топология / Farb, Margalit, A primer on mapping class groups

is simply connected. Later Hatcher–Lochak–Schneps gave an alternate proof for a closely related complex [77], and Wajnryb gave a combinatorial proof of simple connectivity for the original complex [169].

In general, when a group G acts cocompactly on a simply connected complex X with finitely presented vertex stabilizers and finitely gene rated edge stabilizers, the group G is finitely presented (see Proposition 6.6 below). For each orbit of vertices of X, there are relations in G coming from the relations in those vertex stabilizers, for each orbit of edges of X there are relations in G coming from the generators of those edge stabilizers (the relations identify elements of the two vertex stabilizers), and finally there is one relation in G for each orbit of 2–cells in X. See the paper of Ken Brown for details [33].

Since the complex X(Sg ) is defined by topological rules, it follows that Mod(Sg ) acts on X(Sg ). Using the Change of Coordinates Principle it is not hard to see that the action is cocompact; indeed there is a single orbit of vertices and a single orbit of edges. Now, the stabilizer in Mod(Sg ) of a vertex of X(Sg ) is closely related to a braid group. This is because if we cut Sg along the simple closed curves corresponding to a vertex of Sg , the result is a sphere with 2g boundary components, cf. Chapter 9. Therefore, the presentation for a vertex stabilizer can be derived from known presentations of braid groups, or, mapping class groups of genus 0 surfaces. Generating sets for edge stabilizers are obtained similarly.

To give a flavor of the calculation used to get Wajnryb's actua l presentation, we explain how the braid relation comes up in his analysis of the action of

Mod(Sg ) on X(Sg ).

b

. . .

c

Figure 6.10 The simple closed curves ai give a vertex of the cut system complex, and the simple closed curves a, b, and c, along with a2, . . . ag , give a triangle of the complex.

In what follows, we abuse notation, denoting a simple closed curve and its

associated Dehn twist by the same symbol.

Let va be the vertex of X(Sg ) corresponding to the cut system {ai} given in Figure 6.10. We will make use of two particular elements of the stabilizer Gva of va, namely the Dehn twist a and the element s = ba2b, where b is the Dehn twist about the simple closed curve shown in Figure 6.10.

Let eab be the edge of X(Sg ) spanned by the vertices va and vb defined above. One element of the stabilizer Geab of eab is r = aba. Since r interchanges the vertices of eab, it follows that r2 is an element of Gva . In particular, it is the element sa2 Gva . So we obtain the following relation (relation P10 in [169, Theorem 31]):

r2 = sa2.

We now focus on the stabilizer of a 2–cell, namely the triangl e tabc spanned by va, vb, and vc. The element ar does not stabilize va or eab, but it does stabilize tabc, inducing an order 3 rotation of tabc. Thus, (ar)3 is an element of Gva , and again one can write it as a word in the elements s, a Gva , namely, (asa)2. So we have the following relation (relation P11 in [169, Theorem 31]):

(ar)3 = (asa)2.

We can rewrite this last relation using the relation r2 = sa2 and the trivial relations aa−1 = 1 and bb−1 = 1.

(ar)3 = (asa)2

= (ar)3 = a(sa2)sa = (ar)3 = ar2sa

Replacing r with aba and s with ba2b, we find:

a2ba3ba3ba = a2ba2baba2ba = (a2ba2)aba(a2ba) = (a2ba2)bab(a2ba)

= aba = bab

Thus we “see” the braid relation from the action of Mod(Sg ) on X(Sg ); it comes from two relations one gets by flipping edges and by rota ting triangles. Deriving the complete presentation of Mod(Sg ) given in Theorem 6.3 is quite involved; we refer the reader to Wajnryb's paper [169] for details.

It is straightforward to carry out this procedure in the case of the torus. The complex X(T 2) is the Farey Complex (see Section 5.1), and, in fact, the

relations r2 = sa2 and aba = bab already discussed suffice to present the group Mod(T 2) ≈ SL(2, Z).

