A primer on mapping class groups.

*(English)*Zbl 1245.57002
Princeton Mathematical Series. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14794-9/hbk; 978-1-400-83904-9/ebook). xiv, 492 p. (2011).

Let \(S= S_{g,b,n}\) be a connected, oriented, differentiable surface of topological genus \(g\) admitting \(b\geq 0\) boundary components and \(n\geq 0\) punctures. Then there are two fundamental objects that can be attached to such a surface \(S\), namely the mapping class group of \(S\) denoted by \(\text{Mod}(S)\) and the Teichmüller space \(\text{Teich}(S)\) of \(S\). These objects have been studied for several decades in the past, and are now subject to a vast literature in various areas of both classical and contemporary mathematics.

The book under review provides a profound introduction to the fascinating theory relating these two kinds of objects, thereby touching upon the diverse algebraic, combinatorial, analytical and topological facets forming this profuse topic, which is experiencing a great renaissance in these days, in particular with a view toward moduli theory and its modern applications in mathematical physics.

As the authors point out in the preface, their goal in the present book is to explain as many important theorems, examples, and technique as possible, as quickly and directly as possible, while at the same time giving (nearly) full details and keeping the text widely self-contained. Also, the book contains some simplifications of known approaches and proofs, the exposition of some results that are not readily available in the literature, and some new material as well. As for the precise contents, the book consists of three parts titled “Mapping class groups” (Part 1), “Teichmüller space and moduli space” (Part 2), and “The classification and pseudo-Anosov theory” (Part 3). Each part is divided into several chapters and subsections, where the main results of the single chapters are outlined in an “Overview” at the very beginning of the book.

Part 1 covers the basic theory of mapping class groups of surfaces, with the central theme being the relationship between the algebraic structure of the mapping class group \(\text{Mod}(S)\) and the combinatorial topology of the underlying surface \(S\).

Chapter 1 recalls some basic facts about surfaces and hyperbolic geometry, including the working with simple closed curves and their geodesic representatives, their geometric intersection numbers, the notion of isotopy of a surface, and some useful facts from surface topology regarding self-homeomorphisms. Chapter 2 gives the definition of the mapping class group \(\text{Mod}(S)\) of a surface \(S\), which is then followed by first examples and computations in the simplest concrete cases. In this context, the classical algorithmic “Alexander method” is explained and applied, and the fundamental theorem of M. Dehn on the structure of the mapping class group \(\text{Mod}(T^2)\) of the 2-torus \(T^2\) is presented as well. Chapter 3 describes the famous Dehn twists as fundamental mapping classes, namely as the simplest infinite-order elements in \(\text{Mod}(S)\). The authors treat Dehn twists and their action on simple closed curves in great depth, with concrete applications to the structure of \(\text{Mod}(S_{g,0,0})\) for a closed compact surface of genus \(g\geq 3\). Chapter 4 is devoted to the problem of generating the mapping class group \(\text{Mod}(S)\) by Dehn twists, culminating in the celebrated Dehn-Lickorish-Humphries theorem, the construction of varieties simplicial complexes of curves, the famous Birman exact sequence with respect to punctures, and the exhibition of different sets of generators of \(\text{Mod}(S)\) by means of Dehn twists. Chapter 5 turns to presentations and the low-dimensional homology of certain mapping class groups. This includes the Dehn-Johnson “lantern relation” among Dehn twists, J. Harer’s theorem on the triviality of the first homology group \(H_1(\text{Mod}(S_{g,0,0}),\mathbb{Z})\), the explicit finite presentations of special mapping class groups due to B. Wajnryb [Isr. J. Math. 45, 157–174 (1983; Zbl 0533.57002)] and S. Gervais [Trans. Am. Math. Soc. 348, No. 8, 3097–3132 (1996; Zbl 0861.57023)] the proof of the finite presentability via group actions on simplicial complexes, Hopf’s formula for computing \(H_2(G,\mathbb{Z})\) for any group from a finite presentation for G, W. Pitsch’s upper bound on the rank of \(H_2(\text{Mod}(S_{g,0,0}),\mathbb{Z})\), and a new proof of J. Harer’s theorem stating that \(H_2(\text{Mod}(S_{g,0,0}),\mathbb{Z})\) is infinitely cyclic for \(g\geq 4\). Furthermore, the Euler class in the second cohomology group of a surface \(S_{g,1,0}\) is analyzed, and the so-called Meyer signature cocycle in the cohomology group \(H^2(\text{Mod}(S_{g,0,0}),\mathbb{Z})\) is described via \(S_{g,0,0}\)-bundles (surface bundles).

