De Gruyter Studies in Mathematics 44
Editors Carsten Carstensen, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Columbia, Missouri, USA Niels Jacob, Swansea, United Kingdom Karl-Hermann Neeb, Erlangen, Germany
Alex Degtyarev Topology of Algebraic Curves
An Approach via Dessins d’Enfants
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Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen Printed on acid-free paper Printed in Germany www.degruyter.com I dedicate this book to my wife Ay¸seBulut: it is her constant support, patience, and understanding that make my work successful.
Preface
The purpose of this monograph is to summarize, unify, and extend a number of inter- related results that were published or submitted during the last five years in a series of papers by the author, partially in collaboration with Ilia Itenberg, Viatcheslav Khar- lamov, and Nermin Salepci. As often happens in a long research project, the principal ideas have evolved and a few minor mistakes have been discovered, calling for a new, more comprehensive and unified approach to the earlier papers. Furthermore, as the work is still in progress, I am representing older results in a more complete and general form. The monograph also contains several newer results that have never appeared elsewhere. Thus, we complete the analysis of the metabelian invariants of a trigonal curve (see Chapter 6), compute the fundamental groups of all, not neces- sarily maximizing, irreducible simple sextics with a triple point (see Chapter 8; a few new sextics with finite nonabelian fundamental group have been discovered), and es- tablish the quasi-simplicity of most ribbon curves, including M-curves (see Section 10.3.2). The latest achievement is the complete understanding of simple monodromy factorizations of length two in the modular group (see Section 10.2.2; joint work with N. Salepci). In spite of its apparent simplicity, our description of such factorizations has interesting applications to the topology of real trigonal curves and real Lefschetz fibrations; they are discussed in Section 10.3. There also are a few other advances in the study of monodromy factorizations (see, e.g., Section 10.2.3), but in general the situation still remains unclear and the problem seems wild. The dominant theme of the book is the very fruitful close relation between three classes of objects: • elliptic surfaces and trigonal curves in ruled surfaces, see Chapter 3, • skeletons (certain bipartite ribbon graphs), see Chapter 1, and • subgroups of the modular group Γ := PSL(2, Z), see Chapter 2. (Slightly different versions of skeletons appeared in the literature under a number of names, the most well known being dessins d’enfants and quilts.) When restricted to appropriate subclasses, this relation becomes bijective, providing an intuitive combi- natorial and topological framework for the study of trigonal curves, on the one hand, and of subgroups of Γ, on the other. Undoubtedly, both dessins d’enfants and the modular group are amongst the most popular objects of modern mathematics; both are extensively covered in the literature. Thus, the modular group and its subgroups play a central rôle in the theory of modular forms, Moonshine theory, some aspects of number theory and hyperbolic geometry. viii Preface
Dessins d’enfants, apart from Grothendieck’s original idea [83] connecting them (via Bely˘ı maps) to the absolute Galois group Gal(Q¯ /Q), are related (via moduli spaces of curves and the Gromov–Witten theory) to topological field theories and integrable par- tial differential equations. Unfortunately, all of these fascinating topics are beyond the scope of this book. Our primary concern is a straightforward application of dessins to topology of trigonal curves and plane curves with deep singularities: the monodromy of such a curve can be computed, in a purely combinatorial way, in terms of its dessin; as a consequence, the monodromy group is a subgroup of Γ of genus zero and, using the well developed theory of such subgroups, we obtain numerous restrictions on the fundamental group of the curve and its more subtle geometric properties. This idea is summarized in Speculation 5.90 in Chapter 5.
Principal results Originally, my interest in dessins d’enfants was motivated by our work (joint with I. Itenberg and V. Kharlamov) on real trigonal curves and real elliptic surfaces and by my attempts to compute the fundamental groups of plane sextics. These two classes of objects remain the principal geometric applications of the theory. Here is a brief account of the most important statements found in the text.
Plane sextics. We use skeletons to classify irreducible plane sextics with a triple singular point (including non-simple ones) and compute their fundamental groups, see Theorems 7.45, 8.1, and 8.2. Thanks to works by J.-G. Yang [166], I. Shimada [148], and the author [46], the classification of simple plane sextics is close to its completion. This classification relies on the global Torelli theorem for K3-surfaces and is not quite constructive; the ‘visualization’ of sextics, necessary for the detailed study of their geometry, remains an open problem. In the presence of a triple point, this problem is solved by means of the skeletons. A brief survey of the known results concerning plane sextics and their fundamental groups is given in Section 7.2.1 and Section 7.2.3. Another application in this direction is the classification up to deformation and the computation of the fundamental groups of singular plane quintics, see Theorems 7.49 and 7.50. This result is old, but its complete proof has never been published.
Universal trigonal curves and metabelian invariants. The monodromy group of a non-isotrivial trigonal curve over a rational base is a subgroup of genus zero, and any subgroup H ⊂ Γ of genus zero is realized by a certain universal curve, from which any other curve with the monodromy group subconjugate to H is induced, see Section 5.3.1 and Corollary 5.88. These facts impose strong restrictions on the fundamental group and relate the latter to some geometric properties of the curve. (I expect that there should be a reasonable description of all finite quotients of such groups.) As a first step, we obtain universal (independent of the singularities) bounds Preface ix on the Alexander module of a trigonal curve, see Theorems 6.1 and 6.16. Then, as an illustration of Speculation 5.90 (2), we establish a version of Oka’s conjecture for trigonal curves, see Theorem 6.10, and classify the so-called Z-splitting sections of such curves, see Theorem 6.15: any Z-splitting section is induced from a certain universal one. This statement may have further implications to the study of tetragonal curves, hence plane sextics with A type singular points only.
