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 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. 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. [...]... 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. .. 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... realizing Σ as a quotient of H2 , and therefore it is all the possible universal coverings with marking modulo isomorphism This is the Teichm¨ller space of Σ An open question is to investigate whether there is a u 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 π1 (Σ), G ? 13 For example the mapping class. .. 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 ΓΣ when Σ is a higher genus surface Nevertheless, we are able to use his results (although at... forms of a matrix with the above eigenvalues are ( 1 a ) , 0 1 of these matrices by the matrix 1 √ a 1 √ a 0 −1 a 0 −1 , or ( 1 0 ) If we conjugate each 0 1 , we get the matrices ( 1 1 ) , ( 1 −1 ), 0 1 0 1 or ( 1 0 ) respectively Therefore we can choose those as representatives from 0 1 each conjugacy class and hence each parabolic element will belong in the conjugacy class of one of the above matrices... 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. .. , A + A + ¯ bb 1 log(1 + ) πi aa The multiplication is well defined since ¯ bb aa + b¯ = aa (1 + a 1 ba −1¯ ) = eπiA eπiA elog(1+ aa ) b b b¯ b < 1 since The condition |a| 2 − |b|2 = 1 gives us that aa b |a| 2 |b|2 1 − 2 = 2 =⇒ 1 − 2 |a| |a| |a| a 2 = b 1 =⇒ 2 |a| a 2 =1− 1 . 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. 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. 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