Every mapping class group is generated by 6 involutions

Tara E. Brendle and Benson Farb ^{∗}

February 12, 2004

Abstract

Let Mod_{g,b} denote the mapping class group of a surface of genus g with b punctures. Luo asked in [Lu] if there is a universal upper bound, independent of genus, for the number of torsion elements needed to generate Mod_{g,b}. We answer Luo’s question by proving that 3 torsion elements suffice to generate Mod_{g,0}. We also prove the more delicate result that there is an upper bound, independent of genus, not only for the number of torsion elements needed to generate Mod_{g,b} but also for the order of those elements. In particular, our main result is that 6 involutions (i.e. orientation-preserving diffeomorphisms of order two) suffice to generate Mod_{g,b} for every genus g ≥ 3, b = 0 and g ≥ 4, b = 1.

1 Introduction

Let S_{g,b} denote a closed, oriented surface of genus g with b punctures, and let Mod_{g,b} denote its mapping class group, which is the group of homotopy classes of orientation-preserving homeomorphisms preserving the set of punctures. We shall frequently abuse terminology by confusing an individual homeomorphism with its mapping class in Mod_{g,b}.

We begin with a brief survey of some known generating sets for Mod_{g,b} possessing various properties. Dehn [De] produced a finite set of generators of Mod_{g,0}, proving that 2g(g − 1) Dehn twists suffice for g ≥ 3. Lickorish [Li] improved on this result by giving a generating set for Mod_{g,0} consisting of 3g − 1 twists for any g ≥ 1. Humphries [Hu] then showed that a certain subset of Lickorish’s set consisting of 2g+1 twists suffices to generate Mod_{g,0} and that this is in fact the minimal number of twist generators for Mod_{g,0} (here again g ≥ 1). Johnson [Jo] later proved that Humphries’ set also generates Mod_{g,1}.

∗The first author is partially supported by a VIGRE postdoc under NSF grant number 9983660 to Cornell University. The second author is supported in part by NSF grants DMS-9704640 and DMS-0244542.

If one allows generators other than twists, smaller generating sets can be obtained. Lickorish [Li] noted that Mod_{g,0} can be generated with 4 elements, 3 of which are twists and 1 of which has finite order. N. Lu [Lu] found a generating set with 3 elements, 2 of which have finite order. Wajnryb [Wa1] proved that for g ≥ 1, b = 0, 1, the group Mod_{g,b} can be generated by 2 elements, one of which has finite order.

The problem of finding small generating sets, torsion generating sets, and generating sets of involutions (i.e. elements of order two) is a classical one, and has been studied extensively, especially for finite groups (see, e.g.,

[DT] for a survey). In 1971, Maclachlan [Mac] proved that Mod_{g,0} is generated by torsion elements, and deduced from this that moduli space M_{g} is simply-connected as a topological space. These results were later extended to Mod_{g,b}, g ≥ 3, b ≥ 1 by Patterson [Pa]. The question of generating mapping class groups by involutions^{1} was first investigated by McCarthy and Papadopoulos [MP]. Among other results, they proved that for g ≥ 3, Mod_{g,0} is generated by infinitely many conjugates of a certain involution.

Luo [Luo], using work of Harer [Ha], described the first finite set of involutions which generate Mod_{g,b} for g ≥ 3, b ≥ 0. The order of his generating set depends on both g and b; in particular, his set consists of 12g + 2 involutions when b = 0, 1. Luo also gives torsion generators, not necessarily involutions, for all other cases except g = 2, b = 5k + 2. It should be noted that the existence of a generating set for Mod_{g,0} consisting of 4g + 4 torsion elements follows directly from a lemma of Birman [Bi] (see Lemma 3 below). In his paper ([Luo], §1.4), Luo poses the question of whether there is a universal upper bound, independent of g and b, for the number of torsion elements needed to generate Mod_{g,b}. Our first result is a positive answer to Luo’s question for b = 0.

Theorem 1 (Three torsion elements generate). For each g ≥ 1, the group Mod_{g,0} is generated by 3 elements of finite order.