Chapter 6 discusses the symplectic representation of the mapping class group \(\text{Mod}(S_{g,0,0})\) induced by the action of the latter on the homology \(H_1(S_{g,0,0};\mathbb{Z})\), different proofs of the surjectivity of the symplectic representation, the associated Torelli group, and some classical structure theorems on \(\text{Mod}(S_{g,0,0})\) and the Torelli group, respectively, which are basically due to J.-P. Serre, E. L. Grossman, D. Johnson, and others.

Finite subgroups of mapping class groups are studied in Chapter 7, where the Nielsen realization theorem, the Hurwitz \(84(g-1)\) theorem for closed hyperbolic surfaces, Wiman’s \(4g+2\) theorem on the orders of automorphisms of closed hyperbolic surfaces, and a Gauss-Bonnet formula for orbifolds appear as related fundamental results.

Chapter 8 is an exposition of one of the most beautiful results in geometric group theory: the famous Dehn-Nielsen-Baer theorem. This theorem states that the extended mapping class group of \(S_{g,0,0}\) is isomorphic to the group \(\text{Out}(\pi_1(S_{g,0,0}))\) of outer automorphisms of the fundamental group of \(S_{g,0,0}\). Part 1 ends with Chapter 9, which gives a brief introduction to braid groups \(B_n\) as well as a new proof of the Birman-Hilden theorem. The latter says that for \(g\geq 1\), the braid group \(B_{2g+1}\) is isomorphic to some subgroup of the mapping class group \(\text{Mod}(S_{g,1,0})\).

Part 2 of the book provides a concise introduction to Teichmüller theory and the related moduli theory of Riemann surfaces, with emphasis on those aspects of the theory that are most directly related to the mapping class group \(\text{Mod}(S_{g,0,0})\) and its structure as studied in Part 1. In this regard, Part 2 is of much more analytic and geometric nature than the first part, in which the group-theoretical properties of \(\text{Mod}(S_{g,0,0})\) were of principal interest.

In Chapter 10, Teichmüller space \(\text{Teich}(S)\) of a surface \(S\) is introduced as the space of hyperbolic structures on \(S\) up to isotopy. After discussing some other interpretations of \(\text{Teich}(S)\), a proof of the classical theorem by R. Fricke and F. Klein [Lectures on the theory of automorphic functions. First volume; the group-theoretic basis. (Vorlesungen über die Theorie der automorphen Functionen. Erster Band; Die gruppentheoretischen Grundlagen.) Leipzig: B. G. Teubner. XIV (1897; JFM 28.0334.01)] in the form \(\text{Teich}(S_{g,0, 0})\cong\mathbb{R}^{6g-6}\) is given, thereby using the so-called Fenchel-Nielsen coordinates on the corresponding Teichmüller space. This chapter ends with the fundamental “\(9g-9\) Theorem” about hyperbolic metrics on \(S_{g,0,0}\), which establishes an embedding of \(\text{Teich}(S_{g,0,0})\) into \(\mathbb{R}^{9g-9}\) via simple closed curves in the surface.

