Sphere systems in 3-manifolds and arc graphs by Francesca Iezzi Thesis Submitted to the University of Warwick for the degree of Doctor of Philosophy Mathematics March 2016 Contents Acknowledgments iii Declarations vi Abstract vii Introduction 1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Chapter 1 Sphere graphs of manifolds with holes 1 1.1 Some questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Chapter 2 Standard form for sphere systems 5 2.1 Intersection of spheres, Minimal and Standard form . . . . . . . . . 6 2.1.1 Spheres, partitions and intersections . . . . . . . . . . . . . . 7 2.1.2 Minimal and standard form . . . . . . . . . . . . . . . . . . . 15 2.2 Dual square complex . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Inverse construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 The core of two trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.5 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.6 The case where the two systems contain spheres in common . . . . . 56 2.6.1 Constructing a dual square complex in the case where the two sphere systems contain spheres in common . . . . . . . . . . 56 2.6.2 Inverse construction in the general case . . . . . . . . . . . . 58 2.6.3 The core of two trees containing edges in common . . . . . . 60 2.6.4 Consequences in the case of sphere systems with spheres in common . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 i 2.7 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Chapter 3 Sphere and arc graphs 66 3.1 Injections of arc graphs into sphere graphs . . . . . . . . . . . . . . . 67 3.2 Proof of Theorem 3.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.2.1 Defining the retraction: first naive idea and main problems arising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.2.2 Efficient position for spheres and surfaces . . . . . . . . . . . 71 3.2.3 Is the retraction well defined? . . . . . . . . . . . . . . . . . . 76 3.2.4 Proof of Theorem 3.1.2 . . . . . . . . . . . . . . . . . . . . . 81 3.3 Some consequences about the diameter of sphere graphs . . . . . . . 83 3.4 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Appendix A Maps between graphs and surfaces 86 Appendix B A further proof of Lemma 2.2.3 89 Bibliography 96 ii Acknowledgments It seems like it was only yesterday that I left my country, my family and friends, to come to a different place, speaking a language which I was not very familiar with. On the one hand I remember being excited, but on the other hand very sad and scared. Now that this “cycle of my life”has almost come to an end, I look back and I acknowledge that these four years have been a very important part of my life, and that, throughout this period I grew a lot both from an educational point of view and from a personal point of view. I would not be at this stage without the help of a lot of people, who deserve all of my gratitude. First of all, I am really grateful to my advisor, Brian Bowditch, for accepting me as his student, and above all for his kind and patient guidance and advice. I thank him for teaching me a lot and challenging me often. I have learnt a lot from his teaching and from his impressive mathematical talent. I am very thankful to Karen Vogtmann and Arnaud Hilion for agreeing to examine my work, for many useful comments on how to improve my thesis, and for making my thesis defence an instructive and enjoyable experience. I would like to thank Saul Schleimer for very useful and stimulating mathe- matical conversations, and for pointing out a mistake in an early draft of my thesis. I thank Sebastian Hensel for many stimulating discussions. I am thankful to all who have been members of the Geometry and Topology group at Warwick during these four years for many interesting discussions and for the support they have given me. A special mention goes to Beatrice and Federica for their support in the pre-submission and post-submission period. iii I am extremely grateful to all the people who, over these four years and in particular in these last few months, have generously spent part of their time to help me overcome some difficulties in using LaTex and inkscape. IthankmyfellowgraduatestudentsandyoungpostdocsatWarwick,Ithank in particular those with whom I had the possibility to share the joys, but also the difficulties and challenges a graduate student constantly faces. Without them, I would have felt much more lonely over the last four years. I am grateful to all the staff working in the Maths Department at Warwick University, for their efficiency, and for making the department a wonderful work environment. IamverythankfultoTheEngineeringandPhysicalSciencesResearchCoun- cil for founding my doctoral studies. Two years ago I spent seven fruitful and unforgetable months in Tokyo, vis- iting the Tokyo Institute of Technology. This experience was very fruitful workwise and, on the other hand gave me the opportunity to challenge myself, and to experi- ence a different culture. I could never be thankful enough to my advisor, for giving me the opportunity to have this experience, and to Prof Kojima and his lab at the Tokyo Institute of Technology, for welcoming me and helping me in overcoming many of the practical issues. I would like to express my gratitude to Fr Harry and the members of the CatholicSocietyattheUniversityofWarwick. MylifeatWarwickwouldhavelacked something if I had not had the opportunity to be part of this Society. Through the society I met some of my best friends. Moreover, attending the society activities and sharing my experiences with other people helped me a lot in looking at things from a different perspective and in finding the right balance in different aspects of my life. These years would have been much sadder and lonelier without the help and the support of many friends. I apologise for not mentioning them by name, but they are many, and I would risk forgetting to mention somebody. I thank many good friends I have met while studying in Pisa, whom, despite the distance, have iv continued to be people I could count on. I thank friends whom I have met during my studies at Warwick, with whom I have shared happy moments, and to whom I have turned during the most challenging periods to seek support. Some of them have spent a long period here, some of them have spent too short a period here, unfortunately, but all of them have been and are an important part of my life. Finally, I could never thank my family enough. It is thanks to them that I developed the curious personality which led me to undertake this Ph.D. I’d like in particular to thank my parents for continuously supporting me during these last four years, and for helping me to keep my