homomorphisms of the fundamental group of a surface into psu(1,1), and the action of the mapping class group

HOMOMORPHISMS OF THE FUNDAMENTAL GROUP OF A
SURFACE INTO PSU(1, 1), AND THE ACTION OF THE
MAPPING CLASS GROUP
by
Panagiota Savva Konstantinou
A Dissertation Submitted to the Faculty of the
DEPARTMENT OF MATHEMATICS
In Partial Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
In the Graduate College
THE UNIVERSITY OF ARIZONA
2 0 0 6
UMI Number: 3214652
3214652
2006
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2
THE UNIVERSITY OF ARIZONA
GRADUATE COLLEGE
As members of the Final Examination Committee, we certify that we have read the
dissertation prepared by Panagiota Savva Konstantinou
entitled Homomorphisms of the Fundamental Group of a Surface into

PSU(1, 1), and the Action of the Mapping Class Group
and recommend that it be accepted as fulfilling the dissertation requirement for the
Degree of Doctor of Philosophy.
Date: 1 May 2006
Douglas Pickrell
Date: 1 May 2006
Phillip Foth
Date: 1 May 2006
David Glickenstein
Date: 1 May 2006
Douglas Ulmer
Final approval and acceptance of this dissertation is contingent upon the candi-
date’s submission of the final copies of the dissertation to the Graduate College.
I hereby certify that I have read this dissertation prepared under my direction
and recommend that it be accepted as fulfilling the dissertation requirement.
Date: 1 May 2006
Dissertation Director: Douglas Pickrell
3
STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of requirements for an
advanced degree at The University of Arizona and is deposited in the University
Library to be made available to borrowers under rules of the Library.
Brief quotations from this dissertation are allowable without special permission,
provided that accurate acknowledgment of source is made. Requests for permission
for extended quotation from or reproduction of this manuscript in whole or in part
may be granted by the head of the major department or the Dean of the Graduate
College when in his or her judgment the proposed use of the material is in the
interests of scholarship. In all other instances, however, permission must be obtained
from the author.
SIGNED:

Panagiota Savva Konstantinou
4
ACKNOWLEDGEMENTS
I express my deepest graditute to my thesis advisor Doug Pickrell, for his excellence
guidance, and advice, his continual encouragement and incredible patience, and for
caring, during the years of my doctoral work. I want to thank him especially, for
being such an inspiration to me, and for giving me the motivation to continue in
the PhD program.
I thank my Final Defense Committee: Phillip Foth, David Glickenstein, Douglas
Ulmer, for carefully reading through my dissertation and giving me many suggestions
on improving it. I thank my external reviewer, William Goldman, for taking the
time to review my paper, and for helping me with a lot of questions I had over the
last few years. I thank Jane Gilman and Shigenori Matsumoto, for taking the time
to respond to a number of questions I had during this work.
There are also numerous professors and faculty members that I wish to thank
at the University of Arizona and the University of Cyprus, including Jan Wehr,
Larry Grove, Deborah Hughes-Hallet, Maceij Wojtkowski, Tina Deemer, Pantelis
Damianou, Christos Pallikaros, Christodoulos Sofokleous and Evangelia Samiou for
the classes that they have taught and the conversations that we have had.
I thank my close friend Stella Demetriou, without whose help and inspiration
I would never have had the courage to start graduate school in the USA; and my
close friend Guadalupe Lozano, who, though these years, gave me encouragement,
faith, and the strength to not give up.
There are numerous classmates and friends that I would like to thank for their
help in classes and their support during these years. These include Maria Agro-
tis, Lisa Berger, Arlo Caine, Derek Habermas, Selin Kalaycioglu, Alex Perlis and
Sacha Swenson. Also many thanks to the wonderful staff of the Department of
Mathematics.
I give a special thank you to my close friends that have been more than a
family to me here in Tucson: Nakul Chitnis; Luis Garcia-Naranjo; Adam Spiegler;

