A Primer on Mapping Class Groups (Princeton Mathematical Series, 49) - Hardcover

Farb, Benson; Margalit, Dan

Hardcover

ISBN 10: 0691147949 ISBN 13: 9780691147949

Publisher: Princeton University Press, 2011

Synopsis

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students.



A Primer on Mapping Class Groups begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

About this title

About the Author

Benson Farb is professor of mathematics at the University of Chicago. He is the editor of Problems on Mapping Class Groups and Related Topics and the coauthor of Noncommutative Algebra. Dan Margalit is assistant professor of mathematics at Georgia Institute of Technology.

Excerpt. © Reprinted by permission. All rights reserved.

A Primer on Mapping Class Groups

By Benson Farb Dan Margalit

PRINCETON UNIVERSITY PRESS

Copyright © 2012 Princeton University Press
All right reserved.
ISBN: 978-0-691-14794-9

Contents

Preface....................................................................xiAcknowledgments............................................................xiiiOverview...................................................................1PART 1. MAPPING CLASS GROUPS...............................................151. Curves, Surfaces, and Hyperbolic Geometry...............................172. Mapping Class Group Basics..............................................443. Dehn Twists.............................................................644. Generating the Mapping Class Group......................................895. Presentations and Low-dimensional Homology..............................1166. The Symplectic Representation and the Torelli Group.....................1627. Torsion.................................................................2008. The Dehn–Nielsen–Baer Theorem...............................2199. Braid Groups............................................................239PART 2. TEICHMÜLLER SPACE AND MODULI SPACE............................26110. Teichmüller Space.................................................26311. Teichmüller Geometry..............................................29412. Moduli Space...........................................................342PART 3. THE CLASSIFICATION AND PSEUDO-ANOSOV THEORY........................36513. The Nielsen–Thurston Classification..............................36714. Pseudo-Anosov Theory...................................................39015. Thurston's Proof.......................................................424Bibliography...............................................................447Index......................................................................465

Chapter One

Curves, Surfaces, and Hyperbolic Geometry

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 surface S is determined by, and is best understood by, its action on homotopy classes of simple closed curves in S. 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 may be endowed with a constant-curvature Riemannian metric, and we study the relation between curves, the fundamental group, and geodesics. We then introduce the geometric intersection number, which we think of as an "inner product" for simple closed curves. A second fundamental tool is the change of coordinates principle, which is analogous to understanding change of basis in a vector space. After explaining these tools, we conclude this chapter with a discussion of some foundational technical issues in the theory of surface topology, such as homeomorphism versus diffeomorphism, and homotopy versus isotopy.

1.1 SURFACES AND HYPERBOLIC GEOMETRY

We begin by recalling some basic results about surfaces and hyperbolic geometry that we will use throughout the book. This is meant to be a brief review; see [208] or [119] for a more thorough discussion.

1.1.1 SURFACES

A surface is a 2-dimensional manifold. The following fundamental result about surfaces, often attributed to Möbius, was known in the mid-nineteenth century in the case of surfaces that admit a triangulation. Radò later proved, however, that every compact surface admits a triangulation. For proofs of both theorems, see, e.g., [204].

THEOREM 1.1 (Classification of surfaces) Any closed, connected, orientable surface is homeomorphic to the connect sum of a 2-dimensional sphere with g = 0 tori. Any compact, connected, orientable surface is obtained from a closed surface by removing b = 0 open disks with disjoint closures. The set of homeomorphism types of compact surfaces is in bijective correspondence with the set {(g, b) : g, b = 0}.

The g in Theorem 1.1 is the genus of the surface; the b is the number of boundary components. One way to obtain a noncompact surface from a compact surface S is to remove n points from the interior of S; in this case, we say that the resulting surface has n punctures.

Unless otherwise specified, when we say "surface" in this book, we will mean a compact, connected, oriented surface that is possibly punctured (of course, after we puncture a compact surface, it ceases to be compact). We can therefore specify our surfaces by the triple (g, b, n). We will denote by S_g,n a surface of genus g with n punctures and empty boundary; such a surface is homeomorphic to the interior of a compact surface with n boundary components. Also, for a closed surface of genus g, we will abbreviate S_g,0 as S_g. We will denote by ?S the (possibly disconnected) boundary of S.

Recall that the Euler characteristic of a surface S is

X(S) = 2 - 2g - (b + n).

It is a fact that X(S) is also equal to the alternating sum of the Betti numbers of S. Since X(S) is an invariant of the homeomorphism class of S, it follows that a surface S is determined up to homeomorphism by any three of the four numbers g, b, n, and X(S).