Several important aspects of the rich geometry of Teichmüller spaces are explained in Chapter 11. The discussion presented here covers some of the various other interpretations of \(\text{Teich}(S)\) as a classifying space, L. Bers’ approach to Teichmüller theory via quasiconformal maps, O. Teichmüller’s original existence and uniqueness theorems on extremal quasiconformal maps,measured foliations, the Teichmüller metric on \(\text{Teich}(S)\), Teichmüller geodesics, holomorphic quadratic differentials, Beltrami differentials, and the measurable Riemann mapping theorem. Chapter 12 gives a treatment of the moduli space \(M_g\) of compact Riemann surfaces of genus \(g\geq 2\) via Teichmüller theory, that is, by means of the analytic isomorphism \(M_g\cong\text{Teich} (S_g)/\text{Mod} (S_g)\), where \(S_g\) stands for a fixed compact Riemann surface of genus \(g\). Topologically, \(S_g\) is homeomorphic to any surface \(S_{g,0,0}\). The main results of this chapter include Fricke’s theorem on the properly discontinuous action of \(\text{Mod}(S_g)\) on Teichmüller space \(\text{Teich}(S_g)\) the aspherical orbifold structure of the moduli space \(M_g\), Mumford’s compactness criterion for the so-called “\(\varepsilon\)-thick parts” of the moduli space \(M_g\), and the determination of the topology of \(M_g\) (with respect to the induced Teichmüller metric) at infinity. This chapters ends with a proof of the fact that the rational cohomology of the moduli space \(M_g\) is isomorphic to the rational cohomology of the mapping class group \(\text{Mod}(S_g)\), thereby showing that \(M_g\) is very close to being a classifying space for \(S_g\)-bundles.

Finally, the main goal of Part 3 is to study the individual elements of the mapping class group \(\text{Mod}(S)\) more closely.To this end, Chapter 13 is devoted to a proof of the famous Nielsen-Thurston classification theorem for elements in \(\text{Mod}(S_g)\), where \(S_g\) is again a compact Riemann surface of genus \(g\geq 2\). This theorem states that each mapping class in \(\text{Mod}(S_g)\) is either (1) periodic; (2) reducible with respect to simple closed curves; or (3) is a pseudo-Anosov homeomorphism. The authors’ proof uses many of the ideas and results proved earlier in the book, especially in Part 2. In the subsequent Chapter 14, pseudo-Anosov classes in \(\text{Mod}(S_g)\) are investigated in greater depth. The authors give five explicit constructions of pseudo-Anosov homoeomorphisms, describe the orbits of a pseudo-Anosov homeomorphism, explore special properties of those measured foliations that are stable (or unstable) foliations of some pseudo-Anosov elements, and prove a few fundamental facts concerning stretch factors of pseudo-Anosov homeomorphisms.

Part 3 and the book end with Chapter 15, in which W. Thurston’s proof of the Nielsen-Thurston classification theorem is expounded. However, instead of giving a formal treatment, the authors rather try to illustrate (and to sketch) the beautiful circle of ideas underlying Thurston’s approach. This is achieved by explaining a fundamental example in the beginning, together with the crucial combinatorial device of so-called “train tracks”, and giving then a sketch of Thurston’s general theory via Markov partitions, laminations, measured foliations, and other crucial concepts. As for this topic, a very lucid and detailed account can be found in the classic book “Thurston’s Work on Surfaces” by A. Fathi, F. Laudenbach and V. Poénaru [Thurston’s work on surfaces. Transl. from the French by Djun Kim and Dan Margalit. Mathematical Notes 48. Princeton, NJ: Princeton University Press. xiii, 255 p. (2012; Zbl 1244.57005)].

The book comes with a rich bibliography, a carefully compiled index, and with a wealth of illustrating, highly instructive figures in the text. The presentation of the utmost rich and versatile material on mapping class groups and their allied theories stands out by its high degree of clarity, fullness, accuracy, and expository mastery.

No doubt, this primer is the most complete textbook for graduate students, instructors, and researchers who are looking for a profound source in this fascinating area of both classical and current mathematics.