Rosangela Sviercoski; Gabriella, Eleni, Alexandros and Pavlos Michaelidou; Antonio
Colangelo and Mariagrazia Mecoli. I thank them for being on my side all these years,
for believing in me and for making my life in Tucson fun and enjoyable. I also thank
my friends of many years Avra Charalambous, Georgia Papageorgiou and Anna
Sidera for their continual support and the great summers I spent with them.
I thank my grandparents Panagiota and Charalambos Konstantinou, Maria
Zeniou and my spiritual father Michalis Pigasiou for their continual wishes and
prayers; and my loving a supporting family: my mother Andreani, my father Sav-
vas, my brothers Michael-Zenios and Charalambos.
Finally I thank the Department of Mathematics at the University of Arizona.
5
DEDICATION
To my parents, Andreani and Savvas Konstantinou, who made all this possible.
6
TABLE OF CONTENTS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 10
CHAPTER 2 BACKGROUND AND NOTATION . . . . . . . . . . . . . . . 18
2.1 Conjugacy classes of PSU(1, 1) . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Conjugacy classes for the double cover SU(1, 1) . . . . . . . . . . . . 21
2.3 A model for

PSU(1, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Poincar´e rotation number . . . . . . . . . . . . . . . . . . . . . . . . 25
CHAPTER 3 COMMUTATORS IN PSL(2, R) . . . . . . . . . . . . . . . . . 30
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Preliminary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 A description of the Image(R
p

) . . . . . . . . . . . . . . . . . . . . . 36
3.4 Conjecture: The level sets of R
p
are connected . . . . . . . . . . . . . 39
3.5 The image of pairs of elliptic elements under R
1
. . . . . . . . . . . . 43
CHAPTER 4 CHARACTERIZATIONS OF THE TEICHM
¨
ULLER COMPO-
NENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
CHAPTER 5 THE MAPPING CLASS GROUP . . . . . . . . . . . . . . . 52
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.2 The mapping class group for the one-holed torus . . . . . . . . . . . 53
5.3 Elements of the mapping class group of the one-holed torus . . . . . 54
CHAPTER 6 THE ONE-HOLED TORUS . . . . . . . . . . . . . . . . . . . 56
6.1 The one-holed torus, with group element boundary condition. . . . . 56
6.2 Infinitesimal transitivity . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.3 The action of the mapping class group . . . . . . . . . . . . . . . . . 67
CHAPTER 7 BASIC NOTIONS AND SEWING [PX02] . . . . . . . . . . . 74
7.1 The n-holed torus, with group element boundary condition . . . . . . 75
APPENDIX A GOLDMAN’S RESULTS [Gol03] . . . . . . . . . . . . . . . . 77
TABLE OF CONTENTS – Continued
7
APPENDIX B THE COMMUTATOR OF VECTOR FIELDS . . . . . . . . 83
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
8
LIST OF FIGURES
1.1 The one-holed torus, with group element boundary condition . . . . . 15
2.1 Example of elliptic element in PSU(1, 1). . . . . . . . . . . . . . . . . 19

2.2 Example of hyperbolic element in PSL(2, R). . . . . . . . . . . . . . . 20
2.3 Conjugacy classes in PSU(1, 1). . . . . . . . . . . . . . . . . . . . . . 21
2.4 Conjugacy classes in SU(1, 1). . . . . . . . . . . . . . . . . . . . . . . 22
2.5 Conjugacy classes in PSU(1, 1) and

PSU(1, 1) ([DP03]). . . . . . . . 28
2.6 Conjugacy classes in SU(1, 1) and

SU(1, 1). . . . . . . . . . . . . . . 29
3.1 Neighborhood of a “nonclosed point” in G/ conj . . . . . . . . . . . . 41
3.2 Subset of G/ conj with fibers the conjugacy classes . . . . . . . . . . 43
5.1 The one-holed torus, with group element boundary condition . . . . . 54
5.2 The Dehn Twist ([Bir75] page 167). . . . . . . . . . . . . . . . . . . 55
6.1 The one-holed torus, with group element boundary condition . . . . . 56
7.1 The n-holed torus, with group element boundary condition . . . . . . 76
A.1 Fundamental domain for the three-holed sphere . . . . . . . . . . . . 82
9
ABSTRACT
In this paper we consider the action of the mapping class group of a surface on the
space of homomorphisms from the fundamental group of a surface into PSU(1, 1).
Goldman conjectured that when the surface is closed and of genus bigger than one,
the action on non-Teichm¨uller connected components of the associated moduli space
(i.e. the space of homomorphisms modulo conjugation) is ergodic. One approach to
this question is to use sewing techniques which requires that one considers the action
on the level of homomorphisms, and for surfaces with boundary. In this paper we
consider the case of the one-holed torus with boundary condition, and we determine
regions where the action is ergodic. This uses a combination of techniques developed
by Goldman, and Pickrell and Xia. The basic result is an analogue of the result of
Goldman’s at the level of moduli.
10

