NORTHWESTERN UNIVERSITY
Rigidity of Solvable Group Actions
A DISSERTATION
SUBMITTED TO THE GRADUATE SCHOOL
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
for the degree
DOCTOR OF PHILOSOPHY
Field of Mathematics
By
Anne E. McCarthy
EVANSTON, ILLINOIS
June 2006
UMI Number: 3212800
3212800
2006
Copyright 2006 by
McCarthy, Anne E.
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2
c
 Copyright by Anne E. McCarthy 2006
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3
ABSTRACT
Rigidity of Solvable Group Actions
Anne E. McCarthy
This thesis investigates dynamical properties of actions of abelian-by-cyclic groups on
compact manifolds. For a non-singular integer matrix A, let Γ
A
be the fundamental group
of the mapping cylinder of the induced map f
A
on the torus T
n
. The standard actions ρ
λ
of Γ
A
on the circle RP
1
are generated by maps f(x) = λx and g
i
(x) = x + b
i
, where λ is
a real-valued eigenvalue for A, and (β
1
, , β
n
) is the associated eigenvector. It is known
that any analytic action of Γ
A

on the circle is a ramified lift of one of the standard actions
ρ
λ
. This thesis shows that for each analytic action, ρ, there exists R ≥ 2 such that ρ is
C
r
locally rigid for all r ≥ R.
We then consider actions of the groups Γ
A
on compact manifolds of higher dimension
that are generated by C
1
diffeomorphisms close to the identity. We show that any action of
Γ
A
on a surface with non-zero Euler characteristic has a global fixed point. Also, we show
that for any compact manifold M, there are no faithful actions of the Baumslag-Solitar
group generated by diffeomorphisms close to the identity.
4
Acknowledgements
Many thanks are due to my advisor, Amie Wilkinson. Under her guidance and in-
struction, I have been fortunate to have absorbed even a small fraction of he r breadth
and depth in knowledge of mathematics. I also am grateful to have been the sometimes
undeserving beneficiary of both her paitence and generosity. Benson Farb posed many of
the questions that have inspired the work of this thesis. I extend gratitude to Benson for
many illuminating conversation about Sol and BS groups. Keith Burns has enriched my
study of dynamics at Northwestern by overseeing many working seminars in which I have
been priveledged to participate. Several conversations with John Franks have also been
very helpful. I would also like to thank Michael Johnson, Kalman Nanes, Chris Novak,
and all the other graduate students who have taken the time to listen to me talk about

my work, and have given useful feedback on many of the talks I have given in the past
year.
5
Table of Contents
ABSTRACT 3
Acknowledgements 4
Chapter 1. Introduction 7
Chapter 2. Abelian-by-cyclic actions on S
1
15
2.1. Preliminaries 16
2.2. Global Invariant Sets 21
2.3. Local Analysis 28
2.4. The Liouville cocycle and BS(1,n) 38
2.5. Ramified Lifts of Affine Actions 41
2.6. Proof of Theorem 1.0.2 44
2.7. Unfaithful Actions 49
2.8. Actions Not Satisfying the Spectral Gap Condition 51
Chapter 3. Actions of Abelian-by-Cyclic Groups on Manifolds 53
3.1. Preliminaries 53
3.2. Invariant Sets for Actions of Γ
A
Close to the Identity 55
3.3. Rigidity for BS(1, n) close to id 57
3.4. Actions of Subgroups of Aff(R) on S
2
59
6
3.5. Examples of BS(1, n) acting on S
2

60
References 65
7
CHAPTER 1
Introduction
This thesis investigates the dynamics of solvable group actions on compact manifolds.
The actions of nilpotent and solvable groups on one-manifolds have been studied by Kopell
[K], Plante and Thurston [PT], Ghys [G], and Farb and Franks [FF1]. Furthermore,
work of Navas [N1] and Burslem and Wilkinson [BW] gives a classification for solvable
group actions on S
1
under the appropriate regularity hypotheses. We add to this work
by demonstrating a class of solvable groups that exhibit unusual rigidity phenomena. We
then turn to the study of actions of solvable groups on higher dimensional manifolds. This
thesis includes an investigation of actions of abelian-by-cyclic groups on any manifold that
are generated by diffeomorphisms close to the identity.
Given a finitely generated group Γ and a manifold M, a C
r
action of Γ on M is a
homomorphism ρ : Γ → Diff
r
(M). We will commonly refer to this homomorphism as
a representation (into Diff
r
(M)) and use the associated language. The representation
ρ is said to be faithful if ρ is injective. We denote by R
r
(Γ, M) the collection of all
representations of Γ into Diff
r

