F2 ) has no isolated points, the accumulation points of A belong to a Cantor
set embedded in CO(G). Remarkably, we do not know a single example of any group G
such that CO(G) – or even LO(G) – has accumulation points that do not belong to a
Cantor set.

36

Question 6.2. Does there exist a countable group G such that the derived set of CO(G)
is neither empty nor a Cantor set?
Another question that remains open is the following:
Question 6.3. Does there exist an isolated circular order on Fn , for n odd?
Theorem 1.5 reduces this to an essentially combinatorial problem of checking whether
an arrangement of attracting domains gives an action which permutes transitively the
complementary intervals to the minimal set. At this point we do not know a single
example of an isolated circular order, and suspect the answer to Quesiton 6.3 may well be
negative.

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Kathryn Mann
Dept. of Mathematics, University of California, Berkeley
kpmann@math.berkeley.edu
Cristóbal Rivas
Dpto. de Matemáticas y C.C., Universidad de Santiago de Chile
cristobal.rivas@usach.cl

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