CHAPTER 1
INTRODUCTION
Throughout this paper, unless we state otherwise, G will denote an abstract Lie
group which is isomorphic to PSL(2, R). The elements of the group G fall into
three classes: elliptic, parabolic, and hyperbolic. If g ∈ G is elliptic, it is useful
to represent G as the group PSU(1, 1), the group of holomorphic automorphisms
of the unit disk, ∆ ⊂ C; in this case g is conjugate to a rotation of the disk. If
g ∈ G is parabolic or hyperbolic, it is useful to represent G as PSL(2, R), the group
of holomorphic automorphisms of the upper half plane H
2
⊂ C; in this case g is
conjugate to a translation or dilation (this is recalled in more detail in chapter 2).
The fundamental group of G is Z, so that the universal covering map induces an
exact sequence of groups
0 → Z →
˜
G → G → 0. (1.1)
We will construct an explicit model for the universal covering in section 2.3.
Let Σ denote a closed oriented surface with fixed basepoint. Let γ denote the
genus of Σ. The space of homomorphisms Hom

π
1
(Σ), G

is called the representa-
tion variety associated to Σ and G. If we fix a marking of Σ, i.e. a choice of standard
generators of π
1
(Σ), α

1
, β
1
, , α
γ
, β
γ
, then we can identify Hom

π
1
(Σ), G

with the
set
{(g
α
1
, , g
β
γ
) ∈ G
2γ
: [g
α
1
, g
β
1
] [g

α
γ
, g
β
γ
] = 1}.
This is because the group π
1
(Σ) is defined by the single relation [α
1
, β
1
] [α
γ
, β
γ
] = 1.
This clearly displays Hom

π
1
(Σ), G

as an affine algebraic variety. We let H
1
(Σ, G)
denote the space Hom

π
1

(Σ), G

modulo the action of conjugation by G:
G × Hom

π
1
(Σ), G

→ Hom

π
1
(Σ), G

: (g, φ) → conj(g) ◦ φ,
11
where conj(g) denotes the inner automorphism of conjugation by g. This space does
not depend upon the choice of basepoint. The total space H
1
(Σ, G) is not a variety,
because for example, there exist points in this quotient space which are not closed.
(However as we will remark below, these bad points are confined to one component).
Let MCG(Σ) = π
0

Aut(Σ)

denote the mapping class group of Σ (see chapter
5). This group acts naturally on Hom


π
1
(Σ), G

; because Σ is connected, the map-
ping class group can be identified with the isotopy classes (or homotopy classes)
of homeomorphisms which fix our preferred basepoint; with this understood, the
action is given by
MCG(Σ) × Hom

π
1
(Σ), G

→ Hom

π
1
(Σ), G

: ([h], φ) → φ ◦ h
∗
,
where h
∗
is the automorphism of π
1
(Σ) induced by the homeomorphism h (which
fixes our basepoint), and which does not depend upon the choice of h ∈ [h].

In this paper we are interested in the topology of the spaces Hom

π
1
(Σ), G

and
H
1
(Σ, G), and the action of the mapping class group on these spaces. A great deal
is known about this, as we will now recall.
1. What is the topology of H
1
(Σ, G)?
The short exact sequence (1.1) induces a “long exact sequence” of pointed spaces
H
2

Σ, π
1
(G)

H
1

Σ, π
1
(G)

//

H
1
(Σ,
˜
G)
//
H
1
(Σ, G)
BECD
@A
eu
OO
H
0

Σ, π
1
(G)

//
H
0
(Σ,
˜
G)
//
H
0
(Σ, G)

BECD
@A
OO
(1.2)
The connected map
H
1
(Σ, G)
eu
−→ H
2
(Σ, π
1
(G))
∼
=
π
1
(G)
∼
=
Z (1.3)
is given abstractly by mapping the G-bundle associated to a homomorphism to its
Euler class. Milnor ([Mil58]) explicitly realized this map in the following way: Let
12
α
1
, , β
γ
denote generators for π