(M). A point q ∈ M is said to be a global fixed point for
the action ρ if q is fixed by ρ(γ) for all γ ∈ Γ.
The goal of this thesis is to study the structure of R
r
(Γ, M) where Γ is a solvable
group, and M is a compact manifold. The central motivating examples of this thesis are
8
actions on the circle S
1
of the solvable Baumslag-Solitar group:
BS(1, n) = a, b|aba
−1
= b
n
.
The standard affine action of this group on the circle S
1
 RP
1
given by ρ(a) : x → nx
and ρ(b) : x → x + 1. This standard action can be used to generate many more actions
by taking ramified lifts.
An analytic map π :

M → M is said to be a ram ified covering map over the point
p ∈ M, if the restriction of π to

M\{π
−1
(p)} is a regular covering map onto M\{p}. A

map

f :

M →

M is said to be a π-ramified lift of f : M → M if π ◦

f = f ◦ π for the
ramified covering π.
For any ramified covering map π : S
1
→ S
1
there exists a π-ramified lift ρ
n
:
BS(1, n) → Diff
ω
(S
1
) of the standard representation, ρ
n
. It was shown by Burslem and
Wilkinson [BW] that any faithful action ρ : BS(1, n) → Diff
ω
(S
1
) is conjugate to a
π-ramified lift of the standard action ρ

n
. Furthermore, [BW] also prove that all analytic
actions ρ : BS(1, n) → Diff
ω
are locally rigid, as defined below.
The collection of C
r
representations, R
r
(Γ, M) carries a topology. For the group Γ,
fix a generating set γ
1
, , γ
k
, and let
d
C
r
(ρ
1
, ρ
2
) = sup
γ
1
, ,γ
k
d
C
r

(ρ
1
(γ
i
), ρ
2
(γ
i
)).
This topology is independent of the choice of generating set. Representations ρ, ρ

∈
R
r
(Γ, M) are said to be conjugate in Diff
r
(M) if there exists h ∈ Diff
r
(M) such that
h ◦ ρ(γ) = ρ

(γ) ◦ h for all γ ∈ Γ. A representation ρ
0
∈ R
r
(Γ, M) is C
(k,r)
locally rigid
9
if there exists a C

k
-open neighborhood U of ρ
0
in R
r
(Γ, M) such that every ρ ∈ U is
conjugate in Diff
r
(M) to ρ
0
.
Let ρ
0
∈ R
ω
(BS(1, n), S
1
). [BW] show that there exists r ≥ 2 such that ρ
0
is C
(1,r)
locally rigid, but not C
(1,r−1)
locally rigid. The authors [BW] also give a classification of
all analytic actions on S
1
of solvable groups.
Theorem 1.0.1 (Burslem, Wilkinson). Suppose that G < Diff
ω
(S

1
) is solvable. Then
either G is virtually abelian, or G is conjugate in Diff
ω
(S
1
) to a subgroup of a π-ramified
lift of Aff(R), where π : RP
1
→ RP
1
is a ramified cover over ∞.
This thesis shows that many of the results of [BW] about actions of BS(1, n) on S
1
extend to the class of abelian-by-cyclic groups. To any invertible n × n matrix A with
integer entries, one can associate the solvable group
Γ
A
= a, b
1
, b
n
|b
i
b
j
= b
j
b
i

