## via the ordered complex of curves

Silvia Benvenuti

(Communicated by G. Gentili)

Abstract. We describe an algorithm to compute ®nite presentations for the mapping class group of a connected, compact, orientable surface, possibly with boundary and punctures. By an inductive process, such an algorithm, starting from a presentation well known for the mapping class group of the sphere and the torus with ``few'' boundary components and/or punctures, produces a presentation for the mapping class group of any other surface.

Key words.Mapping class group, complex of curves, inductive process.

2000 Mathematics Subject Classi®cation. Primary 20F05; Secondary 20F36

1 Introduction

Let F F_{g;r}^{s} be a connected, compact, oriented surface of genus g withrboundary
components and with a set P fp_{1};. . .;p_{s}g of sdistinguished points, called punc-
tures
g;r;sd0. We denote byH
F;Pthe group of orientation preserving homeo-
morphismsh:F !F which pointwise ®x the boundary ofFand satisfyh
P P.

The mapping class group of F is the group of the isotopy classes of elements
of H
F;P: we denote it byM
F, or by M_{g;r}^{s} , as it clearly depends only (up to
isomorphism) on the genusg, the numberrof boundary components and the number
sof punctures.

In this paper we will be concerned with the problem of ®nding ®nite presentations
forM_{g;r}^{s} .

By a result of Brown (see [4]), we can write down explicitly a ®nite presentation of a groupGacting on a simply connected simplicial complexX, provided we know:

± the structure of the 2-skeleton of the quotient X=G, which must have a ®nite number of 2-cells;

± a ®nite presentation for the isotropy subgroup of a representative of every vertex inX=G;

± a ®nite set of generators for the isotropy subgroup of a representative of every edge inX=G.

Hence, the problem of ®nding a ®nite presentation for the mapping class group of

Harer [11] found a smaller complex with the same properties as the Hatcher±Thurstong;1

complex, and obtained a ®nite (but very unwieldy) explicit presentation forM_{g;}^{0}_{1}; this
presentation was then simpli®ed by Wajnryb [31] in 1983. In 1998, starting from
Wajnryb's result, Gervais [7] and Matsumoto [25] derived independently two simple
presentations ofM_{g;1}^{0} (it must be noticed that Gervais' result concerns more generally
any M_{g;}^{0}_{r}). We recall also a recent paper by Wajnryb [32], where the author gives a
completely self-contained proof of another simple presentation for the groups M_{g;1}^{0}
andM_{g;}^{0}_{0}, still using the cut system complex; an interesting point of this paper is that
the simple connectivity of the complex is shown by elementary methods.

Here, we apply Brown's method to a di¨erent complex, the ordered complex of
curves, a suitable modi®cation of the complex introduced by Harvey in [13], and
deeply studied by several authors (see for example [12, 18, 23, 24, 27, 28]). We get
this way, as a byproduct, a simpler and more direct proof of the presentations given
in [14, 31, 7]. Moreover, our method works for the general case ofM_{g;r}^{s} .

During the preparation of this paper, we were informed of a paper by Hirose [15],
where the author recovers Gervais' presentation exploiting the action of the mapping
class group on a di¨erent ``complex of curves'', involving only non-separating curves
and simplices. We remark here that the main advantage in using our complex of
curves, instead of the Hatcher±Thurston's one or the one used by Hirose, is the fact
that our complex, though having several M_{g;r}^{s} -equivalence classes of vertices (while
the others have a single such class), has only triangular 2-cells, which makes it par-
ticularly powerful in reducing the presentation coming from Brown's method to a
very simple and meaningful one.

The paper is organized as follows: in Section 2 we state a special version of
Brown's result which applies to our case. In Section 3 we de®ne theordered complex
of curves X_{g;r;s}^{ord}, and show that, with the exception of a ®nite number of cases, called
sporadic cases, such a complex is simply connected and admits a natural action of
M_{g;r}^{s} satisfying all the hypotheses needed to apply Brown's Theorem. To be precise,
the sporadic cases are the surfaces F_{g;}^{s}_{r} with g0, rs0;1;2;3;4;5 and g1,
rs0;1;2, and their presentations are well known (see Section 5). In Section 4
we analyze the 2-skeleton ofX_{g;r;s}^{ord} and explain how to produce a ®nite presentation
for the isotropy subgroups of its vertices and edges, provided one knows a presen-
tation for eachM_{g}^{s}0^{0};r^{0} such that
g^{0};r^{0}s^{0}<
g;rs(with the lexicographic order).

