# A Primer on Mapping Class Groups (PMS-49)

Benson Farb
Dan Margalit
Pages: 488
https://www.jstor.org/stable/j.ctt7rkjw

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. Preface
(pp. xi-xii)
Benson Farb and Dan Margalit
4. Acknowledgments
(pp. xiii-xvi)
5. Overview
(pp. 1-14)
6. ### PART 1. MAPPING CLASS GROUPS

• Chapter One Curves, Surfaces, and Hyperbolic Geometry
(pp. 17-43)

A linear transformation of a vector space is determined by, and is best understood by, its action on vectors. In analogy with this, we shall see that an element of the mapping class group of a surfaceSis determined by, and is best understood by, its action on homotopy classes of simple closed curves inS. We therefore begin our study of the mapping class group by obtaining a good understanding of simple closed curves on surfaces.

Simple closed curves can most easily be studied via their geodesic representatives, and so we begin with the fact that every surface...

• Chapter Two Mapping Class Group Basics
(pp. 44-63)

In this chapter we begin our study of the mapping class group of a surface. After giving the definition, we compute the mapping class group in essentially all of the cases where it can be computed directly. This includes the case of the disk, the annulus, the torus, and the pair of pants. An important method, which we call the Alexander method, emerges as a tool for such computations. It answers the fundamental question: how can one prove that a homeomorphism is or is not homotopically trivial? Equivalently, how can one decide when two homeomorphisms are homotopic or not?

Let...

• Chapter Three Dehn Twists
(pp. 64-88)

In this chapter we study a particular type of mapping class called a Dehn twist. Dehn twists are the simplest infinite-order mapping classes in the sense that they have representatives with the smallest possible supports. Dehn twists play the role for mapping class groups that elementary matrices play for linear groups. We begin by defining Dehn twists inSand proving that they have infinite order in Mod(S). We determine many of the basic properties of Dehn twists by studying their action on simple closed curves. As one consequence, we compute the center of Mod(S). At the end of the...

• Chapter Four Generating the Mapping Class Group
(pp. 89-115)

Is there a way to generate all (homotopy classes of) homeomorphisms of a surface by compositions of simple-to-understand homeomorphisms? We have already seen that Mod(T2) is generated by the Dehn twists about the latitude and longitude curves. Our next main goal will be to prove the following result.

THEOREM 4.1 (Dehn–Lickorish theorem)For g≥ 0,the mapping class group Mod(Sg) is generated by finitely many Dehn twists about nonseparating simple closed curves.

Theorem 4.1 can be likened to the theorem that for eachn≥ 2 the group SL(n,$\mathbb{Z}$) can be generated by finitely many elementary matrices....

• Chapter Five Presentations and Low-dimensional Homology
(pp. 116-161)

Having found a finite set of generators for the mapping class group, we now begin to focus on relations. Indeed, one of our main goals in this chapter is to give a finite presentation for Mod(S). In doing so, we will see some beautiful topological ideas, as well as some useful techniques from geometric group theory.

The relations in a groupGare intimately related to the first and second homology groups ofG. Recall that the homology groups ofGare defined to be the homology groups of anyK(G, 1)-space. The first and second homology groups have direct,...

• Chapter Six The Symplectic Representation and the Torelli Group
(pp. 162-199)

One of the fundamental aspects of Mod(Sg) is its action onH1(Sg;$\mathbb{Z}$). The representation Ψ : Mod(Sg) → Aut(H1(Sg;$\mathbb{Z}$)) is like a first linear approximation to Mod(Sg), and we can try to transfer our knowledge of the linear group Aut(H1(Sg;$\mathbb{Z}$)) to the group Mod(Sg).

As we show in Section 6.1, the algebraic intersection number onH1(Sg;$\mathbb{R}$) gives this vector space a symplectic structure. This symplectic structure is preserved by the image of Ψ, and so Ψ can be thought of as a representation

$\Psi :{\text{Mod}}(S_g ) \to {\text{Sp}}(2g,\mathbb{Z})$

into the integral symplectic group. The homomorphism Ψ is called thesymplectic...

• Chapter Seven Torsion
(pp. 200-218)

In this chapter we investigate finite subgroups of the mapping class group. After explaining the distinction between finite-order mapping classes and finite-order homeomorphisms, we then turn to the problem of determining what is the maximal order of a finite subgroup of Mod(Sg). We will show that, forg≥ 2, finite subgroups have order at most 84(g— 1) and cyclic subgroups have order at most 4g+ 2. We will also see that there are finitely many conjugacy classes of finite subgroups in Mod(S). At the end of the chapter, we prove that Mod(Sg) is generated by finitely many...

• Chapter Eight The Dehn—Nielsen—Baer Theorem
(pp. 219-238)

The Dehn—Nielsen—Baer theorem states that Mod(Sg) is isomorphic to an index 2 subgroup of the group Out(π1(Sg)) of outer automorphisms of π1(Sg). This is a beautiful example of the interplay between topology and algebra in the mapping class group. It relates a purely topological object, Mod(Sg), to a purely algebraic one, Out(π1(Sg)). Further, these are related via hyperbolic geometry!