, ab
i
a
−1
=

j
b
a
ij
j
.
This group has a geometric interpretation as the fundamental group of the mapping
cylinder for the map induced by A on T
n
. Note that in the case where A is the 1 × 1
matrix A = [n], we have that Γ
A
= BS(1, n).
A group Γ is said to be abelian-by-cyclic if there exists an exact sequence
1 → A → Γ → Z → 1,
where the group A is abelian, and Z is an infinite cyclic group. Note that the commutator
subgroup [Γ, Γ] is contained in A, so all such Γ are solvable groups. The class of all finitely
10
presented, torsion free, abelian-by-cyclic groups is exactly given by groups of the form Γ
A
.
See [FM] for a nice proof of this.
To each real eigenvalue of A, there corresponds an action of Γ
A

on the circle, given
by ρ
λ
(a) : x → λx and ρ
λ
(b
j
) : x → x + β
j
, where β
j
are the entries of the eigenvector
corresponding to the eigenvalue λ, normalized in such a way that β
1
= 1. We call the
representations ρ
λ
the standard representations of Γ
A
into Aff(R) < Diff
ω
(S
1
). The
classification of real-analytic actions of solvable groups given in [BW] implies that each
action ρ : Γ
A
→ Diff
ω
(S

1
) is conjugate to a π-ramified lift of one of these standard affine
actions.
Our first main result is a classification of representations ρ : Γ
A
→ Diff
r
(S
1
) for r < ω.
The main hypothesis of our result involves the hyperbolicity of the representation in the
following sense. We require that ρ have a spectral gap compatible with the spectrum of
the matrix A, as follows. Let the eigenvalues of A be given as |λ
n
| ≥ ≥ |λ
1
|, and fix
generators a, b
i
 for the group Γ
A
. Let f = ρ(A), and define
σ
+
(ρ) = inf



(f
k

)

(p)


1/k
: p ∈ Fix(f
k
), (f
k
)

(p) ≥ 1

, and
σ
−
(ρ) = sup



(f
k
)

(p)


1/k
: p ∈ Fix(f

k
), (f
k
)

(p) ≤ 1

.
We will say that a representation ρ ∈ R
r
(S
1
) satisfies the C
r
spectral gap condition if
σ
−
(ρ) < (
1
|λ
n
|
)
1
r−1
and σ
+
(ρ) > (
1
|λ

1
|
)
1
r−1
. With this we now can state our main theorem
concerning C
r
actions of abelian-by-cyclic groups on the circle.
Theorem 1.0.2. Suppose that A is a non-singular integer matrix with no eigenvalues
that are roots of unity. For each r ≥ 2, if ρ : Γ
A
→ Diff
r
(S
1
) is a faithful representation
11
that satisfies the C
r
spectral gap condition, then ρ is conjugated by an element of Diff
r
(S
1
)
into a unique conjugacy class of R
ω
(Γ
A
).

The C
r
spectral gap condition serves to prohibit the group Γ from acting via diffeo-
morphisms that are r-tangent to the identity. The classification given by Theorem 1.0.2
does not apply to actions with diffeomorphisms tangent to the identity. Furthermore, in
many cases, including that of BS(1, n) actions that do not satisfy the C
r
spectral gap
condition are not C
(1,r)
locally rigid.
One feature that is common to all faithful actions of the groups Γ
A
on the circle
is the existence of a finite globally invariant set. This and other features enjoyed by
circle actions no longer remain true if we instead consider actions of BS(1, n) on higher
dimensional manifolds. The question of existence of global fixed points for group actions
on surfaces has been studied in many different contexts. The earliest work in this area
concerned global fixed points for actions of Lie groups on closed surfaces with non-zero
Euler characteristic. It was shown by Lima [L] that n commuting vector fields on a
surface Σ
g
of non-zero Euler characteristic have a common singularity. This implies that
any action of the abelian Lie group R
n
on Σ
g
has a global fixed point. It was later shown
by Plante [P] that any action of a nilpotent Lie group on a surface with non-zero Euler
characteristic has a global fixed point. An example of Lima shows that it is not generally

true that every action of a solvable Lie Group on a surface with χ(Σ
g
) = 0 has a global
fixed point.
Combining a local analysis of C
1
diffeomorphisms that are close to the identity, with
the ideas of [L], Bonatti [B] showed that actions of Z
n
on surfaces of non-zero Euler
characteristic that are C
1
-close to the identity have a global fixed point. Using similar
12
techniques, Druck, Fang and Firmo [DFF] proved a discrete version of Plante’s theorem:
Consider a surface with χ(Σ
g
) = 0. Suppose ρ : Γ → Diff
1
ε
(Σ
g
), is an action of a finitely
generated nilpotent group via diffeomorphisms close to the identity, then ρ has a global
fixed point.
Although actions of the group Z
2
that are C
1
close to the identity have global fixed

