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Scl Danny Calegari scl Danny Calegari 1991 Mathematics Subject Classification. Primary 20J05, 57M07; Secondary 20F12, 20F65, 20F67, 37E45, 37J05, 90C05 Key words and phrases. stable commutator length, bounded cohomology, rationality, Bavard’s Duality Theorem, hyperbolic groups, free groups, Thurston norm, Bavard’s Conjecture, rigidity, immersions, causality, group dynamics, Markov chains, central limit theorem, combable groups, finite state automata Supported in part by NSF Grants DMS-0405491 and DMS-0707130. Abstract. This book is a comprehensive introduction to the theory of sta- ble commutator length, an important subfield of quantitative topology, with substantial connections to 2-manifolds, dynamics, geometric group theory, bounded cohomology, symplectic topology, and many other subjects. We use constructive methods whenever possible, and focus on fundamental and ex- plicit examples. We give a self-contained presentation of several foundational results in the theory, including Bavard’s Duality Theorem, the Spectral Gap Theorem, the Rationality Theorem, and the Central Limit Theorem. The con- tents should be accessible to any mathematician interested in these subjects, and are presented with a minimal number of prerequisites, but with a view to applications in many areas of mathematics. Preface The historical roots of the theory of bounded cohomology stretch back at least as far as Poincar´e[167] who introduced rotation numbers in his study of circle diffeomorphisms. The Milnor–Wood inequality [154, 204] as generalized by Sul- livan [193], and the theorem of Hirsch–Thurston [109] on foliated bundles with amenable holonomy groups were also landmark developments. But it was not until the appearance of Gromov’s seminal paper [97] that a number of previously distinct and isolated phenomena crystallized into a coherent subject. In [97] and in [98] Gromov indicated how many important or delicate geometric and algebraic properties of groups could be encoded and (in principle) recovered from their bounded cohomology. The essence of bounded cohomology is that it is a functor from the category of groups and homomorphisms to the category of normed vector spaces and norm-decreasing linear maps. Theorems in bounded cohomology can be restated as algebraic or topological inequalities; rigidity phenomena arise when equality is achieved (see e.g. [31, 93, 149, 45]). A certain amount of activity followed; for example, the papers [6, 27, 115, 150] contain significant new ideas and advanced the subject. But there is a sense in which the promise of the field as suggested by Gromov has not been realized. One major shortcoming is the lack of adequate tools for computing or extracting meaningful information. There are at least two serious technical problems: (1) the failure of the standard machinery of homological algebra (e.g. spec- tral sequences) to carry over to the bounded cohomology context in a straightforward way (2) the fact that in the cases of most interest (e.g. hyperbolic groups) bounded cohomology is usually so big as to be unmanageable Monod’s monograph [157] addresses in a very useful way some of the most serious shortcomings of the subject by largely restricting attention to continu- ous bounded cohomology in contexts where this restriction is most informative. Burger and Monod (see especially [33] and [34]) developed the theory of contin- uous bounded cohomology into a powerful tool, which is of most value to people working in ergodic theory or the theory of lattices (especially in higher-rank) but is less useful for people whose main concern is the bounded cohomology of discrete groups (although Theorem 2 from [34] is an exception). To get an idea of the state of the subject ca. 2000, we quote an excerpt from Burger–Monod [35], p. 19: Although the theory of bounded cohomology has recently found many applications in various fields . for discrete groups it re- mains scarcely accessible to computation. As a matter of fact, vii viii PREFACE almost all known results assert either a complete vanishing or yield intractable infinite dimensional spaces. It is therefore a firm goal of this monograph to try to present results in terms which are concrete and elementary. We pay a great deal of attention to the case of free and surface groups, and present efficient algorithms to compute numerical invariants, whenever possible. It is always hard for an outsider (or even an insider) to get an accurate idea of the critical (internal) questions or conjectures in a given field, whose resolution would facilitate significant progress, and of how the field does or might connect to other threads in mathematics. This monograph has a number of modest aims: (1) to restrict attention and focus to a subfield (namely stable commutator length) which already has a number of useful and well-known applications to a wide range of geometrical contexts (2) to carefully expose a number of foundational results in a way which