THE GERVAIS PRESENTATION

While Wajnryb's presentation (Theorem 6.3) is the most well-known and classical presentation of Mod(S), there are several other useful ones. We now present one due to Gervais. Some of the features of this presentation are: it is fairly easy to write down explicitly, it works for the pure mapping class group of any surface with boundary, and all of the relations are described on uniformly small subsurfaces (tori with at most 3 boundary components). It is important to keep in mind that Gervais's derivation of this presentation is accomplished by starting from Wajnryb's presentation and simplifying the relations there. The same is true for the beautful presentation due to Matsumoto [114], which is phrased in terms of Artin groups, and which we do not discuss here.

The Gervais presentation uses one new relation which we have not seen before.

The star relation. Consider the torus S03 with 3 boundary components d1, d2, and d3. Let c1, c2, c3, and b be isotopy classes of simple closed curves configured as in Figure 6.11. Note that S03 is homeomorphic to a closed regular neighborhood of c1 c2 c3 b (really, the union of 4 representatives).

d3

Figure 6.11 The simple closed curves used in the star relation.

Gervais gives the following relation [61]. If c1, c2, c3, b, d1, d2, and d3 are the isotopy classes of simple closed curves in S03 given in Figure 6.11, then we have:

(Tc1 Tc2 Tc3 Tb)3 = Td1 Td2 Td3 .

As with the lantern relation, this relation can be checked with the Alexander Method. We call b the central curve of the star relation. For any embedding S03 ֒→ S into a surface S, the image of the star relation under the induced homomorphism Mod(S03) → Mod(S) of course gives a relation (betwen the images of the above curves) in Mod(S).

Suppose that S03 is embedded in S in such a way that the isotopy classes c1 and c2 are equal, but distinct from c3. This happens when the image of d3 under the embedding is the trivial isotopy class and the images of d1 and d2 are nontrivial. In this case, the star relation becomes

(Tc21 Tc3 Tb)3 = Td1 Td2 .

We call this a degenerate star relation. We will not need to consider star relations with c1 = c2 = c3. We note that the degenerate star relation is the same as one of the 3–chain relations given in Section 5.4.

Recall that we used the star relation in Section 5.4 to prove Corollary 5.15.

The Gervais presentation. Let S be a compact surface of genus g with n boundary components. We begin by giving the generating set for the Gervais presentation of Mod(S). Each of the generators is a Dehn twist, and so it suffices to list the corresponding simple closed cur ves. The curves are shown in Figure 6.12, where we have drawn S as a torus with g − 1 handles attached and n disks removed.

We start with the top picture in the figure. There is one simple closed curve b which will form the central curve for all of our star relations. There are 2(g − 1) + b simple closed curves {ci} with i(b, ci) = 1. There are 2(g − 1) simple closed curves corresponding to the latitudes and longitudes of the g − 1 handles attached to the central torus. We also include the n boundary components. Finally, for each ordered pair of distinct curves (ci, cj ), there is a simple closed curve ci,j that lies in a neighborhood of ci cj b and that lies in the clockwise direction from ci along b (note that each ci,i+1 has already appeared on the list). The curves ci,j are depicted in the bottom picture of Figure 6.12; there are (2g − 2 + b)(2g − 3 + b) of these curves.

THEOREM 6.4 (Gervais' finite presentation) Let S be a surface of genus g with n boundary components. The group Mod(S) has a presentation with one Dehn twist generator for each simple closed curve shown in Figure 6.12, and with the following relations.

1. All disjointness relations between generators.

b

Figure 6.12 The generators for the Gervais presentation.

2.All braid relations between generators.

3.All star relations between generators, including the degenerate ones, where b is the central curve.

From Theorem 6.4, it is straightforward to write down the presentation explicitly, by listing the generators and relations. For the fi rst two kinds of relations, one needs to find all pairs of generators that are dis joint or that have intersection number 1. The degenerate star relations are given by triples {ci, ci, cj }, where ci 6= cj , and the other star relations are given by triples of distinct ci–curves.