Occasionally, it will be convenient for us to think of punctures as marked points. That is, instead of deleting the points, we can make them distinguished. Marked points and punctures carry the same topological information, so we can go back and forth between punctures and marked points as is convenient. On the other hand, all surfaces will be assumed to be without marked points unless explicitly stated otherwise.

If X(S) = 0] and [partial derivative]S = [empty set], then the universal cover [??] is homeomorphic to R² (see, e.g., [199, Section 1.4]). We will see that, when x(S) < 0, we can take advantage of a hyperbolic structure on [??].

1.1.2 THE HYPERBOLIC PLANE

Let H² denote the hyperbolic plane. One model for H² is the upper half-plane model, namely, the subset of C with positive imaginary part (y > 0), endowed with the Riemannian metric

ds² = [dx² + dy²]/y²,

where dx² + dy² denotes the Euclidean metric on C. In this model the geodesics are semicircles and half-lines perpendicular to the real axis.

It is a fact from Riemannian geometry that any complete, simply connected Riemannian 2-manifold with constant sectional curvature -1 is isometric to H².

For the Poincaré disk model of H², we take the open unit disk in C with the Riemannian metric

ds² = 4 dx² + dy²/(1 - r²)².

In this model the geodesics are circles and lines perpendicular to the unit circle in C (intersected with the open unit disk).

Any Möbius transformation from the upper half-plane to the unit disk is an isometry between the upper half-plane model for H² and the Poincaré disk model of H². The group of orientation-preserving isometries of H² is (in either model) the group of Möbius transformations taking H² to itself. This group, denoted Isom⁺(H²), is isomorphic to PSL(2,R). In the upper half-plane model, this isomorphism is given by the following map:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The boundary of the hyperbolic plane. One of the central objects in the study of hyperbolic geometry is the boundary at infinity of H², denoted by [partial derivative]H². A point of [partial derivative]H² is an equivalence class [?] of unit-speed geodesic rays where two rays ?₁, ?₂ : [0, 8) ? H² are equivalent if they stay a bounded distance from each other; that is, there exists D > 0 so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Actually, if ?₁ and ?₂ are equivalent, then they can be given unit-speed parameterizations so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We denote the union H² [union] [partial derivative]H² by H². The set H² is topologized via the following basis. We take the usual open sets of H² plus one open set U_P for each open half-plane P in H². A point of H² lies in U_P if it lies in P, and a point of [partial derivative]H² lies in U_P if every representative ray ?(t) eventually lies in P, i.e., if there exists T = 0 so that ?(t) [member of] P for all t = T.

In this topology [partial derivative]H² is homeomorphic to S¹, and the union [bar.H²] is homeomorphic to the closed unit disk. The space [bar.H²] is a compactification of H² and is called the compactification of H². In the Poincaré disk model of H², the boundary [partial derivative]H² corresponds to the unit circle in C, and [bar.H²] is identified with the closed unit disk in C.

Any isometry f [member of] Isom(H²) takes geodesic rays to geodesic rays, clearly preserving equivalence classes. Also, f takes half-planes to half-planes. It follows that f extends uniquely to a map [bar.f] : [bar.H²] ? [bar.H². As any pair of distinct points in [partial derivative]H² are the endpoints of a unique geodesic in H², it follows that [bar.f] maps distinct points to distinct points. It is easy to check that in fact [bar.f] is a homeomorphism.

Classification of isometries of H². We can use the above setup to classify nontrivial elements of Isom⁺(H²). Suppose we are given an arbitrary nontrivial element f [member of] Isom+(H²). Since [bar.f] is a self-homeomorphism of a closed disk, the Brouwer fixed point theorem gives that [bar.f] has a fixed point in [bar.H²]. By considering the number of fixed points of [bar.f] in [bar.H²], we obtain a classification of isometries of H² as follows.

Elliptic. If [bar.f] fixes a point p [member of] H², then f is called elliptic, and it is a rotation about p. Elliptic isometries have no fixed points on [partial derivative]H². They correspond to elements of PSL(2,R) whose trace has absolute value less than 2.

Parabolic. If [bar.f] has exactly one fixed point in [partial derivative]H², then f is called parabolic. In the upper half-plane model, f is conjugate in Isom⁺(H²) to z ? z ± 1. Parabolic isometries correspond to those nonidentity elements of PSL(2,R) with trace ±2.

Hyperbolic. If [bar.f] has two fixed points in [partial derivative]H², then f is called hyperbolic or loxodromic. In this case, there is an f-invariant geodesic axis ?; that is, an f-invariant geodesic in H² on which f acts by translation. On [partial derivative]H² the fixed points act like a source and a sink, respectively. Hyperbolic isometries correspond to elements of PSL(2,R) whose trace has absolute value greater than 2.