1
(Σ) satisfying the usual relation,
[α
1
, β
1
] [α
γ
, β
γ
] = 1.
Given a homomorphism φ: π
1
(Σ) → G, choose

φ(α
i
),

φ(β
i
) ∈
˜
G, covering
φ(α
i
), φ(β
i
), respectively. Then [


φ(α
i
),

φ(β
i
)] is independent of the choice of lifts.
Now, [

φ(α
1
),

φ(β
1
)] [

φ(α
γ
),

φ(β
γ
)] covers the identity element, therefore it is an
element of the center of G and hence it is an integer.
Assuming that the genus of Σ is greater than 1, Goldman and Hitchin have
shown that the connected components of the representation variety Hom

π
1

(Σ), G

are separated by the connecting map induced by (1.3),
H
1
(Σ, G) → H
2
(Σ, Z) = Z (1.4)
and the image is the set of integers bounded in magnitude by |χ(Σ)|, where χ(Σ) is
the Euler characteristic. (see [Gol88] Theorem B and [Hit92]). One of the goals of
this paper is to give an almost self-contained proof of this result.
Hitchin has shown that for k = 0, the component of H
1
(Σ, G) with Euler class
k is a smooth manifold which is diffeomorphic to a complex vector bundle of rank
γ − 1 + k over the symmetric product S
2γ−2+k
Σ ([Hit92] Theorem 10.8). Thus in
particular the singular points of H
1
(Σ, G) are confined to the k = 0 component.
However it is not clear how to relate Hitchin’s method to the point of view of this
work.
2. What is the geometric significance of this space?
The connected component that corresponds to the extreme value |χ(Σ)| is iso-
morphic to the set of all possible ways of realizing Σ as a quotient of H
2
, and therefore
it is all the possible universal coverings with marking modulo isomorphism. This is
the Teichm¨uller space of Σ. An open question is to investigate whether there is a

nice geometric description for some part of the other components.
3. What can one say about the action of the mapping class group on H
1
(Σ, G)
or Hom

π
1
(Σ), G

?
13
For example the mapping class group acts properly discontinuously on Te-
ichm¨uller space [GFL00]. At the opposite extreme, for the group SU(2), Gold-
man proved that the mapping class group acts ergodically on H
1

Σ, SU(2)

[Gol97].
In addition Pickrell and Xia generalized this result when K is a connected com-
pact group, [PX02]. Goldman in [Gol03] addresses a closely related question con-
cerning the real part of the moduli space Hom

π
1
(Σ), SL(2, C)

// SL(2, C) where
Σ is the one-holed torus. The group SL(2, C) acts on Hom


π
1
(Σ), SL(2, C)

by
conjugation. The moduli space consists of equivalence classes of elements of
Hom

π
1
(Σ), SL(2, C)

∼
=
SL(2, C) × SL(2, C), where the equivalence class of a ho-
momorphism ρ is defined as the closure of the SL(2, C)−orbit SL(2, C)ρ. For a
synopsis of his results see Appendix A. In a very few lines the idea is as follows:
Let g
α
and g
β
elements in SL(2, C) that correspond to the generators α and β of
π
1
(Σ) and g
c
the group element that corresponds to the boundary. We have that
[g
α

, g
β
] = g
c
. He defines the polynomial
κ(x, y, z) := x
2
+ y
2
+ z
2
− xyz − 2
where
x = tr g
α
, y = tr g
β
, z = tr g
α
g
β
, and κ(x, y, z) = tr g
c
= t.
The space Hom

π
1
(Σ), SL(2, C)


// SL(2, C) identifies with the affine 3-space C
3
(see Appendix A for details). The action of the mapping class group on
Hom

π
1
(Σ), SL(2, C)