points, it is also true that Z
2
acts on the sphere S
2
without global fixed points. Consider
the action generated by two rotations R
x
and R
y
by an angle π about orthagonol axes.
Each of these rotations interchanges the fixed points of the other. There is an invariant
that distinguishes this example from those action with a global fixed point. Handel [H]
generalized the work of [B] by showing that a winding number invariant, w(f, g)detects
the existence of a global fixed points for actions of Z
2
on surfaces of non-zero Euler
characteristic. This result has recently been extended by Franks, Handel and Parwani
[FHP] to all Z
n
actions.
In the spirit of [B] and [DFF], we study actions of abelian-by-cyclic groups generated
by diffeomorphims close to the identity on compact manifolds. Our first main result in
this setting concerns the existence of global fixed points for actions of Γ
A
generated by
diffeomorphisms close to the identity.
Theorem 1.0.3. Let Σ
g
be a compact orientable surface with χ(Σ
g

) = 0. Suppose
A ∈ M
n×n
is a non-singular integer matrix that has no eigenvalues that are roots of
unity. Fix a generating set on Γ
A
. There exists ε > 0 such that any faithful action
ρ : Γ
A
→ Diff
1
(Σ
g
) with d
C
1
(ρ, id) < ε has a global fixed point.
13
The key tool in proving this theorem is the notion of a translation neighborhood as
developed in [B]. In the case that ρ is an action of the group BS(1, n), the estimates of
[B] all us to strengthen this result significantly.
Theorem 1.0.4. Let M be a compact manifold. There exists ε > 0 such if ρ :
BS(1, n) → Diff
1
(M) satisfies d
C
1
(ρ, id) < ε, then ρ is not a faithful action. In particular
ρ(b) = id.
This thesis is broken into two primary components. Chapter 2 addresses actions of

abelian-by-cyclic groups on the circle S
1
, and Chapter 3 investigates actions of abelian-
by-cyclic groups on higher dimensional manifolds.
In section 2.1 we discuss the primary background tools that will be needed in chapter
2. This includes Kopells’ Lemma and distortion estimates, and a specific application that
is central to describing dynamical properties of actions of Γ
A
on S
1
.
The proof of Theorem 1.0.2 requires two primary components. The first is an analysis
of compact globally invariant sets, and the second is a local characterization of faithful
actions. Section 2.2 contains the global analysis. The main result of this se ction, Propo-
sition 2.0.5, says that there is a set S that is a set of global fixed points for a finite index
subgroup Γ < Γ
A
< Diff
r
(S
1
).
Section 2.3 characterizes the dynamics of an action ρ : Γ
A
→ Diff
r
(S
1
) on a half open
interval [q, q

1
). We assume that this interval is such that q is a global fixed point for the
action, and that there are no other global fixed points on (q, q
1
). We prove Proposition
2.0.6. This proposition establishes that a C
r
action that satisfies the C
r
spectral gap
14
condition is C
r
locally conjugate to a ramified lift of a standard affine action. It was sug-
gested in [BW] this the local analysis may be simplified by use of the Schwarzian cocycle.
However, use of this cocycle requires that the diffeomorphism be of class C
3
. In section
2.4 we rectify the regularity shortcomings of this proof, by instead using the Liouville
cocycle. This cocycle approximates the Schwarzian but requires only C
1
regularity.
Section 2.6 contains the proof of Theorem 1.0.2. We verify that by applying the local
conjugacies of Proposition 2.0.6 to intervals in the complement of the set S, we define
a ramified covering. We then devote two sections to investigating actions which do not
satisfy the hypotheses of Theorem 1.0.2. Section 2.7 gives a description of unfaithful
actions ρ : Γ
A
→ Diff
2