The book under review provides a profound introduction to the fascinating theory relating these two kinds of objects, thereby touching upon the diverse algebraic, combinatorial, analytical and topological facets forming this profuse topic, which is experiencing a great renaissance in these days, in particular with a view toward moduli theory and its modern applications in mathematical physics.

As the authors point out in the preface, their goal in the present book is to explain as many important theorems, examples, and technique as possible, as quickly and directly as possible, while at the same time giving (nearly) full details and keeping the text widely self-contained. Also, the book contains some simplifications of known approaches and proofs, the exposition of some results that are not readily available in the literature, and some new material as well. As for the precise contents, the book consists of three parts titled “Mapping class groups” (Part 1), “Teichmüller space and moduli space” (Part 2), and “The classification and pseudo-Anosov theory” (Part 3). Each part is divided into several chapters and subsections, where the main results of the single chapters are outlined in an “Overview” at the very beginning of the book.

Part 1 covers the basic theory of mapping class groups of surfaces, with the central theme being the relationship between the algebraic structure of the mapping class group \(\text{Mod}(S)\) and the combinatorial topology of the underlying surface \(S\).

Chapter 1 recalls some basic facts about surfaces and hyperbolic geometry, including the working with simple closed curves and their geodesic representatives, their geometric intersection numbers, the notion of isotopy of a surface, and some useful facts from surface topology regarding self-homeomorphisms. Chapter 2 gives the definition of the mapping class group \(\text{Mod}(S)\) of a surface \(S\), which is then followed by first examples and computations in the simplest concrete cases. In this context, the classical algorithmic “Alexander method” is explained and applied, and the fundamental theorem of M. Dehn on the structure of the mapping class group \(\text{Mod}(T^2)\) of the 2-torus \(T^2\) is presented as well. Chapter 3 describes the famous Dehn twists as fundamental mapping classes, namely as the simplest infinite-order elements in \(\text{Mod}(S)\). The authors treat Dehn twists and their action on simple closed curves in great depth, with concrete applications to the structure of \(\text{Mod}(S_{g,0,0})\) for a closed compact surface of genus \(g\geq 3\). Chapter 4 is devoted to the problem of generating the mapping class group \(\text{Mod}(S)\) by Dehn twists, culminating in the celebrated Dehn-Lickorish-Humphries theorem, the construction of varieties simplicial complexes of curves, the famous Birman exact sequence with respect to punctures, and the exhibition of different sets of generators of \(\text{Mod}(S)\) by means of Dehn twists. Chapter 5 turns to presentations and the low-dimensional homology of certain mapping class groups. This includes the Dehn-Johnson “lantern relation” among Dehn twists, J. Harer’s theorem on the triviality of the first homology group \(H_1(\text{Mod}(S_{g,0,0}),\mathbb{Z})\), the explicit finite presentations of special mapping class groups due to B. Wajnryb [Isr. J. Math. 45, 157–174 (1983; Zbl 0533.57002)] and S. Gervais [Trans. Am. Math. Soc. 348, No. 8, 3097–3132 (1996; Zbl 0861.57023)] the proof of the finite presentability via group actions on simplicial complexes, Hopf’s formula for computing \(H_2(G,\mathbb{Z})\) for any group from a finite presentation for G, W. Pitsch’s upper bound on the rank of \(H_2(\text{Mod}(S_{g,0,0}),\mathbb{Z})\), and a new proof of J. Harer’s theorem stating that \(H_2(\text{Mod}(S_{g,0,0}),\mathbb{Z})\) is infinitely cyclic for \(g\geq 4\). Furthermore, the Euler class in the second cohomology group of a surface \(S_{g,1,0}\) is analyzed, and the so-called Meyer signature cocycle in the cohomology group \(H^2(\text{Mod}(S_{g,0,0}),\mathbb{Z})\) is described via \(S_{g,0,0}\)-bundles (surface bundles).