// SL(2, C) is commensurable with the action of the group
Aut(κ) on C
3
. Very roughly speaking (among other things) he proves that
• For t < −2 the group Aut(κ) acts properly discontinuous on κ
−1
(t) ∩ R
3
• For −2 ≤ t ≤ 2, there is a compact connected component C
t
of κ
−1
(t) ∩ R
3
;
upon which Aut(κ) acts ergodically; Aut(κ) acts properly discontinuously on
the complement κ
−1
(t) ∩ R
3
\ C

t
;
14
• For t = 2, the action of Aut(κ) is ergodic on the compact subset k
−1
(2) ∩
[−2, 2]
3
and its action is ergodic on the complement κ
−1
(2) \ [−2, 2]
3
;
• For 2 < t ≤ 18, the group Aut(κ) acts ergodically on κ
−1
(t) ∩ R
3
.
• For t > 18 the group Aut(κ) acts properly discontinuous on an open subset of
κ
−1
(t) ∩ R
3
, and ergodically on its complement.
Goldman also showed that there is a very rich geometry underlying the main the-
orem. For t < −2 the level sets correspond to Fricke spaces of one-holed tori with
geodesic boundary. The level set for t = −2 consists of four copies of the Teichm¨uller
space of the puncture torus, together with the origin. The level sets for −2 < t < 2
correspond to Teichm¨uller spaces of singular hyperbolic structures on a torus with
a cone singularity together with a compact component of SU(2) representations.

The space of Hom

π
1
(Σ), G

, depends mainly on π
1
(Σ), and not on Σ itself. Since
the one-holed torus has the same fundamental group as the three holed sphere, for
t > 18 some uniformizations correspond to three-holed spheres.
Now we describe briefly the main problem we are solving in this paper. Let Σ
be the one-holed torus equipped with a basepoint and a boundary component that
is connected to the basepoint as in the figure 1.1. Let Γ
Σ
be the group generated by
the two Dehn twists around s
1
and s
2
in the picture (this is equal to the mapping
class group of Σ-see chapter 5). In this paper we try to understand the action of
Γ
Σ
on Hom

π
1
(Σ), G


(see chapter 6). This is a slightly different question from
what Goldman considers, since he is working with the moduli space and Aut(κ)
rather than the (orientation preserving) mapping class group. It is important for us
that the Dehn twists preserve the orientation of Σ and also they do not move the
boundary loop corresponding to the boundary, since we would like to be able to use
the sewing lemma (see chapter 7) to understand the action of Γ
Σ
when Σ is a higher
genus surface. Nevertheless, we are able to use his results (although at this point I
don’t understand completely his proof) to prove this result.
15
α
α
ββ
c
s
1
s
2
Figure 1.1: The one-holed torus, with group element boundary condition
We can decompose the space of homomorphisms into connected spaces as follows:
Hom

π
1
(Σ), G

=

˜g

c
R
−1
1
( ˜g
c
)
using the lifting of the commutator map:
R
1
: G × G −→
˜
G (g, h) → [˜g,
˜
h] = ˜g
c
The element ˜g
c
projects to g

c
∈ SL(2, R) and depending on the trace of g

c
, the group
Γ
Σ
can act properly discontinuously, or ergodically on R
−1
1

( ˜g
c
).
When the action of Aut(κ) is properly discontinuous on a region of the moduli
space, we are going to be able to use Goldman’s results to prove that the action of
Γ
Σ
on the corresponding region in Hom

π
1
(Σ), G

is properly discontinuous.
To prove ergodicity, we are going to use the technique used in [PX02] where
they prove the ergodicity of the mapping class group on Hom

π
1
(Σ), K

, where K
is a compact group. The group G is noncompact, so we have to use methods which
differ from those in [PX02], (we use machinery developed in [Moo76]). The basic
idea is that Γ
Σ
-ergodicity for almost every boundary condition, is equivalent to a
question of G-transitivity on orbits where G is a continuous group of transformations.
This G transitivity is locally a calculus question about infinitesimal transitivity.
16

But we still need to use Goldman’s results to obtain global transitivity on orbits.
The drawback of this method is that we only obtain results involving almost every
boundary condition. The advantage is that transitivity is easier to study than
ergodicity. There are different types of elements in G, and the [PX02] method only
helps with elliptic type elements. If the tr(g
c
) > 2, Goldman proves a theorem
(theorem 5.2.1 in [Gol03]) that describes a dichotomy on the set of {(x, y, z) ∈ R
3
:
κ(x, y, z) = t}. The theorem roughly speaking says that there exists an element γ
in Aut(κ) so that one of the following occurs:
• γ · (x, y, z) ∈ (−2, 2) × R × R
• γ · (x, y, z) ∈ (−∞, −2)
3
.
Goldman proves ergodicity on the set of triples in κ
−1
(t)∩R
3
where the the first case
occurs. Assuming that we can find γ ∈ Aut(κ) so that γ · (x, y, z) ∈ (−2, 2) ×R ×R,
although Γ
Σ
has fewer elements, we prove that we can find γ