(S
1
), and in Section 2.8 we give an example of an action that does
not satisfy the spectral gap condition, to which our classification theorem 1.0.2 does not
apply.
In Chapter 3 we turn our attention to actions of abelian-by-cyclic groups on higher
dimensional manifolds. In section 3.1 we summarize the local analysis of [B] concerning
C
1
diffeomorphisms close to the identity. Using these methods, we are able to prove
Theorems 1.0.3 and 1.0.4, in sections 3.2 and 3.3, respectively. We conclude this thesis
with a catalog of examples of actions of solvable groups the sphere S
2
. This is the content
of sections 3.4 and 3.5.
15
CHAPTER 2
Abelian-by-cyclic actions on S
1
The primary goal of this chapter is to characterize actions of torsion-free, finitely
presented, abelian-by-cyclic groups on the circle S
1
. We begin by considering the case
where the repres entation ρ : Γ
A
→ Diff
r
(S
1
) is faithful. We will prove Theorem 1.0.2 in

this setting. We then discuss the necessity of certain hypotheses of Theorem 1.0.2.
There are two primary results that will be central to the proof of Theorem 1.0.2. The
first is an analysis of global invariant sets.
Proposition 2.0.5. Let A be a non-singular integer matrix with no eigenvalues that
are roots of unity, and let ρ : Γ
A
→ Diff
r
(S
1
), r ≥ 2 be a faithful action. Suppose
ρ(a) = f and ρ(b
i
) = g
i
. Let k ≥ 0 be the least integer for which τ (f
k
) = τ(g
k
i
) = 0. The
set S =

i
Per(g
i
) = ∅ has the following properties:
(1) g
k
i

|S = id
(2) For each component C of S
c
, g
k
i
(C) = C.
(3) For every 1 ≤ i ≤ n either g
k
i
|C = id or g
k
i
|C has no fixed points.
(4) f(S) = S
(5) For every component C of S
c
, f
k
(C) = C.
(6) ∂S ⊂ Fix(f
k
)
The invariant set ∂S is the set of global fixed points for f
k
and g
k
. To further charac-
terize the dynamics of the action, we must then restrict our attention to S
c

. Proposition
16
2.0.5 implies that each component of S
c
is an interval that is invariant under the action
of Γ
A
k
. We then analyze the action of Γ
A
k
restricted to any such interval. Through this
local analysis we will establish the following characterization. We denote the composition
g
k
1
1
◦ ··· ◦ g
k
n
n
by

j
g
k
j
j
.
Proposition 2.0.6. Suppose A is a non-singular matrix with no eigenvalues that are

roots of unity. Let f, g
i
∈ Diff
r
(S
1
) satisfy [g
i
, g
j
] = 1, and fg
i
f
−1
=

j
g
a
ij
j
. Suppose
that q is a common fixed point for f, g
i
and either f

(q) < (
1
λ
n

)
1
r−1
or f

(q) > (
1
λ
1
)
1
r−1
.
Suppose further that for some 1 ≤ i ≤ n, g
i
(x) = x for all x ∈ (q, q
1
). Then there is a C
r
diffeomorphism α : (q, q
1
) → (−∞, ∞) ⊂ RP
1
such that for all p ∈ (q, α
−1
(0)),
(1)
α(p) = εh(p)
s
,

where h : [q, α
−1
(0)) → [−∞, 0) is a C
r
diffeomorphism, s ∈ Z has 1 ≤ s ≤ r
and ε ∈ {1, −1}, or
(2) αf(p) = λα(p) and αg
i
(p) = α(p) + β
i
,
where λ is a real-valued eigenvalue for the matrix A with corresponding eigenvec-
tor (β
1
, , β
n
).
We will prove these propositions in sections 2.3 and 2.2. These components are com-
bined to complete the proof of Theorem 1.0.2.
2.1. Preliminaries
We begin by introducing some essential background results. We say that g ∈ Diff
r
+
([a, b))
is a contraction if g(a) = a and g(x) < x for all x ∈ (a, b). The following lemma of Kopell
17
about contractions is fundamental to the study of group actions in one dimension. A nice
description of this work is given in [N1].
Lemma 2.1.1 (Kopell). Let g ∈ Diff
2