feet on the ground. A special mention goestomymotherforherpracticalhelpinsomecrucialmomentsoverthelastyears as well as for her neverending support. v Declarations I declare that the material in this thesis is, to the best of my knowledge, my own except where otherwise indicated or cited in the text, or else where the material is widely known. This material has not been submitted for any other degree, and only the material from Chapter 3 has been submitted for peer-reviewed pubblication. All figures were created by the author using inkscape. vi Abstract We present in this thesis some results about sphere graphs of 3-manifolds. IfwedenoteasM theconnectedsumofg copiesofS2×S1, thesphere graph g of M , denoted as S(M ), is the graph whose vertices are isotopy classes of essential g g spheres in M , where two vertices are adjacent if the spheres they represent can be g realised disjointly. Sphere graphs have turned out to be an important tool in the study of outer automorphisms groups of free groups. The thesis is mainly focused on two projects. As a first project, we develop a tool in the study of sphere graphs, via analysing the intersections of two collections of spheres in the 3-manifold M . g Elaborating on Hatcher’s work and on his definition of normal form for spheres ([15]), we define a standard form for two embedded sphere systems (i.e collections of disjoint spheres) in M . g We show that such a standard form exists for any couple of maximal sphere systems in M , and is unique up to homeomorphisms of M inducing the identity g g on the fundamental group. Our proof uses combinatorial and topological methods. We basically show that most of the information about two embedded maximal sphere systems in M g is contained in a 2-dimensional CW complex, which we call the square complex associated to the two sphere systems. The second project concerns the connections between arc graphs of surfaces and sphere graphs of 3-manifolds. If S is a compact orientable surface whose fundamental group is the free group F , then there is a natural injective map i from the arc graph of the surface S g to the sphere graph of the 3-manifold M . It has been proved ([12]) that this map g is an isometric embedding. We prove, using topological methods, that the map i admits a coarsely de- fined Lipschitz left inverse. vii Introduction The main objects of study during my Ph.D have been sphere graphs of 3-manifolds and arc graphs of surfaces. I focused in particular on the connection between these two spaces. Both objects play an important role in Geometric Group Theory, since they actasimportanttoolsinthestudyofsomeofthecentraltopicsinthearea: Mapping Class Groups of surfaces and Outer Automorphisms of Free Groups. Background Given a compact orientable surface S, the Mapping Class Group of S (we will denote it as Mod(S)) is the group of isotopy classes of orientation preserving self- homeomorphismsofthesurface. Thegroupofallisotopyclassesofself-homeomorphisms of S (both orientation preserving and orientation reversing) is often called the ex- tended Mapping Class Group and denoted as Mod±(S). Note that Mod(S) is a subgroup of Mod±(S) of index two. These objects have been among the most important objects of study in Geo- metric topology over the last sixty years. A very important theorem states that the Mapping Class group of a closed orientable surface S is generated by Dehn twists around simple closed curves in S; a proof of this result can be found in [28]. Lick- orish proved ([29]) that a finite number of Dehn twists is sufficient to generate the Mapping Class Group of a closed surface. A lot of progress has been made and inspired by Thurston. KeytoolsinthestudyofsurfaceMappingclassgroupsareTeichmu¨llerspaces and curve complex, which I define below. The Teichmuller space of a surface S is the space of marked hyperbolic met- rics on S up to isotopy. The group Mod(S) acts on the Teichmuller space of S properly discontinuously. I refer to [8] for some more detailed background about surface Mapping Class Groups and Teichmu¨ller spaces. 1 The curve complex of a surface S (usually denoted as C(S)) is the simplicial complex whose vertices are isotopy classes of simple closed curves on the surface S, where k+1 curves span a k-simplex if they can be realised disjointly. This complex was first introduced by Harvey ([14]) as a combinatorial tool for the study of Teichmuller spaces. Ivanov ([23]) proved that, if S is a surface of genus at least two, then the group of simplicial automorphisms of the complex C(S) is the group Mod±(S). Great progress in the study of the geometry of the curve complex has been made by Masur and Minsky ([32] and [33]). In particular they prove ([32]) that the curvecomplexishyperbolic. Shorterproofofthehyperbolicityofthecurvecomplex using combinatorial methods can be found in [3] and [19]. The arc complex of a surface S with boundary is an analogue of the curve complex. Vertices in this complex are isotopy classes of embedded essential arcs in S; again k +1 arcs span a k-simplex if they can be realised disjointly. The arc complex has also been proven to be hyperbolic. Independent proofs can be found in [20], [31] and [19]. In particular, in [19] the authors show that the hyperbolicity constant does not depend on the surface. Automorphism groups of free groups have been an object of study in combi- natorial group theory since the first decades of last century, and has interacted with the study of linear groups GL (Z), and with the study of surface Mapping Class n Groups. I will explain the connections below. On the one hand there is a natural map Aut(F ) → GL (Z). Since inner n n automorphisms of F are contained in the kernel of this map, this map factors n through the group of outer automorphisms, denoted as Out(F ). n Ontheotherhand, ifS isthesurfaceofgenusg withb > 0punctures, then g,b the fundamental group of S is the free group F , where n = 2g+b−1. Therefore n there is a natural map from the extended Mapping Class Group of S to the group Out(F ), mapping a self-homeomorphism of S to its action on π (S). Note that, n 1 since the choice of a base point is arbitrary, this map may not be well defined as a map to the group Aut(F ). This map is injective but is not surjective. The image of n this map is the subgroup of Out(F ) fixing the set of curves surrounding individual n punctures, up to conjugacy (see [8] Theorem 8.8). Both examples mentioned above show that it is natural to study the group Out(F ) i. e. the quotient of the group Aut(F ) by inner homeomorphisms. n n In the very special case where n = 2, it is known that the group Out(F ) is 2 isomorphic to the group GL (Z) and to the group Mod±(S ). 2 1,1 2