∈ Γ
Σ
so that γ


·(g, h) ∈
Ell ×G, where x = tr g, y = tr h and z = tr gh. Given that we have a pair (g, h)
where g is elliptic, we can use the action of the group G to change also h to an
elliptic element and therefore use the [PX02] method. The result on the action of
Γ
Σ
on Hom

π
1
(Σ), G

is going to be very similar to the main result in [Gol03].
We now describe the contents of this paper. In chapter 2 we give some back-
ground on the conjugacy classes of G (this is standard), and we construct a model
for
˜
G. We also introduce the Poincar´e rotation number, and we use this to give a
picture for the conjugacy classes of
˜
G. Very often we will find it useful to have a
picture of the conjugacy classes of the double cover of G: SL(2, R).
In chapter 3 we look at commutators in
˜
G. We use the rotation number to
give a description of the product of commutators. Although the rotation number
is conjugation invariant, it is hard to work with analytically. So we use the ideas
in [EHN81], and we introduce certain estimates m and m that are not conjugation
invariant but are easier to work with. Most of the proofs of these results are taken
17

from [EHN81]. Among other things, we prove that the product of commutators
in
˜
G is bounded by the value |χ(Σ)|. This implies the bound of Wood mentioned
above. We conjecture that the level sets of the commutator map R
p
for p > 1 are
connected. We use an induction argument, which is based on a foundational result
of Goldman [Gol88] (Theorem 7.1) that the level sets of R
1
are connected.
In chapter 4 we give different characterizations of the connected component that
corresponds to the extreme value χ(Σ). We conclude that given a homomorphism
φ : π
1
(Σ) → G that belongs in the extreme component, we can realize Σ as a
quotient of H
2
, by a discrete subgroup of G which is isomorphic to π
1
(Σ). This
subgroup acts freely and properly discontinuously on H
2
. In this chapter we mainly
use results from [Mat87] and [Mil58].
In chapter 5, we state basic facts about the mapping class group. We define Γ
Σ
to be the group generated by the Dehn twists around s
1
and s

2
on the one-holed
torus (see figure 1.1). Notice that Γ
Σ
preserves the orientation.
In chapter 6, we investigate the action of Γ
Σ
on Hom(π
1
(Σ), G), when Σ is the
one holed torus. We start by defining Σ and Γ
Σ
. In section 6.2 we give the result on
infinitesimal transitivity, which is going to be crucial for the proof of the ergodicity
of the mapping class group. In section 6.3 we state the main theorem of this paper.
To proceed the proof of the theorem we state and prove two key lemmas that will
help us use some of Goldman’s results to complete the proof of our theorem.
In chapter 7, we discuss the sewing lemma (the analogue for homomorphisms of
the Seifert-Van Kampen theorem). Our original intension was to investigate the ac-
tion of the mapping class group on components of Hom

π
1
(Σ, G)

corresponding to
k < |χ(Σ)|. This motivates our interest in Hom(π
1
(Σ, G)


(as opposed to H
1
(Σ, G)).
However this has turned out to be far more complicated than we originally imagined.
In Appendix A we give a synopsis of Goldman’s results and in Appendix B we
present a calculation that was necessary during the proof of infinitesimal transitivity
in section 6.2.
18
CHAPTER 2
BACKGROUND AND NOTATION
2.1 Conjugacy classes of PSU(1, 1)
The Lie group PSU(1, 1) is the group of holomorphic automorphisms of the unit
disk.
PSU(1, 1) =





a b
¯
b ¯a


: |a|
2
− |b|
2
= 1, a, b ∈ C




/{±I}
where
PSU(1, 1) ×
¯
∆ −→
¯
∆ : (g, z) −→
az + b
¯
bz + ¯a
.
The Lie group PSL(2, R) is the group of automorphisms of the upper half space. By
conjugating the whole group PSU(1, 1) by the map Φ(z) =
z − i
z + i
, which maps the
unit disk to the upper half space, we obtain PSL(2, R). In the following discussion
we think of having an abstract group G, which we can identify with PSU(1, 1) or
PSL(2, R) when convenient.
For g =

a b
¯
b ¯a

∈ SU(1, 1), the characteristic equation is
λ
2

− (tr g)λ + det g = 0.
First notice that tr g is a real number. Since det g = 1, the characteristic equation
can be written as
λ
2
− (tr g)λ + 1 = 0
and therefore
λ =
(tr g) ±