As we will see in Section 2, at least one of the three torsion generators we give has order depending on g. Subsequent to the original posting of this paper, M. Korkmaz [Ko1] has shown that Mod_{g,b}, b = 0, 1 is generated by two elements, each of order 4g + 2.

1We remind the reader that the only involutions under consideration are orientationpreserving.

Finding a set of generators whose orders are universally bounded and whose cardinality is also universally bounded is more delicate, especially if one wants a generating set consisting of involutions. Our main theorem addresses this.

Theorem 2 (Six involutions generate). For g ≥ 3, b = 0, and for g ≥ 4, b = 1, the group Mod_{g,b} is generated by 6 involutions.

In [Ka], Kassabov builds on our method to extend and improve this result to the case b > 1. Further, in some cases (e.g. g ≥ 8) he proves that 4 involutions suffice to generate Mod_{g,b}.

In §4, we note that Theorem 2 implies that Mod_{g,b} is the quotient of a 6-generator Coxeter group.

Remarks.

1. Since Mod_{1,0} = Z/4Z ∗Z/2Z Z/6Z and H_{1}(Mod_{2,0}, Z) = Z/10Z, it is easy to see that Theorem 2 does not extend to the genus g = 1 or g = 2 cases.

2. As Mod_{g,0} surjects onto the integral symplectic group Sp(2g, Z), it follows from Theorems 1 and 2 that for all g ≥ 3, the group Sp(2g, Z) is generated by 3 torsion elements, and also by 6 involutions.

3. If one allows orientation-reversing involutions, it is possible to use Theorem 1, together with other arguments, to prove that the extended mapping class group, which includes orientation-reversing mapping classes, is generated by 5 involutions.

It would be interesting to obtain the exact bounds for Theorem 2. It is easy to see the lower bound of 3; we do not know how to improve on this bound.

It is a pleasure to thank Dieter Kotschick, Mustafa Korkmaz, and Dan Margalit for their valuable suggestions. We also thank Ian Agol for suggesting that Remark 3 above might be possible, Nathan Broaddus for pointing out the connection with Coxeter groups, and Martin Kassabov who pointed out a redundancy in our original 7-involution generating set, thus reducing the conclusion in Theorem 2 to 6 involutions.

2 Proof that 3 torsion elements generate

In this section we shall only consider the case of the closed surface, i.e., b = 0.

α_{1} β_{1}  γ_{1}  α_{2} β_{2} γ_{2}  α_{3} β_{3} γ_{3} α_{g} β_{g}  γ_{g−1}  ρ_{1}

Figure 1: A circular embedding of the surface together with Lickorish’s 3g−1 twist generators and an involution ρ_{1} which is rotation by π about the axis indicated.

We begin with a lemma of Birman [Bi], and include an adapted version of her proof for completeness. For a simple closed curve x in S_{g,b}, let T_{x} denote the (right-handed) Dehn twist about x.

Lemma 3 (Two torsion elements for a twist). Let x be a simple closed curve which is nonseparating in S_{g,0}, g ≥ 1. Then T_{x} can be written as the product of two torsion elements.

Proof. We will find it convenient to fix a “circular embedding” of the surface S_{g,b}, as seen in Figure 1. Referring to the curves of the same figure, we define

Q = T_{α_{g}}T_{β_{g}}(T_{γ_{g−1}}T_{β_{g−1}})(T_{γ_{g−2}}T_{β_{g−2}}) ⋯ (T_{γ_{1}}T_{β_{1}})T_{α_{1}}  (1)

and

S = QT_{α_{1}}^{−1} = T_{α_{g}}T_{β_{g}}(T_{γ_{g−1}}T_{β_{g−1}})(T_{γ_{g−2}}T_{β_{g−2}}) ⋯ (T_{γ_{1}}T_{β_{1}})  (2)

We say that an ordered set c_{1}, …, c_{n} of simple closed curves on S_{g,b} forms an n-chain if the geometric intersection (c_{k}, c_{k+1}) = 1 for 1 ≤ k ≤ n − 1 and (c_{k}, c_{l}) = 0 if |k−l| ≥ 2. If n is odd, the boundary of a regular neighborhood of any n-chain has two components d_{1} and d_{2}; if n is even the boundary has one component d. The so-called chain relation in Mod_{g,b} tells us that for a given n-chain c_{1}, …, c_{n}, if n is odd we have