It follows from the above classification that if [bar.f] has at least three fixed points in [bar.H²], then f is the identity.

Also, since commuting elements of Isom⁺(H²) must preserve each other's fixed sets in [bar.H²], we see that two nontrivial elements of Isom⁺(H²) commute if and only if they have the same fixed points in [bar.H²].

1.1.3 Hyperbolic Surfaces

The following theorem gives a link between the topology of surfaces and their geometry. It will be used throughout the book to convert topological problems to geometric ones, which have more structure and so are often easier to solve.

We say that a surface S admits a hyperbolic metric if there exists a complete, finite-area Riemannian metric on S of constant curvature -1 where the boundary of S (if nonempty) is totally geodesic (this means that the geodesics in [partial derivative]S are geodesics in S). Similarly, we say that S admits a Euclidean metric, or flat metric if there is a complete, finite-area Riemannian metric on S with constant curvature 0 and totally geodesic boundary.

If S has empty boundary and has a hyperbolic metric, then its universal cover [??] is a simply connected Riemannian 2-manifold of constant curvature -1. It follows that [??] is isometric to H², and so S is isometric to the quotient of H² by a free, properly discontinuous isometric action of p₁(S). If S has nonempty boundary and has a hyperbolic metric, then [??] is isometric to a totally geodesic subspace of H². Similarly, if S has a Euclidean metric, then [??] is isometric to a totally geodesic subspace of the Euclidean plane E².

Theorem 1.2 Let S be any surface (perhaps with punctures or boundary). If x(S) < 0, then S admits a hyperbolic metric. If x(S) = 0, then S admits a Euclidean metric.

A surface endowed with a fixed hyperbolic metric will be called a hyperbolic surface. A surface with a Euclidean metric will be called a Euclidean surface or flat surface.

Note that Theorem 1.2 is consistent with the Gauss-Bonnet theorem which, in the case of a compact surface S with totally geodesic boundary, states that the integral of the curvature over S is equal to 2p_x(S).

One way to get a hyperbolic metric on a closed surface S_g is to construct a free, properly discontinuous isometric action of p₁(S_g) on H² (as above, this requires g = 2). By covering space theory and the classification of surfaces, the quotient will be homeomorphic to S_g. Since the action was by isometries, this quotient comes equipped with a hyperbolic metric. Another way to get a hyperbolic metric on S_g, for g = 2, is to take a geodesic 4g-gon in H² with interior angle sum 2p and identify opposite sides (such a 4g-gon always exists; see Section 10.4 below). The result is a surface of genus g with a hyperbolic metric and, according to Theorem 1.2, its universal cover is H².

We remark that while the torus T² admits a Euclidean metric, the once-punctured torus S1,1 admits a hyperbolic metric.

Loops in hyperbolic surfaces. Let S be a hyperbolic surface. A neighborhood of a puncture is a closed subset of S homeomorphic to a oncepunctured disk. Also, by a free homotopy of loops in S we simply mean an unbased homotopy. If a nontrivial element of p₁(S) is represented by a loop that can be freely homotoped into the neighborhood of a puncture, then it follows that the loop can be made arbitrarily short; otherwise, we would find an embedded annulus whose length is infinite (by completeness) and where the length of each circular cross section is bounded from below, giving in- finite area. The deck transformation corresponding to such an element of p₁(S) is a parabolic isometry of the universal cover H². This makes sense because for any parabolic isometry of H², there is no positive lower bound to the distance between a point in H² and its image. All other nontrivial elements of p₁(S) correspond to hyperbolic isometries of H² and hence have associated axes in H².

We have the following fact, which will be used several times throughout this book:

If S admits a hyperbolic metric, then the centralizer of any nontrivial element of p₁(S) is cyclic. In particular, p₁(S) has a trivial center.

To prove this we identify p₁(S) with the deck transformation group of S for some covering map H² ? S. Whenever two nontrivial isometries of H² commute, it follows from the classification of isometries of H² that they have the same fixed points in [partial derivative]H². So if a [member of] p₁(S) is centralized by ß, it follows that a and ß have the same fixed points in [partial derivative]H². By the discreteness of the action of p₁(S), we would then have that the centralizer of a in p₁(S) is infinite cyclic. If p₁(S) had nontrivial center, it would then follow that p₁(S) ˜ Z. But then S would necessarily have infinite volume, a contradiction.

1.2 SIMPLE CLOSED CURVES

Our study of simple closed curves in a surface S begins with the study of all closed curves in S and the usefulness of geometry in understanding them.

(Continues...)

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Edition details

• Publisher: Princeton University Press
• Publication date: 2011
• Language: English
• ISBN 10: 0691147949
• ISBN 13: 9780691147949
• Binding: Hardcover
• Number of pages: 488