Hence, the method of Section 2 recursively produces a presentation of any non-
sporadicM_{g;r}^{s} , provided we start with a presentation for each sporadic case. In Section
5 we explain such an inductive process and, for the sake of completeness, we recall
a presentation for the non-punctured sporadic cases, that is the basis of the induction
in the situations treated in the last two sections. Section 6 is devoted to the detailed

class vAV and of a representative s
efor every class eAE,such thatM
F_{g;r}is the
free product of the isotropy subgroups of all the s
v's,amalgamated along the isotropy
subgroups of the s
e's.

Since it is possible to ®nd a presentation for the isotropy subgroup of a curve
s
v starting from the knowledge of a presentation for the mapping class group of
the surface obtained fromF_{g;}^{0}_{r}cutting it open alongs
v, if we apply recursively this
theorem to all the non-sporadic subsurfacesF_{g}^{0}0;r^{0}obtained fromF_{g;r}^{0} cutting them open
along generic simple closed curves, we may conclude that all the relations needed
to present the mapping class group are supported in subsurfaces homeomorphic to
sporadic surfaces.

Finally, in Section 7 we show that we recover Gervais' presentation [7] for the
mapping class group of any non-sporadic surface, provided we start the inductive
process with the Gervais presentation for the sporadic subsurfaces. Analogously, once
we have the presentations for the sporadic surfaces according to some ``style'' (e.g. in
terms of Dehn twists [7], or as quotients of Artin groups [25, 20]), our method pro-
duces a presentation of the same ``style'' for every F_{g;}^{s}_{r}. We may then say that the
complex of curves allows to recover by a unique algorithm at least all the simpler
known presentations. Actually, our analysis also suggests the existence of other simple
presentations, where the generators are, besides a number of Dehn twists, a family of
elements having an intrinsic geometric meaning. We will describe these aspects in a
forthcoming paper.

For the reader's convenience, we recall in the Appendix the de®nition of Dehn twist,braid twistandsemitwist.

2 Description of the general method

In [4] Brown describes a general method to get a presentation for a groupGlooking at its action on a simply connected CW-complexX(see also [10] and [3], where similar results are discussed in the setting ofsmall categories without loopsandcomplexes of groups). We describe in this section a particular case of Brown's theorem, under some additional hypotheses both on the complex and on the action. We refer the reader to [4], [10], or [3] for the details.

LetXbe a CW-complex with oriented edges. We denote byE Xthe set of edges ofX, and byV Xthe set of vertices. We de®ne two maps

i;t:E X !V X;

De®nition 2. The action of G on X is called good and orientation compatible (or shortlyXis agood G-CW-complex) if

(i) the action ofGpermutes cells of the same dimension;

(ii) ifgAGleaves a cell invariant, then its restriction to that cell is the identity;

(iii) g i e i g eandg t e t g efor eacheAE Xand for eachgAG.

Given a goodG-CW-complexX, we denote byX the quotient spaceX=G, and by p:X!X the projection.

We suppose that the closure of each 2-cell of Xis an embedded triangle, that is a subcomplex with the induced canonical cell decomposition. Moreover, we suppose that each triangle is determined by its oriented edges, and that the orientation of the edges is such that no triangle has an oriented loop as a boundary; in these hypotheses, we label by f;m;pthe vertices of a 2-cellT, where f <m<p, and we label bya;b;c the edges ofT, whereahf;mi,bhm;pi,chf;pi.