We begin by defining the objects in the statement of the theorem.

Extended mapping class group. LetSbe a surface without boundary. Theextended mapping class group, denoted Mod±(S), is the group of isotopy classes of all...

• Chapter Nine Braid Groups
(pp. 239-260)

In this chapter we give a brief introduction to Artin’s classical braid groupsBn. WhileBnis just a special kind of mapping class group, namely, that of a multipunctured disk, the study ofBnhas its own special flavor. One reason for this is that multipunctured disks can be embedded in the plane, so that elements ofBnlend themselves to specialized kinds of pictorial representations.

The notion of a mathematical braid is quite natural and classical. For instance, this concept appeared in Gauss’s study of knots in the early nineteenth century (see [182]) and in Hurwitz’s 1891 paper...

7. ### PART 2. TEICHMÜLLER SPACE AND MODULI SPACE

• Chapter Ten Teichmüller Space
(pp. 263-293)

This chapter introduces another main player in our story: the Teichmüller space Teich(S) of a surfaceS. Forg≥ 2, the space Teich(Sg) parameterizes all hyperbolic structures onSgup to isotopy. After defining a topology on Teich(S), we give a few heuristic arguments for computing its dimension. The length and twist parameters of Fenchel and Nielsen are then introduced in order to prove that Teich(Sg) is homeomorphic to$\mathbb{R}^{6g - 6}$. At the end of the chapter, we prove the 9g− 9 theorem, which tells us that a hyperbolic structure onSgis completely determined by the lengths...

• Chapter Eleven Teichmüller Geometry
(pp. 294-341)

Teichmüller space Teich(S) was defined in Chapter 10 as the space of hyperbolic structures on the surfaceSmodulo isotopy. But Teich(S) parameterizes other important structures as well, for example, complex structures onSmodulo isotopy and conformal classes of metrics onSup to isotopy.

We would like to have a way to compare different complex or conformal structures onSto each other. A natural way to do this is to search for a quasiconformal homeomorphismf:SSthat is homotopic to the identity map and that has the smallest possible quasiconformal dilatation with respect...

• Chapter Twelve Moduli Space
(pp. 342-364)

The moduli space of Riemann surfaces is one of the fundamental objects of mathematics. It is ubiquitous, appearing as a basic object in fields from low-dimensional topology to algebraic geometry to mathematical physics. The moduli space$\cal{M}$(S) parameterizes, among other things: isometry classes of hyperbolic structures onS, conformal classes of Riemannian metrics onS, biholomorphism classes of complex structures onS, and isomorphism classes of smooth algebraic curves homeomorphic toS.

We will access$\cal{M}$(S) as the quotient of Teich(S) by an action of Mod(S). A key result of this chapter is the theorem (due to Fricke)...

8. ### PART 3. THE CLASSIFICATION AND PSEUDO-ANOSOV THEORY

• Chapter Thirteen The Nielsen—Thurston Classification
(pp. 367-389)

In this chapter we explain and prove one of the central theorems in the study of mapping class groups: the Nielsen—Thurston classification of elements of Mod(S). This theorem is the analogue of the Jordan canonical form for matrices. It states that every$f \in {\text{Mod}}(S)$is one of three special types: periodic, reducible, or pseudo-Anosov. The knowledge of individual mapping classes is essential to our understanding of the algebraic structure of Mod(S). As we will soon explain, it is also essential for our understanding of the geometry and topology of many 3-dimensional manifolds.

We begin this chapter with a classification of...

• Chapter Fourteen Pseudo-Anosov Theory
(pp. 390-423)

The power of the Nielsen–Thurston classification is that it gives a simple criterion for an element$f \in {\text{Mod}}(S)$to be pseudo-Anosov:fis neither finiteorder nor reducible. This fact, however, is only as useful as the depth of our knowledge of pseudo-Anosov homeomorphisms. The purpose of this chapter is to study pseudo-Anosov homeomorphisms: their construction, their algebraic properties, and their dynamical properties.

Anosov maps of the torus. An Anosov homeomorphism of the torusT2is a linear representative of an Anosov mapping class. As discussed in Section 13.1, an Anosov homeomorphism φ :T2T2has an associated Anosov...

• Chapter Fifteen Thurston’s Proof
(pp. 424-446)

In this chapter we give some indication of how Thurston originally discovered the Nielsen—Thurston classification theorem. We begin with a concrete, accessible example that illustrates much of the general theory. We then provide a sketch of how that general theory works. Our goal is not to give a formal treatment as per the rest of the text. Rather, we hope to convey to the reader part of the beautiful circle of ideas surrounding the Nielsen—Thurston classification, including Teichmüller’s theorems, Markov partitions, train tracks, foliations, laminations, and more.

We start by giving an in-depth analysis of a fundamental and...

9. Bibliography
(pp. 447-464)
10. Index
(pp. 465-472)