Chapter 6 discusses the symplectic representation of the mapping class group \(\text{Mod}(S_{g,0,0})\) induced by the action of the latter on the homology \(H_1(S_{g,0,0};\mathbb{Z})\), different proofs of the surjectivity of the symplectic representation, the associated Torelli group, and some classical structure theorems on \(\text{Mod}(S_{g,0,0})\) and the Torelli group, respectively, which are basically due to J.-P. Serre, E. L. Grossman, D. Johnson, and others.

Finite subgroups of mapping class groups are studied in Chapter 7, where the Nielsen realization theorem, the Hurwitz \(84(g-1)\) theorem for closed hyperbolic surfaces, Wiman’s \(4g+2\) theorem on the orders of automorphisms of closed hyperbolic surfaces, and a Gauss-Bonnet formula for orbifolds appear as related fundamental results.

Chapter 8 is an exposition of one of the most beautiful results in geometric group theory: the famous Dehn-Nielsen-Baer theorem. This theorem states that the extended mapping class group of \(S_{g,0,0}\) is isomorphic to the group \(\text{Out}(\pi_1(S_{g,0,0}))\) of outer automorphisms of the fundamental group of \(S_{g,0,0}\). Part 1 ends with Chapter 9, which gives a brief introduction to braid groups \(B_n\) as well as a new proof of the Birman-Hilden theorem. The latter says that for \(g\geq 1\), the braid group \(B_{2g+1}\) is isomorphic to some subgroup of the mapping class group \(\text{Mod}(S_{g,1,0})\).

Part 2 of the book provides a concise introduction to Teichmüller theory and the related moduli theory of Riemann surfaces, with emphasis on those aspects of the theory that are most directly related to the mapping class group \(\text{Mod}(S_{g,0,0})\) and its structure as studied in Part 1. In this regard, Part 2 is of much more analytic and geometric nature than the first part, in which the group-theoretical properties of \(\text{Mod}(S_{g,0,0})\) were of principal interest.

In Chapter 10, Teichmüller space \(\text{Teich}(S)\) of a surface \(S\) is introduced as the space of hyperbolic structures on \(S\) up to isotopy. After discussing some other interpretations of \(\text{Teich}(S)\), a proof of the classical theorem by R. Fricke and F. Klein [Lectures on the theory of automorphic functions. First volume; the group-theoretic basis. (Vorlesungen über die Theorie der automorphen Functionen. Erster Band; Die gruppentheoretischen Grundlagen.) Leipzig: B. G. Teubner. XIV (1897; JFM 28.0334.01)] in the form \(\text{Teich}(S_{g,0, 0})\cong\mathbb{R}^{6g-6}\) is given, thereby using the so-called Fenchel-Nielsen coordinates on the corresponding Teichmüller space. This chapter ends with the fundamental “\(9g-9\) Theorem” about hyperbolic metrics on \(S_{g,0,0}\), which establishes an embedding of \(\text{Teich}(S_{g,0,0})\) into \(\mathbb{R}^{9g-9}\) via simple closed curves in the surface.

Several important aspects of the rich geometry of Teichmüller spaces are explained in Chapter 11. The discussion presented here covers some of the various other interpretations of \(\text{Teich}(S)\) as a classifying space, L. Bers’ approach to Teichmüller theory via quasiconformal maps, O. Teichmüller’s original existence and uniqueness theorems on extremal quasiconformal maps,measured foliations, the Teichmüller metric on \(\text{Teich}(S)\), Teichmüller geodesics, holomorphic quadratic differentials, Beltrami differentials, and the measurable Riemann mapping theorem. Chapter 12 gives a treatment of the moduli space \(M_g\) of compact Riemann surfaces of genus \(g\geq 2\) via Teichmüller theory, that is, by means of the analytic isomorphism \(M_g\cong\text{Teich} (S_g)/\text{Mod} (S_g)\), where \(S_g\) stands for a fixed compact Riemann surface of genus \(g\). Topologically, \(S_g\) is homeomorphic to any surface \(S_{g,0,0}\). The main results of this chapter include Fricke’s theorem on the properly discontinuous action of \(\text{Mod}(S_g)\) on Teichmüller space \(\text{Teich}(S_g)\) the aspherical orbifold structure of the moduli space \(M_g\), Mumford’s compactness criterion for the so-called “\(\varepsilon\)-thick parts” of the moduli space \(M_g\), and the determination of the topology of \(M_g\) (with respect to the induced Teichmüller metric) at infinity. This chapters ends with a proof of the fact that the rational cohomology of the moduli space \(M_g\) is isomorphic to the rational cohomology of the mapping class group \(\text{Mod}(S_g)\), thereby showing that \(M_g\) is very close to being a classifying space for \(S_g\)-bundles.