We ®rst consider the following choices:

(1) for everyvAV
X, we choose a representatives
vAV
X(that isp
s
v v),
and we denote byG_{v}the isotropy subgroup ofs
v,

GvStab s v;

(2) for everyeAE
X, we choose a representatives
eAE
X(that isp
s
e e),
we denote by G_{e} the isotropy subgroup of s
e, and we choose two elements
g_{e;}_{i};g_{e;}_{t}AGsuch that

g_{e;}_{i}
i
s
e s
i
e; g_{e;t}
t
s
e s
t
e

(see Figure 2.1);

(3) for everyTAX2(the 2-skeleton ofX), we choose a representatives
TAX2(that
isp
s
T T). Moreover, we choose, for everyT AX_{2}, the elements g_{T;}_{a},g_{T;b},
g_{T;}_{c}ofGsuch that

g_{T;}_{e}
~e s
e; eAfa;b;cg

where the tilde denotes the lifting in s Tof a vertex (or edge) inT (see Figure 2.1);

(4) we choose a maximal treeTin the 1-skeletonX1. Then, we have the following result.

Theorem 3.Let us suppose that:

(i) X is simply connected;

(ii) the isotropy subgroup of each vertex v is ®nitely presented,G_{v}hS_{v}jR_{v}i;

(iii) the isotropy subgroup of each edge e is ®nitely generated,Gen GeSe; (iv) the quotient XX=G has a ®nite2-skeleton.

Then G is ®nitely presented,and a presentation of G is given by

G

* 6

vAV X

SvU 6

eAE X

E 6

vAV X

RvUR^{
2}UR^{
3}UR^{
4}

+

;

where E is a symbol,associated to the edge e,for each eAE
X,and
R^{
2} fE1jeATg;

R^{
3} fEg_{e;}_{t}gg_{e;t}^{ÿ1}E^{ÿ1}g_{e;i}gg^{ÿ1}_{e;i} jgASe;eAE
Xg;

R^{
4} fg_{c;i}g_{T;c}g^{ÿ1}_{T;a}g_{a;i}^{ÿ1}Ag_{a;}_{t}g_{T;a}g^{ÿ1}_{T;b}g^{ÿ1}_{b;i}Bg_{b;t}g_{T;b}g^{ÿ1}_{T;}_{c}g_{c;t}^{ÿ1}CjT AX2g.

Here and in the following we use the capital letter to indicate the generator associated to the edge denoted with the corresponding lowercase letter(for instance,A is the symbol corresponding to the edge a).

Remark 4. As we said at the beginning of the section, this is a particular case of
a general result by Brown, which holds without the assumptions (ii) and (iii) on
the action ofG, and without restrictive hypotheses on the shape and the boundary
Figure 2.1. A 2-cell ofX (left-hand picture), its representative inXand the meaning of the
elementsg_{e;i};g_{e;}_{t}andg_{T;e}(right-hand picture).

3 The ordered complex of curves

Let FF_{g;}^{s}_{r}be a connected, compact, oriented surface of genus g, withrboundary
components and spunctures,r;sd0; we denote by Pthe set of punctures of F. A
simple closed curvein FnPis an embedding g:S^{1}!FnPwhich does not intersect
the boundary ofF, and two simple closed curvesa;bare said to beisotopic
aFbif
there exists a continuous family h_{t}AH
F;P, tA0;1 such that h_{0} is the identity
andh_{1}ab. A curve is calledgenericif its image does not bound a disk or a disk
with one puncture.

De®nition 5. The complex of curves on F is the simplicial complex X X_{g;}^{s}_{r} of
dimension 3gÿ4rs whose k-simplices are the isotopy classes of families a
fa0;. . .;akg of k1 generic simple closed curves in FnP satisfying the following
conditions:

(i) a_{i}Va_{j}q ifi0j; (disjoint)

(ii) a_{i}6Fa_{j}anda_{i}6Fa^{ÿ1}_{j} ifi0j; (pairwise not isotopic)
(iii) a_{i}6Fany boundary component ofF for alli. (not isotopic to boundary
components)
We call such a family ageneric
k1-family of closed curves.

Notice that the curves are not oriented, and that the families we consider are not
ordered, i.e. the two families a fa0;. . .;akg and b fb_{0};. . .;b_{k}g are equivalent
(i.e. represent the same k-simplex inX) if there exists a permutation sASk1 such
that a_{i}Fb_{s
i}^{G1} for every iAf0;. . .;kg. We denote byX_{k} the k-skeleton ofXand by

a a_{0};. . .;a_{k}the simplex represented by the familya fa_{0};. . .;a_{k}g.