Monodromy factorizations. A long standing question, related to the study of the topology of algebraic and pseudo-holomorphic curves, is whether a simple factor- ization of a given monodromy at infinity is unique up to Hurwitz equivalence. We answer this question in the negative and show that the problem is much wilder than it might seem: in the group as simple as B3, the number of nonequivalent factorizations may grow exponentially in length, see Theorem 10.20. On the other hand, we give a complete classification of Γ-valued factorizations of length two, see Theorems 10.27, 10.30, and 10.32 (joint with N. Salepci). As a by-product, we show that any maximal real elliptic Lefschetz fibration is algebraic, see Theorem 10.88. With elliptic surfaces in mind, we also introduce a new invariant of monodromy factorizations, the so-called transcendental lattice, and study its properties, see Section 10.2.3. As another application, we show that for extremal elliptic surfaces (see Section 10.3.1) and for a certain class of real trigonal curves, including M-curves (see The- orem 10.73), the topological and equisingular deformation classifications are equiv- alent. (Extremal elliptic surfaces are defined over algebraic number fields, and both classifications are also equivalent to the analytic classification in this case.)
Zariski k-plets. We construct a few examples of exponentially large (with respect to appropriate discrete invariants) collections of nonequivalent objects sharing the same combinatorial data. The objects are: extremal elliptic surfaces (see Theorem 9.30), irreducible trigonal curves (the ramification loci of the surfaces above), real trigonal curves (see Example 10.85), and real Lefschetz fibrations (see Example 10.87). All of the examples are essentially based on Theorem 10.20; thus, in each case, the objects differ topologically, constituting the so-called Zariski k-plets. The trigonal curves also share such commonly used invariants as the fundamental group and transcendental lattice, see Addendum 9.36 and Theorem 9.31.
The transcendental lattice. The j-invariant of an extremal elliptic surface is given by its skeleton. We show that the homological invariant, hence the surface itself, can be encoded by an orientation of the skeleton, see Theorem 9.1, and develop a simple algorithm computing the lattice of transcendental cycles and the Mordell–Weil group of the surface in terms of its oriented skeleton, see Theorem 9.6. More generally, the transcendental lattice and the torsion of the Mordell–Weil group of an arbitrary (not necessarily extremal) elliptic surface can be computed in terms of its homological invariant, regarded as a monodromy factorization, see Corollary 9.26. This algorithm x Preface leads us to the definition of transcendental lattice as an invariant of factorizations, see Section 10.2.3, and motivates a topological approach to the study of its arithmetic properties such as the discriminant form and parity.
Contents at a glance The principal concepts are introduced in Part I: we discuss bipartite ribbon graphs and their relation to the subgroups of (appropriate quotients of) the free group F2,see Chapter 1, the modular group Γ and closely related braid group B3, see Chapter 2, and trigonal curves and elliptic surfaces, both complex and real, and their topological counterpart, the so-called Lefschetz fibrations, see Chapter 3. For the reader’s con- venience, I have also included some background material that a topologist may not be familiar with and reproduced the proofs (or at least ideas of the proofs) of a few known statements, which are either difficult to find in the literature or closely related to the main subject. A separate section in Chapter 1 deals with pseudo-trees, which are an important special class of skeletons used later on in the construction of various exponentially large examples and in the study of simple monodromy factorizations. In Chapter 4, we follow [60] and describe the (equivariant) equisingular defor- mation classes of (real) trigonal curves in terms of dessins – certain overdecorated embedded graphs which must be considered up to a number of moves and which can be rather difficult to handle. It turns out that, under some additional extremality assumptions, dessins can be replaced with skeletons, i.q. subgroups of the modular group, making their study feasible. In the real settings, this correspondence between skeletons and deformation classes of curves is made precise in Section 10.3.2. In Chapter 5, we recall the notion of braid monodromy, adjusted to the particular case of curves in ruled surfaces, and the Zariski–van Kampen theorem, computing the fundamental group of such a curve in terms of its monodromy group. The principal result here is a purely combinatorial computation of the braid monodromy of a trigonal curve in terms of its dessin/skeleton (see Section 5.2) and, as an upshot, a strong restriction on the monodromy group of a trigonal curve and the notion of universal curve. The two latter lead us to Speculation 5.90, which is copiously illustrated in Chapter 6. Part II deals with the geometric applications, both old and new. Here, the chapter names are self-explanatory. We compute and study the fundamental groups of trigonal curves and related plane curves (see chapters 6, 7, and 8), discuss the transcendental lattice of an extremal elliptic surface and work out a particular series of examples (see Chapter 9), and make a few steps towards the understanding of Γ-valued monodromy factorizations and their applications to the topology of trigonal curves, elliptic sur- faces, and Lefschetz fibrations (see Chapter 10; for completeness, a few more or less classical results concerning the free groups Fn, symmetric groups Sn, and other braid groups Bn are also cited here). Preface xi
Appendices collect the material that would not fit elsewhere. Appendix A contains a few assorted statements concerning integral lattices and quotient groups, especially the so-called Zariski quotients, which appear as the fundamental groups of algebraic curves. In Appendix B, for comparison and as a very simple application, we discuss bigonal (hyperelliptic) curves in Hirzebruch surfaces. Appendix C is a listing of the GAP code that handles technical details of some proofs. (Shorter ad hoc listings are included into the main text; all GAP files are available for download.) Appendix D is a glossary: we fix the notation and give a brief explanation of a few terms, with the selection based upon the author’s own background and personal preferences.
Reading this book Every effort has been made to produce a text as self-contained and cross-referenced as possible, so that it can be read starting at any point with only a very minimal background from the reader. As usual, the end of a proof is marked with a . Some statements are marked with a , which means that either the statement is trivial (e.g., most corollaries) or its proof has already been explained. If a statement is marked with a , possibly followed by a list of references, its proof is omitted and the reader is directed to the literature. In most cases, the source is cited at the header. We use the commonly accepted symbol := as a shortcut for ‘is defined as’. Most symbols typeset in a special font (bold, Gothic, calligraphic, etc.) represent objects or classes of objects introduced somewhere in the book; they should be found in Section D.2. A brief explanation of other more or less common terms, notations, and concepts used throughout the whole text is given in Section D.1.