([a, b)) be a contraction and suppose that gh =
hg for some h ∈ Diff
1
([a, b)). If h has a fixed point in (a, b), then h = id.
Kopell used this fact to study the centralizers in Diff
2
(S
1
) of contractions. We denote
by Diff
r,∆
+
([a, b)) the collection of orientation-preserving f ∈ Diff
r
([a, b)) that fix a and
have no fixed points in (a, b).
Lemma 2.1.2 (Szekeres, Kopell). For each g ∈ Diff
r,∆
+
([a, b)) there exists an asso-
ciated vector field X
g
: [a, b) → R with no singularities on (a, b) for which the following
properties hold:
(1) X
g
is of class C
2
on (a, b) and C
1

on [a, b).
(2) Denoting the flow generated by X
g
as g
R
= {g
t
: t ∈ R}, we have g = g
1
.
(3) The centralizer of g in Diff
1
([a, b)) is g
R
.
Corollary 2.1.3. Let g ∈ Diff
2
([a, b)) be a contraction that embeds in a C
1
flow g
t
,
with g = g
1
. If h ∈ Diff
1
([a, b)) satisfies hg = gh, then h = g
t
for some t ∈ R.
A common theme in the proofs of these statements is the use of distortion estimates.

Because we will make use of similar estimates, we now demonstrate how such estimates
are obtained.
Lemma 2.1.4 (Distortion Estimates). Let f ∈ Diff
2
+
(M), where M = [a, b), or S
1
.
Suppose there exists an interval I
0
such that all the iterates I
j
= f
j
(I
0
), j ∈ Z are disjoint.
There exists C > 0 such that for all j ∈ Z, if y, y ∈ I
j
, then for all N > 0,
18
1
C
≤
(f
N
)

(y)
(f

N
)

(y)
≤ C.
Proof. Let Q denote the quantity
Q = log
(f
N
)

(y)
(f
N
)

(y)
= log
N−1

i=0
f

(f
i
y)
f

(f
i

y)
=
N−1

i=0
log f

(f
i
y) − log f

(f
i
y).
Since f ∈ Diff
2
+
(S
1
) we know that the function log f

(x) is Lipschitz. Let K =
Lip(log f

). We have
|Q| ≤
N−1

i=0
K d(f

i
(y), f
i
(y)).
Now, since for each i, f
i
(y), f
i
(y) are in the same component f
i
(I
j
), we bound
d(f
i
(y), f
i
(y)) ≤ |I
i+j
|. Since all of these iterates are disjoint, we also know that

i∈Z
|I
i
|
is bounded by the length of the total space, so that
Q ≤ K
N−1

i=0

|f
i
(I)| ≤ K for all N > 0,
which implies that
e
−K
≤
(f
N
)

(y)
(f
N
)

(y)
≤ e
K
for all N > 0.
Setting C = e
K
proves the distortion estimates. 
Kopell’s results allow us to connect the behavior of commuting diffeomorphisms to
that of the common flow into which they embed. It is frequently convenient to embed the
diffeomorphisms into this flow for the characterization of local behavior. In the language
19
of flows, we then can make use of results such as the following lemma, which follows from
results in [K] and [N1].
Lemma 2.1.5 (Conjugate Flows). Let φ

t
: [a
1
, b
1
) → [a
1
, b
1
) and ψ
s
: [a
2
, b
2
) →
[a
2
, b
2
) be C
2
flows with no fixed points in (a
i
, b
i
). Suppose that there exists a C
r
diffeo-
morphism f : [a

1
, b
1
) → [a
2
, b
2
) such that fφ
1
f
−1
= ψ
α
. If the map ψ
α
is a C
2
contraction
then fφ
t
f
−1
= ψ
αt
for all t ∈ R.
Proof. First note that for each t ∈ R, the map fφ
t
f
−1
commutes with ψ

α
:
fφ
t
f
−1
◦ ψ
α
= fφ
t
f
−1
◦ fφ
1
f
−1
= fφ
t+1
f
−1
= fφ
1+t
f
−1
= fφ
1
f
−1
◦ fφ
t

f
−1
= ψ
α
◦ fφ
t
f
−1
.
Since the map fφ
t
f
−1
commutes with ψ
α
for each t ∈ R, by Kopell’s lemma we know
that for each t there must be a β(t) such that fφ
t
f
−1
= ψ
β(t)
.
Note that β(t) is additive, since
ψ
β(t
1
+t
2
)

= fφ
t
1
+t
2
f
−1
= fφ
t
1
f
−1
◦ fφ
t
2
f
−1
= ψ
β(t
1
)+β(t
2
)
.
This implies that for p, q ∈ Z, β(
p
q
t) =
p
q