By Proposition 3.13, one can get a presentation for the case of a surface with punctures by setting each generator corresponding to a Dehn twist about a boundary curve to be trivial.

6.3 PROOF OF FINITE PRESENTABILITY

We now give a proof that Mod(S) is finitely presented. While it is possible to give a proof analogous to our proof of finite generation, we instead choose to introduce a new technique. As a result, we obtain a new proof of finite generation.

The strategy, suggested by Andrew Putman, is to show that the “arc complex” A(S) is contractible, and use the action of Mod(S) on A(S) to build a K(Mod(S), 1) with finite 2–skeleton. It immediately follows that Mod(S) is finitely presented. While this is a simple proof of finite pr esentability, we do not know what explicit finite presentation comes out of thi s approach.

THE ARC COMPLEX

Let S be a compact surface that either has nonempty boundary or has at least one marked point. We define the arc complex A(S) as the abstract simplicial flag complex described by the following data (cf. §5.1).

Vertices. There is one vertex for each free isotopy class of essential simple proper arcs in S.

Edges. Vertices are connected by an edge if the corresponding free isotopy classes have disjoint representatives.

If we take a surface S with nonempty boundary and cap one or more boundary components with a once-marked disk, then A(S) is naturally isomorphic to the arc complex for the capped surface. So in this sense there is no difference between marked points and boundary components in defining the arc complex. When we consider the action of the mapping class group on the arc complex, marked points are more natural than boundary components since Dehn twists about boundary components act trivially on the arc complex.

As a first example, the arc complex of the torus with one bounda ry component is the Farey complex (see Section 5.1).

The most fundamental fact about the arc complex is the following theorem, due to Harer [71].

THEOREM 6.5 Let S be any compact surface with finitely many marked points. If A(S) is nonempty then it is contractible.

The elegant proof we present is due to Hatcher [75]. A number of other mathematicians made various contributions to the circle of ideas surrounding this theorem, including Thurston, Bowditch–Epstein, M umford, Mosher, and Penner.

FINITE PRESENTABILITY VIA GROUP ACTIONS ON COMPLEXES

The group Mod(S) acts by simiplicial automorphisms on the contractible simplicial complex A(S). In order to use this action to analyze Mod(S), we need to apply some geometric group theory.

The following theorem is adapted from Scott–Wall [150]. In t he statement of the theorem, we say that a group G acts on a CW–complex X without rotations if whenever an element g G fixes a cell σ X then g fixes σ pointwise. Any action of a group on a CW–complex can be turned into an action without rotations by barycentrically subdividing the complex. The benefit of an action without rotations is that the quotient ha s a natural CW– complex structure coming from the structure of the original complex.

Proposition 6.6 Let G be a group acting on a contractible CW–complex X without rotations. Suppose that each of the following conditions holds.

1.The quotient X/G is finite.

2.Each vertex stabilizer is finitely presented.

3.Each edge stabilizer is finitely generated.

Then G is finitely presented.

Proof. Let U be any K(G, 1) complex. Consider the contractible complex U × X. Since the action of G on U is free, the diagonal action of G on

complex is called the Borel construction.

We will show that e × has the homotopy type of a complex with

(U X)/G

finite 2–skeleton. Consider the projection

If v is a vertex of X with stabilizer Gv in G, then (U ×v)/Gv is a K(Gv , 1)

preimage of [v] X/G. In other words, over each vertex of X/G there is

in e × a corresponding to that vertex stabilizer. Similarly,

(U X)/G K(π, 1)

lying over each higher-dimensional open cell is the product of a K(π, 1) complex for that cell stabilizer with that open cell.