Theorem 6.If gd1,rsd1,then X_{g;}^{s}_{r}is
2grsÿ4-connected.Moreover,X_{g;0}^{0}
is
2gÿ3-connected and X_{0;r}^{s} is
rsÿ5-connected.

For the proof of this theorem we refer the reader to Harer [12], where the result is proven in the setting of Thurston train tracks theory, or to Ivanov [18], where the same result is proven using Cerf theory. We recall that the 1-connectedness ofXfor gd2 was ®rst proved by Ivanov in [17], where it is derived from the 1-connectedness of the complex of Hatcher±Thurston; instead, the proofs in [18] and [21] are inde- pendent from that.

In particular, except for the cases g0, rs0;1;2;3;4;5 and g1,rs
0;1;2 that we callsporadic, the complex of curvesX_{g;r}^{s} is simply connected (see also
the sketch of a simple proof in [19]).

In order to have an orientation for the edges of X, we ®x a total ordering for
its vertices and we orient each edge accordingly. When we consider oriented edges,
or more generally oriented k-simplices, we use the notation ha_{0};. . .;a_{k}i with the
meaninga_{0}<a_{1}< <a_{k}.

IfaFbandh_{1};h_{2}AH
F;Pare isotopic, then clearlyh_{1}
aFh_{2}
b; therefore the
mapping class groupM
Facts on the set of isotopy classes of simple closed curves,
i.e. onX_{0}, and this action naturally extends to thek-skeleton ofX. Unfortunately this
action is neither good nor orientation compatible, as one can see immediately from
the following example: we considerF_{2;0}^{0} , witha andb as in Figure 3.1. The rotation
of 180 degrees around thez-axis globally ®xes the 1-simplexa;b, but interchanges

aandb, and therefore its restriction toa;bis not the identity.

To overcome this problem, we consider another complex, the ordered complex of curves.

De®nition 7.Theordered complex of curves onF is the simplicial complex, that we
denote byX^{ord}, whosek-simplices are the isotopy classes oforderedfamilies ofk1
generic simple closed curves satisfying the conditions (i), (ii), (iii) of De®nition 5.

The complex X with its orientation can be clearly seen as a subcomplex of X^{ord}
(see Figure 3.2). The action of M
F;P on X extends to a good and orientation
compatible action onX^{ord}: namely, ifa;bis a non-oriented edge ofXwhose image
undergAM
F;Pisg
a;g
b, we set

g ha;bi hg a;g bi:

This can be done since bothhg
a;g
biandhg
b;g
aibelong toX^{ord}. Hence, we
have a goodG-simplicial complexX^{ord}(from now onGwill denote the mapping class
group, unless otherwise stated), and it remains to show that it is simply connected.

Figure 3.1. An element ofM_{2;0}^{0} that leaves a 1-cell invariant but interchanges its endpoints.

Proposition 8.If X is simply connected,then X^{ord} is also simply connected.

Proof.It is su½cient to show that all the loops of type

inX_{1}^{ord} are contractible inX^{ord}: if this is true, each time that a loop inX_{1}^{ord}contains
an edge ofX_{1}^{ord}nX_{1}, we can substitute it with the corresponding edge ofX_{1}, thus we
are done by the simple connectivity ofX. Supposinga<bin the chosen ordering of
the vertices ofXand supposing there is agAX0 such thata;b andgare the vertices
of a triangle in X2, the situation is one of the three described in Figure 3.3. In any
case, inX_{2}^{ord} we have two triangles that allow us to contract the loopabato a point
(such triangles are respectivelyhg;a;biandhg;b;ai,ha;b;giandhb;a;gi,hg;a;bi
andhg;b;ai). Finally, we conclude noticing that in the non-sporadic cases such a g

Figure 3.2. The ordered complex of curves.

always exists, because given two simple closed generic curves, disjoint and not isoto- pic, it is always possible to complete them to a pants decomposition (see Subsection 4.1), hence obtaining a third simple closed generic curve which is disjoint and not isotopic to the other two.