should be accessible to any mathematician interested in the subject, and with a minimal number of prerequisites (3) to develop a number of “hooks” into the subject which invite contribu- tions from mathematicians and mathematics in what might at first glance appear to be unrelated fields (representation theory, computer science, combinatorics, etc.) (4) to highlight the importance of hyperbolic groups in general, and free groups in particular as a critical case for understanding certain basic phe- nomena (5) to give an exposition of some of my own work, and that of my collabora- tors, especially that part devoted to the “foundations” of the subject Recently, there has been an outburst of activity at the intersection of low- dimensional bounded cohomology, low-dimensional dynamics, and symplectic topol- ogy (e.g. [71, 73, 86, 169, 170, 174], and so forth). I have done my best to discuss some of the highlights of this interaction, but I am not competent to delve into it too deeply. Danny Calegari Acknowledgments Thanks to Marc Burger, Jean-Louis Clerc, Matthew Day, Benson Farb, David Fisher, Koji Fujiwara, Ilya Kapovich, Dieter Kotschick, Justin Malestein, Fedor Manin, Curt McMullen, Geoff Mess, Assaf Naor, Andy Putman, Pierre Py, Peter Sarnak, Alden Walker, Anna Wienhard, Dave Witte-Morris and Dongping Zhuang for additions and corrections. Special thanks to Jason Manning for extensive com- ments (especially on Chapters 1 and 2), and for many useful conversations about numerous technical points. Extra-special thanks to Shigenori Matsumoto for his meticulous refereeing, which led to vast improvements throughout the book. Fi- nally, thanks to Tereez for her patience. ix Contents Preface vii Acknowledgments ix Chapter 1. Surfaces 1 1.1. Triangulating surfaces 1 1.2. Hyperbolic surfaces 6 Chapter 2. Stable commutator length 13 2.1. Commutator length and stable commutator length 13 2.2. Quasimorphisms 17 2.3. Examples 20 2.4. Bounded cohomology 26 2.5. Bavard’s Duality Theorem 34 2.6. Stable commutator length as a norm 36 2.7. Further properties 41 Chapter3. Hyperbolicityandspectralgaps 51 3.1. Hyperbolic manifolds 51 3.2. Spectral Gap Theorem 56 3.3. Examples 60 3.4. Hyperbolic groups 64 3.5. Counting quasimorphisms 71 3.6. Mapping class groups 77 3.7. Out(Fn) 83 Chapter 4. Free and surface groups 87 4.1. The Rationality Theorem 87 4.2. Geodesics on surfaces 110 4.3. Diagrams and small cancellation theory 127 Chapter 5. Irrationality and dynamics 137 5.1. Stein–Thompson groups 137 5.2. Groups with few quasimorphisms 143 5.3. Braidgroupsandtransformationgroups 154 Chapter 6. Combable functions and ergodic theory 163 6.1. An example 163 6.2. Groups and automata 172 6.3. Combable functions 174 6.4. Counting quasimorphisms 181 xi xii CONTENTS 6.5. Patterson–Sullivan measures 184 Bibliography 197 Index 205 CHAPTER 1 Surfaces In this chapter we present some of the elements of the geometric theory of 2-dimensional (bounded) homology in an informal way. The main purpose of this chapter is to standardize definitions, to refresh the reader’s mind about the relation- ship between 2-dimensional homology classes and maps of surfaces, and to compute the Gromov norm of a hyperbolic surface with boundary. All of this material is essentially elementary and many expositions are available; for example, [10] covers this material well. We start off by discussing maps of surfaces into topological spaces. One way to study such maps is with linear algebra; this way leads to homology. The other way to study such maps is with group theory; this way leads to the fundamental group and the commutator calculus. These points of view are reconciled by Hopf’s formula; a more systematic pursuit leads to rational homotopy theory. 1.1. Triangulating surfaces A surface is a topological space (usually Hausdorff and paracompact) which is locally two dimensional. That is, every point has a neighborhood which is homeo- morphic to the plane, usually denoted by R2. 1.1.1. The plane. It is unfortunate in some ways that the standard way to refer to the plane emphasizes its product structure. This product structure is topologically unnatural, since it is defined in a way which breaks the natural topological symmetries of the object in question. This fact is thrown more sharply into focus when one discusses more rigid topologies. Example 1.1 (Zariski topology). The product topology on two copies of the affine line with its Zariski topology is not typically the same as the Zariski topology on the affine plane. A closed set in R1 with the Zariski topology is either all of R, or a finite collection of points. A closed set in R2 with the product topology is therefore either all of R2, or a finite union of horizontal and vertical lines and isolated points. By contrast, closed sets in the Zariski topology in R2 include circles, ovals, and algebraic curves of every degree. Part of the bias is biological in origin: Example 1.2 (Primary visual cortex).
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