(T_{c_{1}}T_{c_{2}} ⋯ T_{c_{n}})^{n+1} = T_{d_{1}}T_{d_{2}}

and if n is even, we have

(T_{c_{1}}T_{c_{2}} ⋯ T_{c_{n}})^{2n+2} = T_{d}

The curves defining the twists of Q and S form a (2g+1)-chain and a (2g)- chain, respectively. However, any boundary curve of a regular neighborhood of either chain is null-homotopic in the closed surface. Hence the chain relation tells us that Q and S have orders which divide 2g + 2 and 4g + 2, respectively, in Mod_{g,0}. In fact, Birman notes in her original proof that an “ugly but routine” calculation establishes that these values are precisely the orders of the two elements, with explicit calculations recorded in [BH] (see also [HK] for a shorter proof).

Given any nonseparating simple closed curve x, there is a homeomorphism h such that h(α_{1}) = x. Recall that for h ∈ Mod_{g,b} and a simple closed curve c contained in S_{g,b}, we have

hT_{c}h^{−1} = T_{h(c)}  (3)

Then we have

T_{x} = hT_{α_{1}}h^{−1}
= h(S^{−1}Q)h^{−1}
= (hS^{−1}h^{−1})(hQh^{−1})

which gives the desired result. ⋄

Now to complete the proof of Theorem 1. It is well known that Mod_{1,0} ≅ SL(2, Z) is generated by an element of order 4 and an element of order 6, so we can assume g ≥ 2. Let α_{i} be as above, let Q and S be defined as in (1) and (2), and let

U = T_{α_{1}}T_{α_{2}}^{−1}

Wajnryb [Wa1] showed that Mod_{g,0}, g ≥ 1 is generated by the two maps U and S. In the proof of Lemma 3, we saw that S has finite order in Mod_{g,0}; hence it remains to deal with U, which is the product of two Dehn twists. As above, we have

T_{α_{1}} = S^{−1}Q

From the proof of Lemma 3, we know that

T_{α_{2}} = h(S^{−1}Q)h^{−1}

where h is any map taking the curve α_{1} to α_{2}.

Let ρ_{1} denote the involution which is rotation by π about the axis indicated in Figure 1. Clearly, ρ_{1}(α_{1}) = α_{2} , so we can write

U = T_{α_{1}}T_{α_{2}}^{−1}
= [S^{−1}Q][ρ_{1}(Q^{−1}S)ρ_{1}^{−1}]

Thus we can generate Wajnryb’s two generators, and hence all of Mod_{g,0}, with three torsion elements: Q, S and ρ_{1}. ⋄

Remarks.

1. We can replace the involution ρ_{1} in the above proof with a rotation of order g.

2. Wajnryb [Wa1] also shows that the two maps U and S generate Mod_{g,1}. However the proof of Theorem 1 does not go through for b = 1 since Lemma 3 fails in this case.

We also note that, using Wajnryb’s two generators S, U, we can generate Mod_{g,b} when b = 0, 1 with a set consisting of two involutions and one element of order 4g + 2. Since U = T_{α_{1}}T_{α_{2}}^{−1}, and ρ_{1}(α_{2}) = α_{1}, we have that U = T_{α_{1}}(ρ_{1}T_{α_{1}}^{−1}ρ_{1}). Shifting the parentheses, we have that U is the product of the involutions ρ_{1} and its conjugate T_{α_{1}}ρ_{1}T_{α_{1}}^{−1}. We will see the usefulness of this “parentheses shifting” technique in the next section.

3 A universal bound on involutions

For the remainder of the paper, we assume that g ≥ 3 and b = 0, 1. For simplicity of exposition, we provide explicit arguments only for b = 0. In the case b = 1 the arguments are the same, although some involutions must be replaced with certain conjugates which move the puncture to a fixed point of the involution.