4 The action ofM_{g;}^{s}_{r} onX^{ord}

In the previous section we showed that in the non-sporadic cases the ordered complex
of curves X^{ord}X_{g;r;s}^{ord} of a surfaceF F_{g;r}^{s} satis®es the hypotheses of Theorem 3;

now we give the main tools to algorithmically carry on the method. More precisely we describe

(i) how to constructX_{2}^{ord}, the 2-skeleton of the quotientX^{ord}=M_{g;r}^{s} ;

(ii) how to ®nd a ®nite presentation for the isotropy subgroup of a vertexvAV
X^{ord}
and how to ®nd a ®nite set of generators for the isotropy subgroup of an edge
eAE
X^{ord}.

4.1 The structure ofX_{2}^{ord}.Let us come back to the non-ordered complex of curvesX:

we want to ®nd a method to determine whether two classes a;bAX are in the same
M_{g;r}^{s} -orbit, and to ®nd a representative for eachM_{g;}^{s}_{r}-orbit. Leta fa_{1};. . .;a_{k}gbe a
generick-family of closed curves on a punctured surfaceFF_{g;r}^{s} . We denote byFa

the natural compacti®cation ofFn
6_{i1}^{k} ai, and byr_{a}:Fa!F the continuous map
induced by the inclusion ofFn
6_{i1}^{k} a_{i}inF. LetNbe a connected component ofF_{a},
andg:S^{1}!qNa boundary curve ofN. We say thatgisan exterior boundary curve
ofNifr_{a}gis a boundary component ofF. For each curvea_{i}:S^{1}!Fin the family
athere are two distinct boundary curvesg;g^{0}:S^{1}!qF_{a}such thatr_{a}gr_{a}g^{0}a_{i},
and two situations are possible: either g and g^{0} are boundary curves of the same
connected component NofFa (in that case we say that ai is a non-separating limit
curveofN), orgis a boundary component ofNandg^{0} is a boundary component of
a di¨erent connected componentN^{0}(in that case we say thatai is aseparating limit
curveofNandN^{0}) (see Figure 4.1).

We are now able to state the following proposition, whose proof is trivial:

Figure 3.3. Contracting a loopaba.

Proposition 9. If a a1;. . .;akand b b_{1};. . .;b_{h}are two simplices of X,then

aisM_{g;r}^{s} -equivalent tobif and only if
(1) kh;

(2) there exists a one to one correspondence between the components of Fa and those
of F_{b};

(3) there exists a permutationsASk such that,for every pair
N;N^{0}where N is any
component of Fa and N^{0}the corresponding component of Fb,we have:

### .

_{g N }

_{g N}

^{0}

_{;}

_{s N }

_{s N}

^{0}

_{;}

_{r N }

_{r N}

^{0}

_{,}where we denote by g Nthe genus, by s Nthe number of punctures and by r Nthe number of boundary components of N;

### .

_{if}

_{g}is an exterior boundary curve of N there exist an exterior boundary curveg

^{0}of N

^{0}such thatr

_{a}gr

_{b}g

^{0};

### .

_{if}

_{a}

_{i}is a separating limit curve of N,thenb

_{s i}is a separating limit curve of N

^{0};

### .

_{if}

_{a}

_{i}is a non-separating limit curve of N,thenb

_{s i}is a non-separating limit curve of N

^{0}.

Now, in order to ®nd a family of representatives forM_{g;r}^{s} -orbits inX, we introduce
the notion ofpants decomposition.

Apair of pants of type I is a (surface homeomorphic to a) disk with 2 punctures
(i.e.F_{0;1}^{2} ), apair of pants of type IIis an annulus with one puncture (i.e.F_{0;}^{1}_{2}), and a
pair of pants of type IIIis a sphere with 3 holes (i.e.F_{0;}^{0}_{3}).

We say that the familyadetermines apants decompositionofFif each component
NofFa, with set of puncturesNVr^{ÿ1}_{a} fpunctures ofFg, is a pair of pants (see Figure
4.3). Then, it is easy to check thatF_{g;}^{s}_{r}admits a pants decomposition, provided that

g;r;sBf 0;0;0; 0;0;1; 0;0;2; 0;0;3; 0;1;0; 0;1;1; 0;2;0; 1;0;0g:

In particular every non-sporadic surface admits a pants decomposition.

