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

As members of the Final Examination Committee, we certify that we have read thedissertation prepared by Panagiota Savva Konstantinouentitled Homomorphisms of the Fundamental Group of a S

byPanagiota Savva Konstantinou

A Dissertation Submitted to the Faculty of theDEPARTMENT OF MATHEMATICS

In Partial Fulfillment of the Requirements

For the Degree ofDOCTOR OF PHILOSOPHY

In the Graduate CollegeTHE UNIVERSITY OF ARIZONA

2 0 0 6

3214652 2006

UMI Microform Copyright

All rights reserved This microform edition is protected against unauthorized copying under Title 17, United States Code.

ProQuest Information and Learning Company

300 North Zeeb Road P.O Box 1346 Ann Arbor, MI 48106-1346

by ProQuest Information and Learning Company

As members of the Final Examination Committee, we certify that we have read thedissertation prepared by Panagiota Savva Konstantinou

entitled Homomorphisms of the Fundamental Group of a Surface intoPSU(1, 1), and the Action of the Mapping Class Group

and recommend that it be accepted as fulfilling the dissertation requirement for theDegree of Doctor of Philosophy

STATEMENT BY AUTHOR

This dissertation has been submitted in partial fulfillment of requirements for anadvanced degree at The University of Arizona and is deposited in the UniversityLibrary 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 permissionfor extended quotation from or reproduction of this manuscript in whole or in partmay be granted by the head of the major department or the Dean of the GraduateCollege when in his or her judgment the proposed use of the material is in theinterests of scholarship In all other instances, however, permission must be obtainedfrom the author

SIGNED:

Panagiota Savva Konstantinou

I express my deepest graditute to my thesis advisor Doug Pickrell, for his excellenceguidance, and advice, his continual encouragement and incredible patience, and forcaring, during the years of my doctoral work I want to thank him especially, forbeing such an inspiration to me, and for giving me the motivation to continue inthe PhD program.

I thank my Final Defense Committee: Phillip Foth, David Glickenstein, DouglasUlmer, for carefully reading through my dissertation and giving me many suggestions

on improving it I thank my external reviewer, William Goldman, for taking thetime to review my paper, and for helping me with a lot of questions I had over thelast 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, PantelisDamianou, Christos Pallikaros, Christodoulos Sofokleous and Evangelia Samiou forthe 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 myclose 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 theirhelp in classes and their support during these years These include Maria Agro-tis, Lisa Berger, Arlo Caine, Derek Habermas, Selin Kalaycioglu, Alex Perlis andSacha Swenson Also many thanks to the wonderful staff of the Department ofMathematics

I give a special thank you to my close friends that have been more than afamily to me here in Tucson: Nakul Chitnis; Luis Garcia-Naranjo; Adam Spiegler;Rosangela Sviercoski; Gabriella, Eleni, Alexandros and Pavlos Michaelidou; AntonioColangelo 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 AnnaSidera for their continual support and the great summers I spent with them

I thank my grandparents Panagiota and Charalambos Konstantinou, MariaZeniou and my spiritual father Michalis Pigasiou for their continual wishes andprayers; 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

To my parents, Andreani and Savvas Konstantinou, who made all this possible

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) 21g 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(Rp) 36

3.4 Conjecture: The level sets of Rp are connected 39

3.5 The image of pairs of elliptic elements under R1 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

APPENDIX B THE COMMUTATOR OF VECTOR FIELDS 83REFERENCES 85

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]).g 28

2.6 Conjugacy classes in SU(1, 1) and SU(1, 1).g 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

In this paper we consider the action of the mapping class group of a surface on thespace 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 tothis 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 weconsider the case of the one-holed torus with boundary condition, and we determineregions 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 ofGoldman’s at the level of moduli

CHAPTER 1

INTRODUCTION

Throughout this paper, unless we state otherwise, G will denote an abstract Liegroup which is isomorphic to PSL(2, R) The elements of the group G fall intothree 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 H2 ⊂ C; in this case g isconjugate 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 anexact sequence of groups

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 thegenus of Σ The space of homomorphisms Hom π1(Σ), G

is called the tion variety associated to Σ and G If we fix a marking of Σ, i.e a choice of standardgenerators of π1(Σ), α1, β1, , αγ, βγ, then we can identify Hom π1(Σ), G

representa-with theset

where conj(g) denotes the inner automorphism of conjugation by g This space doesnot depend upon the choice of basepoint The total space H1(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) = π0 Aut(Σ)

denote the mapping class group of Σ (see chapter5) This group acts naturally on Hom π1(Σ), G

; because Σ is connected, the ping class group can be identified with the isotopy classes (or homotopy classes)

map-of homeomorphisms which fix our preferred basepoint; with this understood, theaction is given by

MCG(Σ) × Hom π1(Σ), G

→ Hom π1(Σ), G

: ([h], φ) → φ ◦ h∗,where h∗ is the automorphism of π1(Σ) induced by the homeomorphism h (whichfixes 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 π1(Σ), G

and

H1(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 H1(Σ, G)?