3.1 Writing Dehn twists as a product of involutions

We begin by recalling the so-called lantern relation in Mod_{g,b}. This relation was discovered by Dehn [De] in the 1930s and was rediscovered by Johnson [Jo] over forty years later. For convenience, we will use the notation X in addition to T_{x} to denote the (right-handed) Dehn twist about the simple closed curve x in S_{g,b}.

a_{1}  a_{2}  a_{3} a_{4} x_{1}  x_{3} x_{2}
Figure 2: The Lantern Relation: X_{1}X_{2}X_{3} = A_{1}A_{2}A_{3}A_{4}.

Referring to the four boundary curves a_{1}, a_{2}, a_{3} and a_{4} of the surface S_{0,4} together with the interior curves x_{1}, x_{2} and x_{3}, as shown in Figure 2, the lantern relation is the following relation amongst the corresponding Dehn twists:

X_{1}X_{2}X_{3} = A_{1}A_{2}A_{3}A_{4}  (4)

The symmetry of Figure 2 clearly shows that the twists on the lefthand side of (4) can be cyclically permuted. Moreover, each twist on the right-hand side commutes with all other twists in the relation since the corresponding curves are disjoint from all others in the lantern. Thus we can rewrite the relation in the following way, which will be convenient for our purposes:

A_{4} = (X_{1}A_{1}^{−1})(X_{2}A_{2}^{−1})(X_{3}A_{3}^{−1})  (5)

The following lemma improves on a result of Luo [Luo] and Harer [Ha]; they showed that a Dehn twist about a nonseparating curve can be written as a product of six involutions.

Lemma 4 (Four involutions for a twist). Let c be a simple closed curve which is nonseparating in S_{g,b}, g ≥ 3. Then T_{c} can be written as the product of four involutions.

Proof. We begin with the argument of Luo and Harer, which we include here for completeness. If g ≥ 3, we can find a lantern as in Figure 2 embedded in S_{g,b} such that the given curve c plays the role of a_{4} and such that the complement of the lantern is connected and its four boundary curves are distinct and nonseparating in S_{g,b}. Let us call such an embedding a

a_{1} a_{2} a_{3} a_{4}  a_{4} x_{2} x_{1} x_{3}  J_{1}

Figure 3: The pair swap involution J_{1}, taking (a_{1}, x_{1}) to (a_{2}, x_{2}). Here the boundary curve a_{4} on the left is identified with that on the right.

good lantern. We now observe that for each j, 1 ≤ j ≤ 3, the pair (x_{j}, a_{j}) consists of two disjoint, nonseparating simple closed curves such that the complement of the union x_{j} ∪ a_{j} is connected in S_{g,b} for each 1 ≤ j ≤ 3. Then for each j we can find an involution I_{j} such that I_{j}(x_{j}) = a_{j}.

By (3) we can write

X_{j}A_{j}^{−1} = X_{j}(I_{j}X_{j}^{−1}I_{j}) = (X_{j}I_{j}X_{j}^{−1})I_{j}

for each 1 ≤ j ≤ 3. Note that each X_{j}I_{j}X_{j}^{−1} is an involution, so that each X_{j}A_{j}^{−1}, 1 ≤ j ≤ 3, can be realized as a product of two conjugate involutions. Hence T_{c} = A_{4} is the product of six involutions by (5).

To improve upon this result, consider the surface shown in Figure 3. If the curves labelled a_{4} are identified in the obvious way, then the four curves a_{i} together bound a good lantern with interior curves x_{j} as shown. Given the non-separating curve c = a_{4}, we can find a good lantern with a_{4} as a boundary component. We can choose its remaining labels a_{i}, x_{j}, 1 ≤ i, j ≤ 3, in such a way that there is a homeomorphism of S_{g,b} to the surface of Figure 3 taking this good lantern to the lantern of Figure 3. It is clear now that there exists an involution J_{1} of S_{g,b} which takes the pair (a_{1}, x_{1}) to the pair (a_{2}, x_{2}). We note that the surface of Figure 3 could be embedded in R^{3} in such a way that the involution J_{1} of the surface is a restriction of an isometry of R^{3}; one can simply add a tube connecting the two ends which encloses the surface of Figure 3. (We shall abuse notation throughout by using J_{1} to refer to both the involution of S_{g,b} as well as its conjugate which is the involution of the surface of Figure 3.)