(tr g)
2
− 4
2
.
There are three cases:
19
• If | tr g| < 2, g has two non-real eigenvalues λ and
¯
λ with |λ| = 1. Since
the eigenvalues are distinct, g is diagonalizable over C. Every element g in
SU(1, 1) with that property is called elliptic and g will belong in the conjugacy
class of
g =


λ 0
0
¯
λ



∈ SU(1, 1), where spectrum g = {λ,
¯
λ}.
Geometrically, this representative corresponds to the M¨obius transformation
η : z −→ λ
2
z, that rotates the unit disk by 2φ where λ = e
iφ
. (See figure 2.1)
Figure 2.1: Example of elliptic element in PSU(1, 1).
• If | tr g| > 2 the above equation has two distinct real solutions. If λ is an
eigenvalue of g,
1
λ
should be the other eigenvalue since det g = 1. Every
element g of PSU(1, 1) with the property | tr g| > 2 is called hyperbolic. In
thinking about hyperbolic elements, it is convenient to work with PSL(2, R).
Since g has two distinct eigenvalues, it can be diagonalized over R and g will
belong in the conjugacy class of


λ 0
0 λ
−1


∈ SL(2, R), where spectrum g = {λ,
1

λ
}.
20
This representative geometrically corresponds to the M¨obius transformation
that dilates the upper half plane by the map η : z −→ λ
2
z where λ is real (See
figure 2.2).
Figure 2.2: Example of hyperbolic element in PSL(2, R).
• If | tr g| = 2 the element g is called parabolic. Thinking about parabolic ele-
ments, it is convenient to work with PSL(2, R). Since | tr g| = 2, g has one real
eigenvalue λ = 1 or λ = −1. The possible Jordan canonical forms of a matrix
with the above eigenvalues are (
1 a
0 1
) ,

−1 a
0 −1

, or (
1 0
0 1
). If we conjugate each
of these matrices by the matrix

1
√
a
1

0
√
a

, we get the matrices (
1 1
0 1
) , (
1 −1
0 1
),
or (
1 0
0 1
) respectively. Therefore we can choose those as representatives from
each conjugacy class and hence each parabolic element will belong in the con-
jugacy class of one of the above matrices. Geometrically, these representatives
will correspond to the M¨obius transformation η(z) = z + 1, η(z) = z − 1, or
η(z) = z respectively.
One can arrive at this classification using geometry. Given any map of
¯
∆ →
¯
∆, there
is at least one fixed point. Elliptic elements have a single fixed point in the interior
of the unit disk. Parabolic elements have one fixed point on S
1
(the boundary of
the unit disk), and hyperbolic elements have two fixed points on S
1

. Every g ∈ G,
21
is conjugate to one of the above cases. For the picture of the conjugacy classes for
Par
Hyp
Ell
Par
+
−
|tr|
2
I
Figure 2.3: Conjugacy classes in PSU(1, 1).
PSU(1, 1) see figure 2.3, where Ell, Hyp, Par denote the sets of elliptic, hyperbolic
and parabolic elements of G respectively. Each point of the real line corresponds to a
conjugacy class of PSL(2, R) with representative

λ 0
0
1
λ

∈ PSL(2, R), where λ ∈
R. Each point of the circle corresponds to a conjugacy class of PSU(1, 1) with
representative

e
iφ
0
0 e

−iφ

where φ ∈ [0, 2π]. The bifurcation point corresponds to
the three conjugacy classes Par
±
and I, with representatives [(
1 1
0 1
)], [(
1 −1
0 1
)] and
[(
1 0
0 1
)] respectively.
Note that Par
±
are not closed points. If we approach that point from above,
limit points are Par
+
and the identity, if we approach that point from below, limit
points are Par
−
and the identity.
2.2 Conjugacy classes for the double cover SU(1, 1)
In many cases, it is also useful to look at the conjugacy classes in SU(1, 1), the
double covering space of PSU(1, 1) (see figure 2.4).
2.3 A model for