The short exact sequence (1.1) induces a “long exact sequence” of pointed spaces

α1, , βγ denote generators for π1(Σ) satisfying the usual relation,

[α1, β1] [αγ, βγ] = 1

Given a homomorphism φ : π1(Σ) → G, choose φ(α]i),φ(β]i) ∈ G, covering˜φ(αi), φ(βi), respectively Then [φ(α]i),φ(β]i)] is independent of the choice of lifts.Now, [^φ(α1),φ(β]1)] [φ(α^γ),^φ(βγ)] covers the identity element, therefore it is anelement of the center of G and hence it is an integer

Assuming that the genus of Σ is greater than 1, Goldman and Hitchin haveshown that the connected components of the representation variety Hom π1(Σ), G
are separated by the connecting map induced by (1.3),

and the image is the set of integers bounded in magnitude by |χ(Σ)|, where χ(Σ) isthe Euler characteristic (see [Gol88] Theorem B and [Hit92]) One of the goals ofthis paper is to give an almost self-contained proof of this result

Hitchin has shown that for k 6= 0, the component of H1(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 S2γ−2+kΣ ([Hit92] Theorem 10.8) Thus inparticular the singular points of H1(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 thiswork

2 What is the geometric significance of this space?

The connected component that corresponds to the extreme value |χ(Σ)| is morphic to the set of all possible ways of realizing Σ as a quotient of H2, and therefore

iso-it is all the possible universal coverings wiso-ith marking modulo isomorphism This isthe Teichm¨uller space of Σ An open question is to investigate whether there is anice geometric description for some part of the other components

3 What can one say about the action of the mapping class group on H1(Σ, G)

or Hom π1(Σ), G

?

For example the mapping class group acts properly discontinuously on ichm¨uller space [GFL00] At the opposite extreme, for the group SU(2), Gold-man proved that the mapping class group acts ergodically on H1 Σ, SU(2)

Te-[Gol97]

In addition Pickrell and Xia generalized this result when K is a connected pact group, [PX02] Goldman in [Gol03] addresses a closely related question con-cerning the real part of the moduli space Hom π1(Σ), SL(2, C)

com-// SL(2, C) where

Σ is the one-holed torus The group SL(2, C) acts on Hom π1(Σ), SL(2, C)

byconjugation The moduli space consists of equivalence classes of elements ofHom π1(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 asynopsis of his results see Appendix A In a very few lines the idea is as follows:Let gα and gβ elements in SL(2, C) that correspond to the generators α and β of

π1(Σ) and gc the group element that corresponds to the boundary We have that[gα, gβ] = gc He defines the polynomial

κ(x, y, z) := x2+ y2+ z2− xyz − 2

where

x = tr gα, y = tr gβ, z = tr gαgβ, and κ(x, y, z) = tr gc = t

The space Hom π1(Σ), SL(2, C)

// SL(2, C) identifies with the affine 3-space C3

(see Appendix A for details) The action of the mapping class group onHom π1(Σ), SL(2, C)

// SL(2, C) is commensurable with the action of the groupAut(κ) on C3 Very roughly speaking (among other things) he proves that

• For t < −2 the group Aut(κ) acts properly discontinuous on κ−1(t) ∩ R3

• For −2 ≤ t ≤ 2, there is a compact connected component Ct of κ−1(t) ∩ R3;upon which Aut(κ) acts ergodically; Aut(κ) acts properly discontinuously onthe complement κ−1(t) ∩ R3\ Ct;

• 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) ∩ R3

• For t > 18 the group Aut(κ) acts properly discontinuous on an open subset of

κ−1(t) ∩ R3, and ergodically on its complement

Goldman also showed that there is a very rich geometry underlying the main orem For t < −2 the level sets correspond to Fricke spaces of one-holed tori withgeodesic boundary The level set for t = −2 consists of four copies of the Teichm¨ullerspace of the puncture torus, together with the origin The level sets for −2 < t < 2correspond to Teichm¨uller spaces of singular hyperbolic structures on a torus with

the-a cone singulthe-arity together with the-a compthe-act component of SU(2) representthe-ations.The space of Hom π1(Σ), G

, depends mainly on π1(Σ), and not on Σ itself Sincethe 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 ΓΣ be the group generated bythe two Dehn twists around s1 and s2 in the picture (this is equal to the mappingclass group of Σ-see chapter 5) In this paper we try to understand the action of

ΓΣ on Hom π1(Σ), G

(see chapter 6) This is a slightly different question fromwhat 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 usthat the Dehn twists preserve the orientation of Σ and also they do not move theboundary loop corresponding to the boundary, since we would like to be able to usethe sewing lemma (see chapter 7) to understand the action of ΓΣ when Σ is a highergenus surface Nevertheless, we are able to use his results (although at this point Idon’t understand completely his proof) to prove this result

Figure 1.1: The one-holed torus, with group element boundary condition

We can decompose the space of homomorphisms into connected spaces as follows:

The element ˜gc projects to g0

c ∈ SL(2, R) and depending on the trace of g0

c, the group

ΓΣ can act properly discontinuously, or ergodically on R1−1( ˜gc)

When the action of Aut(κ) is properly discontinuous on a region of the modulispace, we are going to be able to use Goldman’s results to prove that the action of