β(t). By continuity, we conclude that β(at) =
aβ(t) for all a ∈ R. Since β is linear, and β(1) = α, we can then conclude that β(t) =
αt. 
In what follows, we will be specifically interested in the dynamics of an action of an
abelian-by-cyclic group Γ
A
. A central idea to this exposition is that in a neighborhood of
a global fixed point, we may associate a flow to this action. By applying both Kopell’s
20
lemma and the conjugate flow lemma we will see that such an action must demonstrate
very specific dynamics.
Corollary 2.1.6. Let f, g
i
∈ Diff
2
(S
1
) satisfy the relations fg
i
f
−1
=

j
g
a
ij
j
and
[g

i
, g
j
] = 1. Suppose there exist points q, q
1
∈ S
1
such that each g
i
fixes q, but has no fixed
points on the interval (q, q
1
). There exists a flow γ
t
such that g
i
= γ
β
i
, where (β
1
, , β
n
) is
an eigenvector for A = {a
ij
}, with corresponding real-valued eigenvalue λ. Furthermore,
the time-λ map γ
λ
is as smooth as the maps f, g

i
.
Proof. By replacing each g
i
with its inverse, we may assume without loss of generality
that g
1
is a contraction. By Kopell’s lemma, each g
i
is the time-β
i
map of the flow γ for
which g
1
= γ
1
. Written in terms of this flow, the group relations become
fγ
β
i
f
−1
= γ
P
j
a
ij
β
j
.

In particular, by setting λ =

j
a
1j
β
j
, we have that fg
1
f
−1
= γ
λ
. Since f conjugates
the flow γ
t
to itself, we may apply Lemma 2.1.5, which implies that fγ
t
f
−1
= (fγ
1
f
−1
)
t
.
So we may write
fγ
β

i
f
−1
= γ
β
i
λ
.
We equate these two different expressions for fγ
β
i
f
−1
to conclude that

j
a
ij
β
j
= λβ
j
,
which shows that λ and β
j
must satisfy the relation
A







β
1
.
.
.
β
n






= λ






β
1
.
.
.
β
n







.
21
Therefore, the values for λ and β
j
must be determined by the eigenvalues and eigen-
vectors for A. Note that

j
β
j
a
1j
= λ is a real valued eigenvalue for A. Additionally,
since γ
λ
= fg
1
f
−1
, it is clear that the time-λ map of the flow is as smooth as f and g
1
. 
2.2. Global Invariant Sets
Let f, g in Diff

2
(S
1
) satisfy fg
i
f
−1
=

j
g
a
ij
j
and [g
i
, g
j
] = 1. In this section we will
examine how these relations determine the structure of the sets Per(f) and Per(g
i
). The
primary goal of this section is to prove Proposition 2.0.5.
To simplify arguments, we will in several places replace the maps g
i
and f with their
inverses or higher powers. Note that if we replace each g
i
by g
k

i
, the relation fg
k
i
f
−1
=

j
g
ka
ij
j
holds. Also, replacing f by f
k
we have that f
k
g
i
f
−k
=

j
g
(A
k
)
ij
j

. Where (A
k
)
ij
are the entries of the matrix A
k
.
Lemma 2.2.1. Suppose that 1 is not an eigenvalue for A. Then each g
i
has rational
rotation number.
Proof. Because rotation number is a conjugacy invariant, we have
τ(g
i
) = τ (fg
i
f
−1
) = τ (

j
g
a
ij
j
).
Also, because the g
i
commute,
τ(


j
g
a
ij
j
) =

j
a
ij
τ(g
j
) (mod 1).
22
Therefore, there exist {k
1
, k
n
} ∈ Z such that






τ(g
1
)
.