In fact, given a particular choice of labels for a good lantern, a similar process yields 6 distinct “pair swaps” of order 2, i.e., involutions taking one pair (a_{i}, x_{j}) to another pair (a_{k}, x_{l}), assuming i ≠ k, j ≠ l. In particular, there is a conjugate involution J_{2} taking the pair (a_{1}, x_{1}) to the pair (a_{3}, x_{3}). This is clear if we act on the surface of Figure 3.1 by the homeomorphism

α β  γ R_{g}

Figure 4: A generating set for Mod_{g,b}.

which interchanges the “tube” containing the curve a_{2} with the tube containing a_{3}, fixing the curve x_{1} while simultaneously moving the curve x_{3} to the current position of x_{2}.

We can now rewrite (5) as

T_{c} = A_{4} = [(X_{1}I_{1}X_{1}^{−1})I_{1}][J_{1}(X_{1}I_{1}X_{1}^{−1})I_{1}J_{1}][J_{2}(X_{1}I_{1}X_{1}^{−1})I_{1}J_{2}] (6)

and hence T_{c} is a product of four involutions. ⋄

3.2 Proof of the main result: 6 involutions generate

Lickorish [Li] proved that Dehn twists about the 3g − 1 curves given in Figure 1 suffice to generate Mod_{g,b}. In the same paper, he also described a generating set for Mod_{g,b} consisting of four elements, one of which is not a Dehn twist. We shall now give a rigorous description of such a generating set. Let R_{g} ∈ Mod_{g,b} denote clockwise rotation of S_{g,b} by \frac{2π}{g} about the axis which is perpendicular to the plane of the page and intersects the surface twice in the center of our “circle of handles”. Applying Equation (3) to R_{g} conjugating the twists T_{α}, T_{β} and T_{γ} (referring to the curves of Figure 4), we see that the group generated by R_{g} together with these three twists contains

{R_{g}^{m}T_{α}R_{g}^{−m}, R_{g}^{m}T_{β}R_{g}^{−m}, R_{g}^{m}T_{γ}R_{g}^{−m} : 0 ≤ m ≤ g − 1}

Hence the four elements T_{α}, T_{β}, T_{γ}, R_{g} generate all of Lickorish’s twist generators and thus they generate all of Mod_{g,b}. Clearly, one could replace any of the three twists generators T_{c}, c ∈ {α, β, γ}, with R_{g}^{k}T_{c}R_{g}^{−k} = T_{R_{g}^{k}(c)} for any k ∈ Z and the resulting set still generates Mod_{g,b}.

L_{2} L_{1}  ρ_{1} ρ_{2}

Figure 5: Two involutions generating R_{g}.

Consider the rotation R_{g}. For i = 1, 2, let ρ_{i} denote rotation in R^{3} by π about the axis labelled L_{i}, as shown in Figure 5. Note that if the genus g is even, then L_{1} intersects the surface in six points and L_{2} in two points. If the genus g is odd, then L_{1} and L_{2} each intersect the surface in four points. In both cases, we observe that

R_{g} = ρ_{1}ρ_{2}  (7)

Combining (7) with Lemma 4, together with the fact proved above that Mod_{g,b} is generated by R_{g} and the three twists T_{α}, T_{β}, T_{γ}, we have that Mod_{g,b} can be generated by 2 + 3 · 4 = 14 involutions.

We now show how to reduce this number. Since, as previously noted, we can replace T_{β} with T_{R_{g}^{2}(β)} in our generating set, we shall abuse notation by using β to refer to the curve R_{g}^{2}(β) for the remainder of the proof. As shown in Figure 6, we can embed a good lantern in our surface which contains α, β, and γ as three of its four boundary components. In order to motivate our choices of notation, we first observe that, with this particular choice of a good lantern, we can choose x_{1} to be the curve labelled as such in Figure 6. If we then let α play the role of a_{1}, we see that ρ_{1}(a_{1}) = x_{1}, and thus ρ_{1} plays the role of I_{1} in the proof of Lemma 4.