Acknowledgements I would like to thank my colleagues Norbert A’Campo, Mouadh Akriche, Igor Dol- gachev, Ergün Elçin, Sergey Finashin, Alexander Klyachko, Anton Klyachko, Ana- toly Libgober, Viatcheslav Nikulin, Mutsuo Oka, Stepan Orevkov, Ichiro Shimada, Muhammed Uludag,˘ and Özgün Ünlü, with whom I had numerous discussions while working on this project and earlier papers and who kindly coped with my ignorance in their fields of expertise. My special gratitude goes to my co-authors Ilia Itenberg, Viatcheslav Kharlamov, and Nermin Salepci, who generously shared many ideas during our work on joint projects. Some of these ideas underlie whole sections of the book. This book would never have appeared had it been not for Michael Efroimsky, who encouraged me and finally convinced me to undertake this tremendous task. I had a chance to represent a few selected topics in a mini-course given at Faculté des Sciences de Bizerte, and I wish to thank the audience for their hospitality, patience, and valuable suggestions that helped me improve the clarity of the exposition. xii Preface
Modern research is unthinkable without software and Internet. I would like to mention and express my gratitude to: • founders, maintainers, and contributors of Wikipedia, the free encyclopædia, • Donald Knuth, the person who created TEX and made the art of mathematical typesetting accessible to the general public, • Alexander Simonic, the developer of WinEdt, a text editor that makes the joy of TEX truly joyful, • the creators of GAP [76], a symbolic computation package without which some of the results of this book could not have been obtained in finite time, • the creators of GLE, a software package that helps one replace frustrating mouse based picture drawing with the excitement of writing code and chasing bugs. The final version of the manuscript was prepared during my sabbatical stay at l’Instutut des Hautes Études Scientifiques. I wish to extend my gratitude to this insti- tution and its friendly staff for their hospitality and excellent working conditions. Ankara, December 2011 Alex Degtyarev Bilkent University Contents
Preface vii I Skeletons and dessins 1 Graphs 3 1.1 Graphsandtrees ...... 3 1.1.1 Graphs...... 3 1.1.2 Trees ...... 6 1.1.3 Dynkin diagrams ...... 7 1.2 Skeletons ...... 9 1.2.1 Ribbon graphs ...... 9 1.2.2 Regions...... 12 1.2.3 The fundamental group ...... 16 1.2.4 First applications ...... 22 1.3 Pseudo-trees ...... 26 1.3.1 Admissibletrees ...... 26 1.3.2 The counts ...... 31 1.3.3 The associated lattice ...... 36 2 The groups Γ and B3 41 2.1 The modular group Γ := PSL(2, Z) ...... 41 2.1.1 The presentation of Γ ...... 41 2.1.2 Subgroups ...... 47 2.2 The braid group B3 ...... 50 2.2.1 Artin’s braid groups Bn ...... 50 2.2.2 The Burau representation ...... 54 2.2.3 The group B3 ...... 57 3 Trigonal curves and elliptic surfaces 63 3.1 Trigonal curves ...... 63 3.1.1 Basic definitions and properties ...... 63 3.1.2 Singularfibers...... 71 3.1.3 Special geometric structures ...... 76 xiv Contents
3.2 Elliptic surfaces ...... 79 3.2.1 Thelocaltheory ...... 79 3.2.2 Compact elliptic surfaces ...... 83 3.3 Realstructures...... 90 3.3.1 Real varieties ...... 91 3.3.2 Real trigonal curves and real elliptic surfaces ...... 96 3.3.3 Lefschetzfibrations...... 101 4 Dessins 109 4.1 Dessins...... 109 4.1.1 Trichotomicgraphs...... 109 4.1.2 Deformations...... 115 4.2 Trigonal curves via dessins...... 118 4.2.1 The correspondence theorems ...... 118 4.2.2 Complexcurves...... 120 4.2.3 Genericrealcurves...... 131 4.3 First applications ...... 137 4.3.1 Ribbon curves ...... 137 4.3.2 Elliptic Lefschetz fibrations revisited ...... 142 5 The braid monodromy 146 5.1 TheZariski–vanKampentheorem ...... 146 5.1.1 The monodromy of a proper n-gonal curve ...... 146 5.1.2 The fundamental groups ...... 152 5.1.3 Impropercurves:slopes ...... 158 5.2 The case of trigonal curves ...... 164 5.2.1 Monodromy via skeletons ...... 164 5.2.2 Slopes ...... 170 5.2.3 Thestrategy...... 173 5.3 Universalcurves ...... 177 5.3.1 Universalcurves ...... 177 5.3.2 The irreducibility criteria ...... 179
II Applications 6 The metabelian invariants 183 6.1 Dihedral quotients ...... 183 6.1.1 Uniform dihedral quotients ...... 183 6.1.2 Geometric implications ...... 187 Contents xv