Figure 4.1. Exterior boundary curves g, separating limit curves a2 and non-separating limit curves a1.

A generic k-family determines a pants decomposition of F if and only if k 3grsÿ3, i.e. if and only if such a family represents a simplex of maximal dimension in the complex of curves; moreover, the number of pants in any decom- position is 2grsÿ2.

Given a generic k-family a fa_{1};. . .;a_{k}g, it can be proven that we can always
complete it to a pants decomposition of F, i.e. there exist generic closed curves
fak1;. . .;a3grsÿ3gsuch thatfa1;. . .;a3grsÿ3gdetermines a pants decomposition
of F. Hence, to ®nd the representatives of the M_{g;r}^{s} -orbits ofX we need to look at
the subfamilies of the pants decompositions, more precisely:

(1) we take the disjoint unionF~ofn_{i}pants of typei,n_{I}n_{II}n_{III} 2grsÿ2,
2n_{I}n_{II} s, we choose 3grsÿ3 boundary curves of F~ and we glue the
connected components ofF~identifying the curves of each pair: the identi®cation
space we obtain is homeomorphic to F, the set of 3grsÿ3 curves which
are the projection of the chosen pairs is a pants decomposition of this space, and
everyM_{g;}^{s}_{r}-orbit of pants decompositions has a representative which is obtained
this way.

Therefore, listing all the possible ways of choosing the 3grsÿ3 pairs and
eliminating the choices giving rise toM_{g;r}^{s} -equivalent pants decompositions, we
get a representative for theM_{g;}^{s}_{r}-orbit of each maximal simplex;

Figure 4.2. Pair of pants.

Figure 4.3. A pants decomposition.

(2) Then, to get the representatives for the M_{g;r}^{s} -orbits of the k-simplices of X, we
just consider all the subfamilies of k1 elements of every representative of a
M_{g;}^{s}_{r}-orbit of pants decompositions, and we eliminate the choices giving rise to
M_{g;}^{s}_{r}-equivalent families.

This shows that the number of pants decompositions (up to M_{g;}^{s}_{r}-equivalence) is

®nite; in particular, the 2-skeleton ofX is also ®nite.

If we considerX^{ord} instead ofX, the only di¨erence introduced by the ordering is
that we do not allow permutations as in Proposition 9 above: for example, in Figure
4.4 the two families fa;bgandfb;gg, though M_{g;}^{s}_{r}-equivalent as edges ofX, are not
M_{g;r}^{s} -equivalent as edges ofX^{ord}. Hence, the ®niteness of the 2-skeleton of the quo-
tient, required in Theorem 3, is preserved.

4.2 The presentation of the stabilizers.Consider an exact sequence of groups
0!K!^{i} G!^{p} H !0

and assume thatK(that we identify withi
KinG) andHhave the presentations
KhG_{K}jR_{K}i and H hG_{H}jR_{H}i:

For eachyAG_{H}we choose an elementy~AGsuch thatp
~y y, and for each relation
ry_{1}. . .y_{m}AR_{H} we set~r~y_{1}. . . ~y_{m}AG; as the sequence is exact, for everyrAR_{H}
there exists a wordw_{r}in the elements ofG_{K}such that~rw_{r}inG. Moreover, for each
xAG_{K}and for each yAG_{H}there exists a wordg_{x;}_{y}AGsuch thatyx~~ y^{ÿ1}g_{x;}_{y}inG.

We omit the simple proof of the following Lemma:

Lemma 10.With the notation as above,G admits the presentation
G hG_{K}U 6

yAGH

~

yjR^{
1}UR^{
2}UR^{
3}i
where

R^{
1}R_{K};

R^{
2} f~rw^{ÿ1}_{r} jrAR_{H}g;

R^{
3} f~yx~y^{ÿ1}g^{ÿ1}_{x;y}jxAG_{K};yAG_{H}g.