Now, in order to make further use of our work in proving Lemma 4, we assign to the curve γ the role of a_{4}, so that a_{1} and a_{4} are separated by x_{1} from a_{2} and a_{3}. Of the two remaining curves, we now assign to β the role of a_{3}, and the remaining curve we label a_{2}. We also choose interior curves

α = a_{1} β = a_{3} γ = a_{4}  x_{1} ρ_{1} a_{2}

Figure 6: A good lantern embedded with α, β and γ as three boundary curves and x_{1} an interior curve.

x_{2} and x_{3} so the labels of the good lantern of Figure 6 match those of the pair swaps defined in the previous section. Thus we have

T_{γ} = A_{4} = (X_{1}T_{α}^{−1})(X_{2}A_{2}^{−1})(X_{3}T_{β}^{−1})  (8)
= (X_{1}A_{1}^{−1})(X_{2}A_{2}^{−1})(X_{3}A_{3}^{−1})  (9)
= [(X_{1}I_{1}X_{1}^{−1})I_{1}][J_{1}(X_{1}I_{1}X_{1}^{−1})I_{1}J_{1}][J_{2}(X_{1}I_{1}X_{1}^{−1})I_{1}J_{2}]
= [(X_{1}ρ_{1}X_{1}^{−1})ρ_{1}][J_{1}(X_{1}ρ_{1}X_{1}^{−1})ρ_{1}J_{1}][J_{2}(X_{1}ρ_{1}X_{1}^{−1})ρ_{1}J_{2}]

Given the two involutions ρ_{1}, ρ_{2} used to generate R_{g}, we have used only three new involutions to write T_{γ}, namely X_{1}ρ_{1}X_{1}^{−1}, J_{1}, and J_{2}. Now (8) and (9) can be rewritten, respectively, as

T_{β} = A_{3} = (X_{1}T_{α}^{−1})(X_{2}A_{2}^{−1})(X_{3}T_{γ}^{−1})
= (X_{1}A_{1}^{−1})(X_{2}A_{2}^{−1})(X_{3}A_{4}^{−1})  (10)

We now observe that we can swap (a_{i}, x_{j}) with (a_{k}, x_{j}) for any i, j, k by using a slight variation on the embedding of Figure 3. The involution J_{3} shown in Figure 7 interchanges (a_{3}, x_{3}) with (a_{4}, x_{3}), as well as (a_{1}, x_{3}) with (a_{2}, x_{3}), (a_{1}, x_{1}) with (a_{2}, x_{1}), and finally (a_{3}, x_{1}) with (a_{4}, x_{1}). Note that in the first two cases, the a_{i}-curves that are interchanged are not separated in the lantern by the fixed curve x_{3}, but in the last two cases, the a_{i}-curves being swapped are separated by the fixed curve x_{1}. Thus any possible configuration of such pair swaps can be achieved, although a different embedding may be required in order to align the x_{j}-curve to be fixed in the swap either

x_{1}  x_{3}  a_{1} a_{4} a_{3} a^{2}  J_{3}

Figure 7: A pair swap J_{3} which takes (a_{3}, x_{3}) to (a_{4}, x_{3}); both x_{1} and x_{3} are interior lantern curves fixed by J_{3}.

“horizontally” or “vertically”, depending, respectively, on whether the two a_{i}-curves one wishes to swap are separated by the x_{j}-curve, or lie on the same side.

We now have that the product J_{3}J_{2} takes the pair (x_{1}, a_{1}) to the pair (x_{3}, a_{4}) and we can rewrite (10) as

T_{β} = A_{3} = (X_{1}A_{1}^{−1})[J_{1}(X_{1}A_{1})^{−1})J_{1}][(J_{3}J_{2})(X_{1}A_{1}^{−1})(J_{2}J_{3})]

Since (X_{1}A_{1}^{−1}) = (X_{1}ρ_{1}X_{1}^{−1})ρ_{1} as above, we see that we need just one new involution, J_{3}, in order to generate T_{β}. Similarly, we can find a pair-swap involution J_{4} taking the pair (x_{1}, a_{1}) to (x_{1}, a_{4}), so that we can write