6.2 The Alexander module ...... 190 6.2.1 Statements ...... 190 6.2.2 Proof of Theorem 6.16: the case N 7 ...... 193 6.2.3 Congruence subgroups (the case N 5) ...... 196 6.2.4 The parabolic case N =6...... 199 7 A few simple computations 203 7.1 Trigonal curves in Σ2 ...... 203 7.1.1 Proper curves in Σ2 ...... 203 7.1.2 Perturbations of simple singularities ...... 207 7.2 Sextics with a non-simple triple point ...... 213 7.2.1 A gentle introduction to plane sextics ...... 213 7.2.2 Classification and fundamental groups ...... 220 7.2.3 Asummaryoffurtherresults ...... 221 7.3 Plane quintics ...... 224 8 Fundamental groups of plane sextics 227 8.1 Statements ...... 227 8.1.1 Principalresults...... 227 8.1.2 Beginningoftheproof ...... 228 8.2 A distinguished point of type E ...... 231 8.2.1 A point of type E8 ...... 232 8.2.2 A point of type E7 ...... 238 8.2.3 A point of type E6 ...... 244 8.3 A distinguished point of type D ...... 259 8.3.1 A point of type Dp, p 6 ...... 259 8.3.2 A point of type D5 ...... 263 8.3.3 A point of type D4 ...... 269 9 The transcendental lattice 275 9.1 Extremal elliptic surfaces without exceptional fibers ...... 275 9.1.1 The tripod calculus ...... 275 9.1.2 Proofsandfurtherobservations...... 277 9.2 Generalizations and examples ...... 281 9.2.1 A computation via the homological invariant ...... 281 9.2.2 Anexample...... 284 10 Monodromy factorizations 288 10.1 Hurwitz equivalence ...... 288 10.1.1 Statement of the problem ...... 288 10.1.2 Fn-valued factorizations ...... 291 10.1.3 Sn-valued factorizations ...... 292 xvi Contents
10.2 Factorizations in Γ ...... 297 10.2.1 Exponential examples ...... 297 10.2.2 2-factorizations ...... 301 10.2.3 The transcendental lattice ...... 307 10.2.4 2-factorizations via matrices ...... 313 10.3 Geometric applications ...... 316 10.3.1 Extremal elliptic surfaces ...... 316 10.3.2 Ribbon curves via skeletons ...... 318 10.3.3 MaximalLefschetzfibrationsarealgebraic ...... 323
Appendices A An algebraic complement 329 A.1 Integral lattices ...... 329 A.1.1 Nikulin’stheoryofdiscriminantforms...... 329 A.1.2 Definite lattices ...... 331 A.2 Quotient groups ...... 335 A.2.1 Zariski quotients ...... 335 A.2.2 Auxiliary lemmas ...... 336 A.2.3 Alexander module and dihedral quotients ...... 337
B Bigonal curves in Σd 340
B.1 Bigonal curves in Σd ...... 340 B.2 Plane quartics, quintics, and sextics ...... 344 C Computer implementations 346 C.1 GAP implementations ...... 346 C.1.1 Manipulating skeletons in GAP ...... 346 C.1.2 ProofofTheorem6.16 ...... 352 D Definitions and notation 359 D.1 Common notation ...... 359 D.1.1 Groups and group actions ...... 359 D.1.2 Topology and homotopy theory ...... 360 D.1.3 Algebraicgeometry...... 362 D.1.4 Miscellaneous notation ...... 364 D.2 Index of notation ...... 365 Bibliography 369 Index of figures 379 Index of tables 382 Index 383 Part I Skeletons and dessins
Chapter 1 Graphs
1.1 Graphs and trees
In this introductory section, we merely fix terminology and notation; most proofs are omitted. We make no attempt to develop a complete combinatorial theory of graphs (this subject is extensively covered in the literature, see, e.g., [82]), nor do we try to adhere consistently to the combinatorial approach. Our main purpose is to make an easy transition to Definition 1.14 in Section 1.2.1, upon which graphs remain geometric objects and are treated using topology and geometric intuition rather than formal combinatorics.
1.1.1 Graphs Geometrically, a graph is a locally finite CW-complex of dimension one. Thus, we allow infinite graphs; we do however require that the valency of each vertex must be finite. Combinatorially, a graph can be defined as follows.
Definition 1.1. A graph is a collection G =(V,E, ,ι),whereV and E are sets, possibly infinite, : E → E is a free involution, and ι: E → V is a map with the preimage of each point finite. The elements of the sets Vtx G := V and End G := E are called the vertices and edge ends of G, respectively, and ι: E → V is called the incidence map.Thestar of a vertex v ∈ V is the pull-back ι−1(v); its cardinality deg v := Card ι−1(v) is called the valency or degree of v. An edge of a graph G is an orbit of ; thus, is the map sending an edge end to the opposite end of the same edge. A directed edge is an ordered orbit of . For a directed edge e =(e0,e1), we let e(0) := ι(e0), e(1) := ι(e1),and−e =(e1,e0). Since each orbit consists of two elements, one can identify a directed edge e =(e0,e1) with its end e0 ∈ E.Anorientation of G is a choice of a direction of each edge of G;in other words, it is an equivariant map o:EndG → Z2, o ◦ (e)=o(e)+1mod 2.A pair (G, o) is called a directed graph. By convention, o(e0)=0for an edge (e0,e1) directed according to the orientation o.
Remark 1.2. In the literature, one can find a great deal of combinatorial definitions of graphs. Usually, they are defined as a pair of sets, vertices and edges, related in an appropriate way. Our definition in terms of edge ends rather than edges is intended 4 Chapter 1 Graphs to avoid complications caused by loops, i.e., edges with both ends attached to the same vertex. In the presence of loops, the star of a vertex should be regarded as a set of edge ends rather than edges, and one needs two distinctive ends of each edge to define an orientation. In fact, each graph has a canonical bipartite subdivision, see Remark 1.6 below, where edge ends become edges and the definition becomes more conventional. Alternatively, an edge end can be regarded as a directed edge, as explained in the previous paragraph. For this reason, we will freely use all commonly accepted intuitive terminology related to graphs; some of the notions are formally (re-)introduced further in this chapter.