Figure 4.4. G-equivalence inXvsG-equivalence inX^{ord}.

a is an
nÿ1-simplexha_{1};. . .;a_{n}iofX^{ord};

Stab^{}
a is the subgroup of Stab
acontaining the elements leaving invariant each
element ofawith its orientation;

H_{a} is the image of p_{1} in the group L
n, which is the group of the linear
transformations f AGL
R^{n} such that f
e_{i} Ge_{i} for each e_{i}, where
fe1;. . .;engis the canonical basis ofR^{n};

p_{1} is the natural homomorphism from Stab
atoL
nde®ned as follows: let
gbe an element of Stab
a, and lethAH
F;Prepresentingg; we set

p_{1}
ge_{i} ei if h
ai ai,
ÿei if h
ai a^{ÿ1}_{i} ;

M Fa is the mapping class group of the surfaceFa.

Let us show the exactness of the two sequences: as far as (4.1) is concerned, we just remark that by an easy analysis case by case it is possible to describeHa exactly (it is su½cient to consider the orientation preserving homeomorphisms of F, ®xing the support of each component ofa, and possibly changing the orientation of some component).

Regarding (4.2), ®rst of all we remark that, if N_{1};. . .;N_{r} are the connected
components ofF_{a}, we have

M Fa M N1 M Nr:

Then we recall the map r_{a}:Fa!F, which induces a homomorphism of groups
r_{a}_{}:M
F_{a} !M
F, whose image is exactly Stab^{}
a(in other words,p_{2}is
r_{a}_{}).

Since it is possible to prove (see [29]) that, ifg_{i}andg_{i}^{0}are the boundary curves ofF_{a}
such thatr_{a}g_{i}r_{a}g_{i}^{0}a_{i}, then kerp_{2}is generated byfC_{1}
C_{1}^{0}^{ÿ1};. . .;C_{n}
C_{n}^{0}^{ÿ1}g
and it is a free Abelian group of rankn, sequence (4.2) is exact.

Using (4.2), we ®nd a presentation for Stab^{}
a; then, applying Lemma 10 to (4.1),
we get a presentation for the isotropy subgroup ofa.

Remark 11. Actually we are interested only in the presentations for the isotropy subgroups of the vertices and 1-simplices. Moreover, we will see in Section 6 that in the non-punctured case, the presentations for the vertices are enough.

As an example, and since in Section 7 we will use explicitly the presentations of the isotropy subgroups of the vertices, we describe them in detail. We say that a

because there exists an element of Stabs
v0 reversing the orientation of the non-
separating curves
v0. On the contrary, for the separating verticesH_{s
v} is 0 except
in one case, when
g;r;s
2k;0;2h and v is such that F_{s
v} has two connected
components, both homeomorphic toF_{k;1}^{h} . In this case, we denote such a vertex byvsym,
referring to it asthe symmetric separating vertex, and, since there exists an element of
Stabs
vsymreversing the orientation ofs
vsym, we conclude thatH_{s
v}_{sym}_{}isZ2.

Hence, ifaiss v0, (4.1) and (4.2) become

0!Stab^{}
s
v_{0} !Stab
s
v_{0} !Z_{2}!0;

0!Z!M
F_{gÿ1;r2}^{s} !Stab^{}
s
v_{0} !0;
4:3

while ifaiss
v_{sym}they become

0!Stab^{}
s
vsym !Stab
s
vsym !Z2!0;

0!Z!M
F_{k;1}^{h} M
F_{k;1}^{h} !Stab^{}
s
v_{sym} !0:
4:4

Finally, if a is a representative for any separating and non-symmetric vertex, we have the exact sequence

0!Z!M
F_{g}^{s}0;^{0}r^{0} M
F_{gÿg}^{sÿs}0^{0};rÿr^{0}2 !Stab
s
v !0:
4:5

Thus, if we know the presentations

M
F_{gÿ1;}^{s} _{r2} hG_{0}jR_{0}i;

M
F_{k;1}^{h} hG_{1}jR_{1}i;

M
F_{g}^{s}0;^{0}r^{0} hG_{2}jR_{2}i;

M
F_{gÿg}^{sÿs}0^{0};rÿr^{0}2 hG3jR3i;

Lemma 10 allows us to conclude that

Stabs
v_{0} hG_{0}UcjR_{0}UR_{g}Ufc^{2}w_{c}^{2}gUR_{m}i;
4:6

Stabs
v_{sym} hG_{1}UGf_{1}UrjR_{1}URf_{1}UR_{c}UR_{g}Ufr^{2}w_{r}^{2}gUR_{m}i;
4:7