.
.
τ(g
n
)






= A






τ(g
1
)
.
.
.
τ(g
n
)







+






k
1
.
.
.
k
n






.
This equation has the solution







τ(g
1
)
.
.
.
τ(g
n
)






= (A −I)
−1






k
1
.
.
.
k

n






.
Since {k
1
, , k
n
} and the entries of A are all integers, this guarantees that τ(g
i
) ∈
Q 
Since all g
i
have rational rotation number, there exists k ∈ Z such that τ (g
k
i
) = 0 for
all 1 ≤ i ≤ n. We are interested in studying the structure of the sets Per(g
i
) = Fix(g
k
i
).
We will therefore replace each g
i

by g
k
i
, and assume for the remainder of this section that
τ(g
i
) = 0.
Lemma 2.2.2. Suppose [g, h] = 1. Then Fix(g) is an h-invariant set, and vice versa.
Proof. Let x ∈ Fix(h). We see that
hg(x) = gh(x) = g(x).
Hence g(x) ∈ Fix(h). 
Corollary 2.2.3. There exists x ∈ S
1
such that g
i
(x) = x for all 1 ≤ i ≤ n.
23
Proof. Let x
1
∈ Fix(g
1
) and consider the orbit {g
k
2
(x
1
)|k ∈ Z}. By Lemma 2.2.2,
each point in this orbit is fixed by g
1
. Since τ(g

2
) = 0, this orbit must accumulate to a
point x
2
which is fixed by g
2
. By continuity, g
1
must also fix x
2
. We continue this process:
consider the orbit of x
i
under g
i+1
to find a point x
i+1
that is fixed by g
j
for all j ≤ i + 1.
It is clear that this process terminates. 
We now consider the set of all common fixed points S =

i
Fix(g
i
). This set S is
the object that is characterized by Proposition 2.0.5. The analysis of this set will give a
description of global invariant sets for actions of Γ
A

on S
1
.
Proof of Proposition 2.0.5. It is clear that Property (1) follows from the defini-
tion of the set S. Before proving properties (2) and (3), we require a preliminary result.
Lemma 2.2.4. For each i = j, g
j
preserves the components of (Fix(g
i
))
c
. (i.e., if I
is a component of (Fix(g
i
))
c
, then g
j
(I) = I.)
Proof. Fix(g
i
) is a closed set, so that (Fix(g
i
))
c
is a union of open intervals. Suppose
that I = (a, b) is a component of (Fix(g
i
))
c

that is not fixed by g
j
. Since g
j
has rotation
number zero, all iterates {g
k
j
(I)|k ∈ Z} are disjoint and accumulate to some point x
0
=
lim
k→∞
g
k
j
(a), which is fixed by both g
i
and g
j
.
We see that g

i
(x
0
) = 1 as follows. Since g
i
is of class C
1

, we know that the derivative
g

i
(x
0
) exists. Using the limit definition, g

i
(x
0
) can be computed along any convergent
subsequence of points x
i
→ x
0
. So we compute along the orbit g
k
j
(a) → x
0
, which implies
that
g

i
(x
0
) = lim
k→∞

g
i
(g
k
j
(a)) − g
i
(x
0
)
g
k
j
(a) − x
0
= 1,
where the final equality holds because x
0
and g
k
j
(a) are fixed by g
i
for all k.
24
Now we note that since g
j
is of class C
2
, and the iterates of I under g

j
are disjoint,
Lemma 2.1.4 applies to g
j
. Hence, there exists C
1
> 0 such that for all y, y ∈ I, and for
all N > 0,
1
C
1
≤
(g
N
j
)

(y)
(g
N
j
)

(y)
≤ C
1
.
Our distortion estimates will also allow us to bound derivatives of g
i
.

Lemma 2.2.5. There exists C > 0 such that for all y ∈ I, N > 0,
1
C
≤ (g
N
i
)

(y) ≤ C.
Proof. Let y ∈ I. Applying the chain rule to the function g
n
j
g
N
i
= g
N
i
g
n
j
at the point
y, we get that for all n, N > 0,
(g
N
i
)

(y) =
(g

n
j
)

(y))
(g
n
j
)

(g
N
i
(y))
(g
N
i
)

(g
n
j
(y)).
Since I is a component of (Fix(g
i
))
c
, we know that g
N
i

(y) ∈ I. Making use of the above
distortion estimates,
1
C
(g
N
i
)

(g
n
j
y) ≤ (g
N
i
)

(y) ≤ C(g
N
i
)

(g
n
j
y)
for all N, n > 0. We now let n → ∞ so that g
n
j
y → x

0
. Since g

i
(x
0
) = 1, this proves the
lemma. 