Finally, the main goal of Part 3 is to study the individual elements of the mapping class group \(\text{Mod}(S)\) more closely.To this end, Chapter 13 is devoted to a proof of the famous Nielsen-Thurston classification theorem for elements in \(\text{Mod}(S_g)\), where \(S_g\) is again a compact Riemann surface of genus \(g\geq 2\). This theorem states that each mapping class in \(\text{Mod}(S_g)\) is either (1) periodic; (2) reducible with respect to simple closed curves; or (3) is a pseudo-Anosov homeomorphism. The authors’ proof uses many of the ideas and results proved earlier in the book, especially in Part 2. In the subsequent Chapter 14, pseudo-Anosov classes in \(\text{Mod}(S_g)\) are investigated in greater depth. The authors give five explicit constructions of pseudo-Anosov homoeomorphisms, describe the orbits of a pseudo-Anosov homeomorphism, explore special properties of those measured foliations that are stable (or unstable) foliations of some pseudo-Anosov elements, and prove a few fundamental facts concerning stretch factors of pseudo-Anosov homeomorphisms.

Part 3 and the book end with Chapter 15, in which W. Thurston’s proof of the Nielsen-Thurston classification theorem is expounded. However, instead of giving a formal treatment, the authors rather try to illustrate (and to sketch) the beautiful circle of ideas underlying Thurston’s approach. This is achieved by explaining a fundamental example in the beginning, together with the crucial combinatorial device of so-called “train tracks”, and giving then a sketch of Thurston’s general theory via Markov partitions, laminations, measured foliations, and other crucial concepts. As for this topic, a very lucid and detailed account can be found in the classic book “Thurston’s Work on Surfaces” by A. Fathi, F. Laudenbach and V. Poénaru [Thurston’s work on surfaces. Transl. from the French by Djun Kim and Dan Margalit. Mathematical Notes 48. Princeton, NJ: Princeton University Press. xiii, 255 p. (2012; Zbl 1244.57005)].

The book comes with a rich bibliography, a carefully compiled index, and with a wealth of illustrating, highly instructive figures in the text. The presentation of the utmost rich and versatile material on mapping class groups and their allied theories stands out by its high degree of clarity, fullness, accuracy, and expository mastery.

No doubt, this primer is the most complete textbook for graduate students, instructors, and researchers who are looking for a profound source in this fascinating area of both classical and current mathematics.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

57-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes |

30-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable |

30F10 | Compact Riemann surfaces and uniformization |

30F60 | Teichmüller theory for Riemann surfaces |

57M35 | Dehn’s lemma, sphere theorem, loop theorem, asphericity (MSC2010) |

20F38 | Other groups related to topology or analysis |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

57M07 | Topological methods in group theory |

57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |

20F36 | Braid groups; Artin groups |

14H15 | Families, moduli of curves (analytic) |

##### MathOverflow Questions:

Dehn-Nielsen-Baer Theorem for surfaces with boundary and puncturesMaximum order of MCG finite order elements

Modern references on hyperbolic groups

Pseudo-Anosov Matrices of surfaces (reference request)