T_{α} = A_{1} = (X_{1}T_{β}^{−1})(X_{2}A_{2}^{−1})(X_{3}T_{γ}^{−1})
= (X_{1}A_{4}^{−1})(X_{2}A_{2}^{−1})(X_{3}A_{3}^{−1})
= [J_{4}(X_{1}A_{1}^{−1})J_{4}][J_{1}(X_{1}A_{1}^{−1})J_{1}][J_{2}(X_{1}A_{1}^{−1})J_{2}]

with J_{4} being the only new involution required to write T_{γ}.

Thus our count for the number of involutions used to generate each of the four elements R_{g}, T_{γ}, T_{β}, T_{α} stands, respectively, at 2 + 3 + 1 + 1 = 7 involutions. Finally, as pointed out to us by M. Kassabov, the involution J_{4} is in fact redundant, and can be taken to be the product J_{2}J_{3}J_{2}. Thus we can generate with 6 involutions, as claimed. ⋄

4 Final remarks

Recall that an Artin group A is generated by elements x_{1}, …, x_{n}, subject to the following relations. Let m_{s,t} = {2, 3, 4, …, ∞}. Then for each s ≠ t, the elements x_{s} and x_{t} satisfy the relations:

(x_{s}x_{t})^{m_{s,t}/2} = (x_{t}x_{s})^{m_{s,t}/2}  if m_{s,t} is even
(x_{s}x_{t})^{(m_{s,t}−1)/2} = (x_{t}x_{s})^{(m_{s,t}−1)/2} if m_{s,t} is odd

Here m_{s,t} = ∞ means that there is no relation between the generators x_{s} and x_{t}. The Coxeter group \overline{A} associated to an Artin group A is the quotient of A by the extra relation x_{i}^{2} = 1 for all i = 1, … n. We thank Nathan Broaddus for pointing out the following corollary of our main theorem.

Corollary 5 (Mod_{g,b} is a Coxeter quotient). For g ≥ 3, b = 0, and for g ≥ 4, b = 1, the group Mod_{g,b} can be realized as a quotient of a Coxeter group on 6 generators.

There has been some effort to understand various mappings of Artin groups both into and onto mapping class groups. For example, Wajnryb studied classes of Artin groups which do not inject into mapping class groups in [Wa2], while Matsumo [Mat] as well as Labruere and Paris [LP] have given explicit presentations of mapping class groups as quotients of certain Artin groups. However, we are not aware of similar investigations of the relationship between Coxeter groups and mapping class groups.

The results of this paper beg several further questions. For example, besides the previously raised question of whether we can do better than 6 involutions, we can also ask whether there exists a constant C, perhaps such that C = C(g), so that every element of Mod_{g,b} can be written as a product of at most C torsion elements ^{2} . Furthermore, what kinds of relations exist amongst the torsion (or involution) generators? In particular, what kind of Coxeter groups arise in the context of Corollary 5?

References

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[BH] J. Birman and H. Hilden, Mapping class groups of closed surfaces and covering spaces, Ann. of Math. Stud. No. 66 (1971), p. 81-115.

2After initial circulation of this paper, M. Korkmaz [Ko2] and D. Kotschick [Kot] answered the latter question in the negative by proving that no such C exists. Both arguments build on a theorem of Endo-Kotschick [EK].

[De] M. Dehn (tr. J. Stillwell), The group of mapping classes, Papers on Group Theory and Topology, Springer-Verlag, 1987 (Die Gruppe der Abbildungsklassen, Acta Math., Vol. 69 (1938), p. 135-206).

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[EK] H. Endo and D. Kotschick, Bounded cohomology and non-uniform perfection of mapping class groups, Invent. Math. 144 (2001), no. 1, 169–175.

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Tara E. Brendle: Dept. of Mathematics, Cornell University 310 Malott Hall Ithaca, NY 14853 E-mail: brendle@math.cornell.edu

Benson Farb: Dept. of Mathematics, University of Chicago 5734 University Ave. Chicago, Il 60637 E-mail: farb@math.uchicago.edu