PSU(1, 1)
Recall that PSU(1, 1) is the group of holomorphic automorphisms of the unit disk,
and we can represent this group as:
22
Hyp
Ell
Ell
Par Par
Par
Par
− Hyp+
Par
+
−
+
−
+
Par−
−I I
I
Hyp
Figure 2.4: Conjugacy classes in SU(1, 1).
PSU(1, 1) =





a b
¯

b ¯a


: |a|
2
− |b|
2
= 1, a, b ∈ C



/{±I}
The Lie group PSU(1, 1) acts on the unit circle, and the action is given by
g · z =
az + b
¯
bz + ¯a
where g =


a b
¯
b ¯a


∈ PSU(1, 1) and z = e
2πit
, t ∈ R. In terms of the universal
covering,
Z −→ R −→ S

1
t −→ e
2πit
we can realize the universal covering of D = Homeo
+
(S
1
) as
˜
D = {Homeo
+
(R) :
˜
f(t + 1) =
˜
f(t) + 1}.
where
0 −→ Z −→
˜
D −→ D −→ 0
˜
f −→ f
23
and
R −→ R t →
˜
f(t)
↓ ↓ ↓ ↓
S
1

−→ S
1
e
2πit
→ e
2πi
˜
f(t)
˜
D is in fact contractible:
I ×
˜
D −→
˜
D
(t,
˜
f(t)) −→ st + (1 − s)
˜
f(t)
One can use this approach to abstractly realize

PSU(1, 1), the universal covering of
PSU(1, 1). However we will construct an explicit model.
By definition as a set,

PSU(1, 1) =




˜g =




a b
¯
b ¯a


, A


∈ SU(1, 1) × C : e
πiA
= a



.
Then (

PSU(1, 1), *) is a group with the multiplication * defined as follows: Let
g, g

∈ SU(1, 1). Then g =

a b
¯
b ¯a


and g

=

a

b

¯
b

¯
a


where |a|
2
−|b|
2
= 1, |a

|
2
−|b

|
2
= 1,
and a, a


, b, b

∈ C. We define
(g, A) ∗ (g

, A

)
def
=

gg

, A + A

+
1
πi
log(1 +
b
¯
b

aa

)

The multiplication is well defined since
aa


+ b
¯
b

= aa

(1 + a
−1
ba
−1
¯
b

) = e
πiA
e
πiA

e
log(1+
b
¯
b

aa

)
.
The condition |a|

2
− |b|
2
= 1 gives us that




b
¯
b

aa





< 1 since
|a|
2
|a|
2
−
|b|
2
|a|
2
=
1

|a|
2
=⇒ 1 −




b
a




2
=
1
|a|
2
=⇒




b
a




2

= 1 −
1
|a|
2
< 1
24
Similarly




b
−1
a





< 1 and therefore log(1 +
bb

aa

) is well defined. Although we will not
do so here, it can be checked that ∗ is a multiplication. We have
0 −→ Z →

PSU(1, 1) −→ PSU(1, 1) −→ 0
n −→


−1 0
0 −1

n
, inπ

(g, A) −→ ±g
As we said before, PSU(1, 1) acts on the unit circle and, therefore,

PSU(1, 1) should
act on the real line. Given ˜g ∈

PSU(1, 1),
R −→ R t −→ ˜g · t
↓ ↓ ↓ ↓
S
1
−→ S
1
z −→
az + b
¯
bz + ¯a
where formally
˜g · t =
1
2πi
log(
az + b

¯
bz + ¯a
) where z = e
2πit
, t ∈ R
and we must make sense of the logarithm (especially how it depends on t and A).
Proposition 2.3.1 The action of ˜g = (g, A) on R is given by
˜g · t = t + Re A +
1
π
Im log(1 + w)
where w = a
−1
z
−1
b.
Proof First consider the special case of diagonal elements of

PSU(1, 1):
˜g = ((
a 0
0 ¯a
) , A) where a = e
πiA
and A ∈ R.
Then
˜g · t =
1
2πi
log


az
¯a

=
1
2πi
log

e
2πiA
· e
2πit

= t + A,