A morphism between two graphs Gi =(Vi,Ei, i,ιi), i =1, 2, is a pair of maps ϕ: V1 → V2 and ψ : E1 → E2 preserving the graph structure, i.e., such that ψ ◦ 1 = 2 ◦ ψ and ψ ◦ ι1 = ι2 ◦ ϕ. A morphism of directed graphs (Gi, oi) must commute with the orientation, i.e., o2 ◦ψ = o1.Asubgraph of a graph G =(V,E, ,ι) is a pair (V ,E ) of subsets, V ⊂ V , E ⊂ E, which is a graph with respect to the restricted maps, i.e., such that (E ) ⊂ E and ι(E ) ⊂ V . A subgraph G is called induced if E = ι−1(V ) ∩ (ι−1(V )); in other words, an edge of G belongs to G if and only if so do its both ends. An induced subgraph is said to be spanned by its vertices. The inclusion G → G of a subgraph G is a morphism. We freely apply to graphs notions and terms used for CW-complexes; thus, we will speak about connected graphs, connected components, the fundamental group (cf. Theorem 1.4 below), etc. Sometimes, to avoid confusion, the CW-complex represent- ing a graph G is referred to as the geometric realization of G and is denoted by |G|. Formally, the geometric realization of G =(V,E, ,ι) can be constructed as follows: Direct each edge and let |G| be the quotient of the union ((E/ )×I)V by the equiv- alence relation (e, 0) ∼ e(0), (e, 1) ∼ e(1) for each edge e ∈ E/ . In this construc- tion, each vertex v is represented by a 0-cell and each directed edge e is represented by a closed 1-cell e × I, which comes with a characteristic map φe : e × I →|G|. Note that a connected graph is at most countable.
Definition 1.3. A walk in a graph G is a finite sequence γ =(v; e1,...,ek), k 0, where v is a vertex and ei are directed edges of G, such that v = e1(0) (if k>0) and ei(1) = ei+1(0) for all i =1,...,k− 1.Theinitial point of γ isγ(0) := v;the terminal point is defined via γ(1) := ek(1) if k>0 or γ(1) := v if k =0. A walk γ is called closed if γ(0) = γ(1). Note that, if (G, o) is a directed graph, the edges constituting a walk do not need to be directed according to o.Ifthisis the case, one speaks about a directed walk. We reserve the term path for a topological path in the geometric realization of the graph. A combinatorial path is a walk (v; e1,...,ek) such that no vertex appears more than once in the sequence v = e1(0),...,ek(0). A closed combinatorial path is called a cycle. Similar to walks, one can speak about directed paths and cycles. Section 1.1 Graphs and trees 5
The geometric realization of a walk γ =(v; e1,...,ek) is the path |γ|: I →|G| =0 | | → | | = · · defined as follows: if k ,then γ : t v is constant; otherwise, γ φe1 ... φek is the product of the characteristic maps of its edges, regarded as paths. · =( ) =( ) The product γ γ of two walks γ v ; ei and γ v ; ej is defined whenever (1) = (0) ( ) γ γ and is the walk obtained by the concatenation v ; ei,ej .Theinverse −1 of a walk γ =(v; e1,...,ek) is the walk γ :=(γ(1); −ek,...,−e1). One has
γ · γ (0) = γ (0),γ · γ (1) = γ (1); γ−1(0) = γ(1),γ−1(1) = γ(0).
The product of any two closed walks starting at the same vertex v is always well defined and is again a closed walk starting at v; the inverse of a closed walk starting at v is also a closed walk starting at v. An elementary contraction of a walk γ is the walk obtained from γ by removing a consecutive pair of the form e, −e from the sequence of edges of γ. Two walks are said to be equivalent if they are related by a finite sequence of elementary contractions and inverse operations. The following statement is almost straightforward.
Theorem 1.4. With respect to the product, the set of the equivalence classes of all closed walks at a fixed vertex v of a graph G forms a group, called the (combinatorial) fundamental group of G and denoted by π1(G,v). The geometric realization map γ →|γ| induces an isomorphism π1(G,v)=π1(|G|,v). [82]
The equivalence class of a closed walk γ at v is denoted by [γ] ∈ π1(G,v).Any walk γ in G defines the translation isomorphism
Tγ : π1(G,γ(0)) → π1(G,γ(1));
−1 it is given by [α] → [γ · α · γ]. It follows that, as an abstract group, π1(G,v) depends only on the connected component containing v. Thus, if G is connected, one can speak about the abstract group π1(G), which is well defined up to isomorphism.
Bipartite graphs A bipartite graph is a graph G equipped with a partition of its vertices into two classes, i.e.,amapp:VtxG → Z2, such that the composition p ◦ :EndG → Z2 is an orientation. The vertices in the pull-backs p−1(0) and p−1(1) are called • (black) and ◦ (white), respectively. For a ‘variable’ type of vertices we commonly use the symbol ∗, which takes values in the set {•, ◦}. Alternatively, a bipartite graph can be defined combinatorially in terms of its edges rather than edge ends, as follows. 6 Chapter 1 Graphs
Definition 1.5. A bipartite graph is a collection G =(V•,V◦, E,ι•,ι◦),whereV•, V◦,andE are sets, possibly infinite, and ι• : E→V• and ι◦ : E→V◦ are maps such that the preimage of each point is finite.
The elements of the sets Vtx• G := V•,Vtx◦ G := V◦,andEdgG := E are called, respectively, the •-and◦-vertices of G and its edges.Thestar of a vertex and its valency are defined as in the case of ordinary graphs, using ι• and ι◦ instead of ι:the −1 star of a ∗-vertex v is the set ι∗ (v) ⊂E, ∗ = •, ◦, and its valency is the cardinality −1 deg v := Card ι∗ (v). For the formal passage from Definition 1.5 to 1.1, one should let V = V• V◦ and E = E×{•, ◦} and define : (e, •) → (e, ◦) → (e, •) and ι: (e, ∗) → ι∗(e) for ∗ = •, ◦. Note that each bipartite graph G is canonically directed: by convention, we let e(0) = ι•(e) and e(1) = ι◦(e) for an edge e ∈E.