As a result, we see that e × has the structure of a “complex of

(U X)/G

spaces,” with each vertex space a K(Gv , 1) for a vertex stabilizer Gv and each edge space a K(Ge, 1) for an edge stabilizer Ge. That is, the space

e × is obtained inductively as follows: we start with the disjoint

(U X)/G

union of the K(Gv , 1) spaces; then, we take the K(Ge, 1) spaces, cross them with intervals, and glue them to the K(Gv , 1) spaces via any map in the unique homotopy class of maps determined by the inclusion Ge ֒→ Gv . This process is repeated inductively (and analogously) on higher dimensional skeleta.

We make the following observation: if each space in the complex of spaces is replaced with another space to which it is homotopy equivalent (i.e., another K(π, 1) space), the homotopy type of the resulting complex does not change. In other words, the “homotopy colimit” is well-defin ed [76, Prop 4G.1].

Since the stabilizer Gv of each vertex v is assumed to be finitely presented, each K(Gv , 1) space can be chosen to have finite 2–skeleton. Since the stabilizer of each edge e is assumed to be finitely generated, each K(Ge, 1) space can be chosen to have finite 1–skeleton. For the stabili zer Gf of each 2-cell f , the K(Gf ) space can be chosen to have finite 0–skeleton, since for any group H there is a K(H, 1) with a single vertex).

There are three ways that –cells arise in the complex of spaces e × :

2 (U X)/G via 2–cells of K(Gv , 1) spaces, 1–cells of K(Ge, 1) spaces, and 0–cells of K(Gf , 1) spaces. As discussed above, each of these spaces can be chosen to have finite 2–skeleton, 1–skeleton and 0–skeleton, respe ctively. Since the quotient X/G is finite, the resulting complex of spaces has finitely many

We remark that the proof of Proposition 6.6 can be slightly modified to work in the case where X is only assumed to be simply connected, as opposed to contractible. Actually, the complex of curves C(S) is simply connected (but not contractible) for most S. The reason we use the arc complex in our application of Proposition 6.6 is simply because it is easier to prove that A(S) is simply connected.

PROOF THAT THE MAPPING CLASS GROUP IS FINITELY PRESENTED

We are now ready to prove the following theorem.

THEOREM 6.7 If S is a compact surface with finitely many marked points, then the group Mod(S) is finitely presented.

Proof. We first reduce the problem to the case of Sg,n with n > 0 marked points. Suppose we can prove the theorem in this case. We now explain how to deduce the theorem in the case that S has nonempty boundary, and then the case where S is closed.

Let S be a compact surface with n > 0 boundary components, and assume that S is not the disk D2. Also assume by induction that for any compact surface with n−1 boundary components, the mapping class group is finitely presented. We recall Proposition 3.13, which states that if S is the surface obtained from a surface S by capping a boundary component β with a oncemarked disk, then the following sequence is exact:

where Mod (S ) is the subgroup of Mod(S ) consisting of elements that fix the marked point coming from the capping operation. By the inductive hypothesis, we have that Mod(S ) is finitely presented. Since Mod (S ) has finite index in Mod(S ), it is also finitely presented. Since the extension of a finitely presented group by a finitely presented group is fi nitely presented, it follows from Proposition 3.13 that Mod(S) is finitely presented.

A similar argument to the above, using the Birman exact sequence, shows that Mod(Sg,0) is finitely presented if Mod(Sg,1) is, since the quotient of a finitely presented group by a finitely generated group is finit ely presented.

We have thus reduced the proof to showing that Mod(Sg,n) is finitely presented when n > 0. We may assume that (g, n) 6= (0, 1) because we already know Mod(S0,1) = 1. Since a group is finitely presented if and only if any of its finite index subgroups is finitely presented, it suffice s to prove that PMod(Sg,n) is finitely presented. We make the inductive hypothesis that PMod(Sg′,n′) is finitely presented when g′ < g or when g′ = g and n′ < n.

We would like to apply Proposition 6.6. By Theorem 6.5, the arc complex A(Sg,n) is contractible. Therefore its barycentric subdivision A′(Sg,n), on which PMod(Sg,n) acts without rotations, is also contractible. Note