ΓΣ on the corresponding region in Hom π1(Σ), G

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 everyboundary condition The advantage is that transitivity is easier to study thanergodicity There are different types of elements in G, and the [PX02] method onlyhelps with elliptic type elements If the tr(gc) > 2, Goldman proves a theorem(theorem 5.2.1 in [Gol03]) that describes a dichotomy on the set of {(x, y, z) ∈ R3 :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)∩R3 where the the first caseoccurs Assuming that we can find γ ∈ Aut(κ) so that γ · (x, y, z) ∈ (−2, 2) × R × R,although ΓΣhas fewer elements, we prove that we can find γ0 ∈ ΓΣso that γ0·(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 anelliptic element and therefore use the [PX02] method The result on the action of

ΓΣ on Hom π1(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 ground on the conjugacy classes of G (this is standard), and we construct a modelfor ˜G We also introduce the Poincar´e rotation number, and we use this to give apicture for the conjugacy classes of ˜G Very often we will find it useful to have apicture of the conjugacy classes of the double cover of G: SL(2, R)

back-In chapter 3 we look at commutators in ˜G We use the rotation number togive 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 conjugationinvariant but are easier to work with Most of the proofs of these results are taken

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 mentionedabove We conjecture that the level sets of the commutator map Rp for p > 1 areconnected We use an induction argument, which is based on a foundational result

of Goldman [Gol88] (Theorem 7.1) that the level sets of R1 are connected

In chapter 4 we give different characterizations of the connected component thatcorresponds to the extreme value χ(Σ) We conclude that given a homomorphism

φ : π1(Σ) → G that belongs in the extreme component, we can realize Σ as aquotient of H2, by a discrete subgroup of G which is isomorphic to π1(Σ) Thissubgroup acts freely and properly discontinuously on H2 In this chapter we mainlyuse results from [Mat87] and [Mil58]

In chapter 5, we state basic facts about the mapping class group We define ΓΣ

to be the group generated by the Dehn twists around s1 and s2 on the one-holedtorus (see figure 1.1) Notice that ΓΣ preserves the orientation

In chapter 6, we investigate the action of ΓΣ on Hom(π1(Σ), G), when Σ is theone holed torus We start by defining Σ and ΓΣ In section 6.2 we give the result oninfinitesimal 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 willhelp 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 ofthe Seifert-Van Kampen theorem) Our original intension was to investigate the ac-tion of the mapping class group on components of Hom π1(Σ, G)

corresponding to

k < |χ(Σ)| This motivates our interest in Hom(π1(Σ, G)

(as opposed to H1(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 wepresent a calculation that was necessary during the proof of infinitesimal transitivity

in section 6.2

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 unitdisk

z + i, which maps theunit 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) orPSL(2, R) when convenient

λ = (tr g) ±

p(tr g)2 − 4

There are three cases:

• If | tr g| < 2, g has two non-real eigenvalues λ and ¯λ with |λ| = 1 Sincethe eigenvalues are distinct, g is diagonalizable over C Every element g inSU(1, 1) with that property is called elliptic and g will belong in the conjugacyclass of

 ∈ SU(1, 1), where spectrum g = {λ, ¯λ}

Geometrically, this representative corresponds to the M¨obius transformation

η : z −→ λ2z, that rotates the unit disk by 2φ where λ = eiφ (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 aneigenvalue of g, 1

λ should be the other eigenvalue since det g = 1 Everyelement g of PSU(1, 1) with the property | tr g| > 2 is called hyperbolic Inthinking 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 willbelong in the conjugacy class of

This representative geometrically corresponds to the M¨obius transformationthat dilates the upper half plane by the map η : z −→ λ2z where λ is real (Seefigure 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 ments, it is convenient to work with PSL(2, R) Since | tr g| = 2, g has one realeigenvalue λ = 1 or λ = −1 The possible Jordan canonical forms of a matrixwith the above eigenvalues are (1 a

ele-0 1) , −1 a

0 −1

, or (1 0

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 S1 (the boundary ofthe unit disk), and hyperbolic elements have two fixed points on S1 Every g ∈ G,

is conjugate to one of the above cases For the picture of the conjugacy classes for

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, hyperbolicand parabolic elements of G respectively Each point of the real line corresponds to aconjugacy class of PSL(2, R) with representative hλ 0

0 1 λ

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, limitpoints 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), thedouble 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:

b 0 a¯ 0

where |a|2−|b|2 = 1, |a0|2−|b0|2 = 1,and a, a0, b, b0 ∈ C We define

(g, A) ∗ (g0, A0)def= gg0, A + A0+ 1

πilog(1 +

b¯b0

aa0)
The multiplication is well defined since

aa0 + b¯b0 = aa0(1 + a−1ba0−1¯b0) = eπiAeπiA 0

elog(1+aa0b¯b0).The condition |a|2− |b|2 = 1 gives us that

b¯b

0

aa0

< 1 since

ba

...

3.3 A description of the Image(Rp)

We start by proving the first part of the. .. self-contained

Proposition 3.1.1

We present the proof of the first and the third part of. .. consider the special case of diagonal elements ofPSU(1, 1):^

since |a| 2