Remark 1.6. An ordinary graph G =(V,E, ,ι) can be regarded as a bipartite graph: one lets V• = V , V◦ = E/ , E = E, ι• = ι and takes for ι◦ the natural projection E → E/ . We call G¯ =(V•,V◦, E,ι•,ι◦) the bipartite subdivision of G; the edges of G¯ represent directed edges of G. Intuitively, when passing from G to G¯ , one declares the vertices of G black and inserts a bivalent ◦-vertex in the middle of each edge of G. Conversely, a bipartite graph G¯ is the bipartite subdivision of an ordinary graph if and only if all ◦-vertices of G¯ are bivalent. In view of this correspondence, and since bipartite graphs are our primary concern, we introduce most notions for bipartite graphs only, referring to Definition 1.5 or Definition 1.14 below.
Convention 1.7. In the drawings of bipartite graphs we routinely omit bivalent ◦- vertices, assuming that such a vertex is to be inserted at the center of each edge con- necting two •-vertices.
1.1.2 Trees
A tree as a simply connected graph, i.e., a connected graph T with π1(T)=0.Dueto Theorem 1.4 and the Whitehead theorem [163, 164], a graph T is a tree if and only if its geometric realization |T| is contractible. Yet another equivalent definition is that a tree is a connected graph without cycles. As a straightforward application of Zorn’s lemma, any connected graph G contains a maximal tree; in fact, a subtree T ⊂ G is maximal if and only if it contains all vertices of G. Since |T| is contractible, the quotient projection |G|→|G|/|T| is a homotopy equivalence. On the other hand, the quotient |G|/|T| is a CW-complex of dimension one and with a single 0-cell |T|/|T|, hence a wedge of circles. By the Seifert–van Kampen theorem, the fundamental group π1(G)=π1(|G|/|T|) is the free group generated by the circles constituting the wedge. Section 1.1 Graphs and trees 7
Corollary 1.8. For a connected graph G, the geometric realization |G| is homotopy equivalent to a wedge of circles and the group π1(G) is free of rank 1 − χ(|G|). As a consequence of Corollary 1.8, one obtains a simple proof of the Nielsen– Schreier theorem and Schreier index formula. Purely algebraic proofs of these state- ments take several pages, see, e.g., [114].
Theorem 1.9 (Nielsen–Schreier). A subgroup H of a finitely generated free group G is free. If the index n :=[G : H] is finite, then H is of finite rank rk H = n(rk G − 1) + 1.
Proof. (This proof was discovered by Max Dehn, see [115].) One has G = π1(W ), where W is the wedge of rk G circles, regarded as a graph. A subgroup H ⊂ G defines a topological covering W˜ → W , which is also a graph. Hence, H is free. If the degree n of the covering is finite, counting cells yields χ(W˜ )=nχ(W ), which gives the index formula.
Given a tree T, any two distinct vertices u, v of T can be connected by a unique path γ. Define the vertex distance dist(u, v) as the number of edges constituting γ, and extend this definition to dist(u, u)=0for any vertex u. It is immediate that dist is indeed a metric on the set of vertices of T.
1.1.3 Dynkin diagrams Dynkin diagrams are a very special family of graphs that keep appearing in various unrelated areas of mathematics. Our interest is due to the fact that Dynkin diagrams classify simple singularities, singular elliptic fibers, and definite lattices generated by vectors of square (−2). In this graph related chapter, we only list the so-called simply laced Dynkin diagrams, both elliptic and affine, and discuss their relation to definite lattices. Other applications and a detailed description of the corresponding root systems can be found in [27]. Simply laced Dynkin diagrams constitute two infinite series, Ap, p 1,andDq, q 4, and three exceptional classes E6, E7,andE8, see Figure 1.1. Each diagram appears in two forms, elliptic and affine; the latter is usually marked with a tilde. For a characterization (one of the many) of the simply laced Dynkin diagram, let us assign a lattice LG to each finite loop free graph G. As a free abelian group, let 2 LG := Zv, the summation running over all vertices v ∈ Vtx G; the square v of each generator v ∈ Vtx G equals −2, and the product u · v, u = v, of two distinct generators is the number of edges of the graph connecting u and v.
Proposition 1.10. For a connected finite loop free graph G, the lattice LG is negative definite (semidefinite) of and only if G is a simply laced Dynkin diagram (respectively, affine Dynkin diagram). In the affine case, the kernel ker LG is generated by the vector z := nvv,wherenv are the coefficients shown in Figure 1.1. 8 Chapter 1 Graphs
111 1 Ap : A˜ p :
1
1 1 11 2 2 Dq : D˜ q : 1 1
12321 E6 : E˜ 6 : 2 1
1234321 E7 : E˜ 7 : 2
24654321 E8 : E˜ 8 : 3
Figure 1.1. Simply laced Dynkin diagrams.
Proof. The lattices LG associated with (affine) Dynkin diagrams are indeed (semi-)definite and the kernels ker LG are as stated. Hence, the affine Dynkin dia- grams are extremal in the sense that adding an extra vertex would produce an ele- ment of positive square: for such a vertex u one would have u · z 1 and hence (u +2z)2 2. Thus, affine Dynkin diagram exhaust the semidefinite case, and it remains to notice that elliptic Dynkin diagrams are all graphs that do not contain an affine one.
Convention 1.11. The lattice LG associated with a Dynkin diagram G is denoted by the same symbol as G (see also Convention A.12). Sometimes, we extend this notation 2 by letting A0 := D0 :=0, D1 := Zx, x = −4, D2 :=2A1,andD3 := A3. In each assignment, the right hand side of the expression exposes certain behavior expected from its left hand side. For example, the following proposition, which gives an explicit description of the A and D type lattices, extends to these generalizations.
Proposition 1.12 (see [27]). The lattices Ap and Dp can be defined as follows: ⊥ 2 Ap−1 =(v1 + ···+ vp) ⊂ Bp, Dp = {x ∈ Bp | x =0mod 2}.
In other words, Ap−1 is the orthogonal complement of the ‘minimal’ characteristic vector v := v1 + ···+ vp ∈ Bp, and Dp is its orthogonal complement modulo 2 or, equivalently, Dp is the maximal even sublattice of Bp. Section 1.2 Skeletons 9
⊥ Proof. The orthogonal complement (v1+···+vp) ⊂ Bp is generated by the vectors vi−vi+1, i =1,...,p−1, and these vectors form a canonical basis for Ap−1.ForDp, observe that Bp is odd and, hence, its maximal even sublattice is of index 2.Onthe other hand, the vectors −v1 − v2 and vi − vi+1, i =1,...,p− 1, do generate an even sublattice of index 2, and these vectors form a canonical basis for Dp.
1.2 Skeletons
The notion of skeleton is the principal concept used throughout the book: we will employ skeletons to classify trigonal curves, compute their fundamental groups, con- struct examples, etc. Crucial is Definition 1.14, which establishes a correspondence between skeletons and subgroups of certain groups.
1.2.1 Ribbon graphs Recall that a cyclic order on a finite set X is a transitive Z-action on X:givenx ∈ X, the elements succ x := x ↑ (+1) and pred x := x ↑ (−1) are the immediate successor and predecessor of x, respectively. A ribbon graph is a graph equipped with a cyclic order on the star of each vertex. A typical example is a graph G with a proper immersion |G| → S to an oriented surface S: the cyclic order on the star of a vertex v is given by the counterclockwise rotation (with respect to the given orientation of S) about v.
Convention 1.13. In the figures, we use the standard convention, drawing ribbon graphs in the plane, possibly with self-intersections, and assuming the ‘blackboard thickening’, i.e., the cyclic order induced from the plane. The bipartite subdivision of a ribbon graph is again a ribbon graph, as the star of a bivalent ◦-vertex has a unique cyclic order. For bipartite graphs, the definition of a ribbon graph structure can be combined with Definition 1.5 and simplified.
Definition 1.14. A bipartite ribbon graph is a triple S =(E, nx, op),whereE is a set and nx, op: E→Eare two automorphisms of E with finite orbits. A connected bipartite ribbon graph is also called an (abstract) skeleton. A pair (S,e),whereS is a skeleton and e ∈E, is called a marked skeleton. The sets of •-and◦-vertices of a bipartite ribbon graph are defined as the orbit sets V• = E/ nx and V◦ = E/ op, respectively, the incidence maps ι• and ι◦ being the canonical projections. In most applications, ◦-vertices are at most bivalent and can be regarded as edges, see Convention 1.7, whereas •-vertices are usually trivalent. Thus, the notation ‘op’ and ‘nx’ stands for ‘opposite’ and ‘next’ (with respect to the cyclic order), respectively. We will also use the uniform notation • := nx and ◦ := op. 10 Chapter 1 Graphs
The valency of a vertex is the cardinality of the corresponding orbit. Consider a pair 2 t :=(t•,t◦) ∈ N . A skeleton S is said to be a skeleton of type t,orat-skeleton,if the valency of each ∗-vertex of S divides t∗, ∗ = •, ◦. If the valency of each ∗-vertex equals t∗,thenS is called regular,ort-regular. S =(E ) =12 A morphism between two bipartite ribbon graphs i i, nxi, opi , i , ,is amapϕ: E1 →E2 commuting with nx and op, i.e., such that ϕ ◦ nx1 = nx2 ◦ ϕ and ϕ ◦ op1 = op2 ◦ ϕ. A morphism of marked skeletons is required, in addition, to take the distinguished edge e1 of S1 to the distinguished edge e2 of S2. The group of automorphisms of a skeleton S is denoted by Aut S. We regard Aut as a subgroup of the permutation group S(Edg S).
Warning 1.15. Certainly, any morphism of (bipartite) ribbon graphs induces, in a canonical way, a morphism of the underlying (bipartite) plain graphs. However, the converse is not true. For example, any subgraph S ⊂ S of a bipartite ribbon graph inherits a bipartite ribbon graph structure (e.g., from the minimal supporting surface Supp S); however, the inclusion S → S is not a morphism unless S is a union of whole components of S.
Ribbon graphs as G-sets Let G := x, y be the free group on two generators. Throughout this chapter, we reserve the notation x, y for the chosen pair of generators of G, sometimes using the uniform notation g• := x and g◦ := y adapted to the treatment of bipartite graphs. According to Definition 1.14, a bipartite ribbon graph S is merely a set E with an action of G such that both x and y have finite orbits. For a technical reason, we extend nx and op to a right G-action E×G →E, denoted by (e, g) → e ↑ g;by definition, x acts via nx and y acts via op. The graph is connected if and only if this action is transitive; thus, a marked skeleton (S,e) can be identified with the quotient set H\G,whereH = stabG e is the stabilizer of e in G. Furthermore, a morphism of bipartite ribbon graphs is merely a morphism of G-sets. In particular, a morphism ϕ of skeletons is uniquely determined by the image of any edge of the source and has a well defined degree deg ϕ := Card ϕ−1(e),wheree is any edge of the target. A finite bipartite ribbon graph S =(E, nx, op) can be represented in GAP [76] by a pair of permutations nx, op ∈ S(E). A few simple functions manipulating skeletons and computing various invariants are shown in Listing C.2 and explained further in this chapter. Thus, functions "Black", "Blacks", and "BlackNo" compute, respec- tively, the set of •-vertices, numbers of •-vertices by valencies, and the total number of •-vertices; the White* counterparts of these functions provide the same informa- tion about the ◦-vertices. The minimal, in the obvious sense, type t of a skeleton S is given by "Type", and "IsRegularSkeleton" decides whether S is t-regular. The connected components of a bipartite ribbon graph S are handled by "IsConnected", "Components", "Component", and "ExtractComponent". Section 1.2 Skeletons 11
For the obvious reasons, in GAP, instead of the infinite group G we deal with the finite subgroup of S(E) generated by nx and op. The stabilizer stab(e),regardedasa subgroup of S(E),isgivenby"stab". If S is a skeleton, the stabilizers of all edges of S form a whole conjugacy class of subgroups; we call this class the stabilizer of S and denote it by StabG S.The requirement that all orbits of both x and y must be finite translates into the following condition on the representatives H ∈ StabG S: (∗) for each g ∈ G, there is a pair (r, s) ∈ N2 such that xr, ys ∈ g−1Hg. Next few statements are immediate consequence of elementary theory of discrete group actions, see, e.g.,[90].
Theorem 1.16. The functors (S,e) → stabG e and H → (H\G,H\H) establish an equivalence between the category of marked skeletons and morphisms and that of subgroups H ⊂ G satisfying (∗) and inclusions. A skeleton S = H\G is of type t ∈ N2 if and only if H contains the normal subgroup generated by xt• and yt◦ .
2 t• t◦ ∼ Given a type t ∈ N , denote Gt = x, y | x = y =1 = Zt• ∗ Zt◦ .There is a canonical epimorphism G Gt. Clearly, the G-action on the set of edges of a skeleton of type t factors through Gt; we use the notation stabt and Stabt for the stabilizers regarded as (conjugacy classes of) subgroups of Gt. Note that any subgroup of Gt lifts to a subgroup H ⊂ G satisfying (∗).
2 Corollary 1.17. Given a type t ∈ N , the functors (S,e) → stabt e ⊂ Gt and H → (H\Gt,H\H) establish an equivalence between the category of marked t- skeletons and morphisms and that of subgroups H ⊂ Gt and inclusions.
Corollary 1.18. The maps S → StabG S, [[ H]] → H\G establish a canonical one- to-one correspondence between the set of isomorphism classes of skeletons and that of conjugacy classes of subgroups H ⊂ G satisfying (∗).
2 Corollary 1.19. Given a type t ∈ N , the maps S → Stabt S and [[ H]] → H\Gt establish a canonical one-to-one correspondence between the set of isomorphism classes of t-skeletons and that of conjugacy classes of subgroups H ⊂ Gt. If a skeleton S is fixed, the isomorphism classes of marked skeletons (S,e) are in a one-to-one correspondence with the orbits of Aut S.
Corollary 1.20. The conjugacy class [[ H]] of a subgroup H ⊂ G satisfying (∗) is in a canonical bijection with the set of orbits of the group Aut(H\G). In particular, Aut(H\G) is transitive if and only if H is normal in G.
Corollary 1.21. Given a subgroup H ⊂ G satisfying (∗), there is a canonical group isomorphism Aut(H\G)=NG(H)/H. 12 Chapter 1 Graphs
Proof. Any G-endomorphism of the homogeneous G-set H\G is the multiplication Hx → Hgx by a fixed element g ∈ G (such that Hg is the image of the coset H\H under the endomorphism). This multiplication map is well defined if and only −1 if g Hg ⊂ H, i.e.,ifg ∈NG(H); it is the identity if and only if g ∈ H.
Corollary 1.22. A subgroup H ⊂ G satisfying (∗) is normal if and only if the group Aut(H\G) acts transitively on the set of edges of the skeleton H\G. Corollary 1.21 is used in function "Aut", see Listing C.2, computing the automor- phism group Aut S.
1.2.2 Regions A region in a bipartite ribbon graph S =(E, nx, op) is an orbit of the element xy ∈ G. Each region is a Z-set: we let 1 ∈ Z act via (xy)−1. The cardinality of a region R is called its width wd R. (In the arithmetical theory of the modular group, instead of regions one speaks about cusps and cusp widths; this, and the fact that the term ‘degree’ is way too overused, explains our terminology.) A region R of finite width n is also referred to as an n-gon or an n-gonal region, the ‘corners’ being the •-ends of the edges constituting R. The set of regions of S is denoted by Reg S. We also use the S ∈ N¯ S notation Regw for the set of all regions of a given width w and Reg0 for the set of all regions of finite width. For a finite skeleton S,thesetRegS is computed by function "Reg", see Listing C.2; the numbers of regions by sizes are given by "Regs", and the total number of regions is "RegNo". With the modular group in mind, see page 45, an element of G conjugate to a power (xy)n, n ∈ N, is called parabolic or unipotent of width n. The following statement is a paraphrase of the definition. (For a subgroup H ⊂ G, a parabolic element g−1(xy)ng ∈ H is called minimal if g−1(xy)mg/∈ H for 0 Lemma 1.23. For a subgroup H ⊂ G,theH-conjugacy classes of minimal parabolic elements of H are in a canonical one-to-one correspondence with the finite regions of the skeleton H\G. The width of a minimal parabolic element of H is equal to the width of the corresponding region. We do not define the boundary ∂R of a region R as a formal object, but we do use it as an intuitive concept. Thus, we say that the •-and◦-ends of all edges e ∈ R belong to ∂R, and so do all (directed) edges of the form e and −e ↑ x, e ∈ R.Ifthe ribbon graph structure is induced from a proper embedding |S| → S to an oriented surface S, see Section 1.2.1, regions in the sense of our definition are the boundary components (properly understood) of the connected components of the complement S |S| (or rather the cut of S along |S|), and the above convention concerning the boundary of a region agrees with the common sense. For example, in Figure 1.2, −i the boundary of R = reg e0 is formed by the edges ei := e0 ↑ (xy) , i ∈ Z,that = ↑ −1 = ↑ constitute R, as well as the edges ei : ei y ei+1 x, which apriorido Section 1.2 Skeletons 13