Stabs v hG2UG3jR2UR3URcURgi; 4:8

where we identify each Stab^{} with its image in Stab, we denote by hGf_{1}jRf_{1}i the
presentation of the second copy ofM
F_{k;1}^{h} in M
F_{s
v}_{sym}_{},w_{c}^{2} andw_{r}^{2} are as in the
notation of Lemma 10, and where

c is an element of Stabs
v_{0}reversing the orientation ofs
v_{0}(the~1 correspond-
ing to the generator 1 ofZ_{2} in (4.3) with the notation of Lemma 10);

r is an element of Stabs
v_{sym} reversing the orientation of s
v_{sym} (the ~1 corre-
sponding to the generator 1 ofZ2 in (4.4));

Rg is the relation identifying the Dehn twists along the two boundary components
ofF_{s
v}corresponding to the curves
v;

Rm are the relations of type R^{
3} with the notation of the Lemma 10, ``mixing''
Stab^{}withZ2;

Rc are the relations saying that the generators of the mapping class group of one of
the connected components ofF_{s
v}commute with those of the other inM
F_{s
v}.

It is always possible to choose forcthe semitwist along c relative to s
v0(see the
appendix for its de®nition), and forrthe rotation ofparound thezaxis (see Figure
4.5). Hence, the relationr^{2}w_{r}^{2}becomesr^{2}(i.e.r^{2}1), while it is not di½cult to prove
thatc^{2}is the Dehn twist along the curveg, and thus to ®nd the expression forw_{c}^{2}.

5 The inductive process and the sporadic surfaces

The general method presented in Section 2 applies to the ordered complex of curves
of any surface F_{g;}^{s}_{r}, provided this complex is simply connected, and provided we
have a ®nite presentation for the isotropy subgroups of a representative in everyM_{g;}^{s}_{r}-
equivalence class of its vertices and edges.

Subsection 4.2 describes how to produce a presentation for the isotropy subgroup
of any generic orderedk-family of closed curvesa, if a presentation is known for the
mapping class group of the surface obtained byF_{g;}^{s}_{r}cutting it open alonga. Moreover,
if such a presentation is ®nite, the same is true for the resulting one of the isotropy
subgroup.

Cutting open F_{g;}^{s}_{r} along a representative for every vertex and every edge, we ®nd
subsurfacesF_{g}^{s}0;r^{0} ^{0} with
g^{0};r^{0}s^{00}<
g;rs(with the lexicographic ordering), and
when these surfaces are not sporadic we can apply again the previous argument to
compute a presentation for their mapping class group.

Figure 4.5. The curvess v0andgand the elementr.

Repeating recursively these steps for any non-sporadic F_{g}^{s}0;^{0}r^{0}, we ®nally arrive
at sporadic surfaces: hence if we know a ®nite presentation for the mapping class
group of each sporadic surface (as a basis of this inductive process), we inductively
get a ®nite presentation forM_{g;r}^{s} .

The sporadic surfaces F_{g}^{s}0;r^{0}^{0} which may appear in this process are the sphere with
r^{0}s^{0}3;4;5 and the torus withr^{0}s^{0}1;2 (by De®nition 7). All these cases are
well known:M
F_{0;3}^{0} Z^{3}, generated by the Dehn twists along the boundary com-
ponents, and in general it is a classical result (see for example [1, 11] or [8]), that, for
rd1,M
F_{0;}^{0}_{r} Prÿ1Z^{rÿ1}, wherePn is the group of pure braids withn strings. In
particular, in the cases we are concerned with,

### .

_{M F}

_{0;4}

^{0}

_{ }

_{P}

_{3}

_{}

_{Z}

^{3}, and it admits a presentation with

Generators:fA1;A2;A3;D12;D13;D23gwhere the curvesai;dij are represented in Figure 5.1 (we recall that, if not otherwise stated, we denote a curve by a lowercase letter, and the Dehn twist along the same curve by the corresponding capital letter).

Relations: A_{i}A_{j} A_{j}A_{i} for alli;j
A_{i}D_{jk} D_{jk}A_{i} for alli;j;k
D_{23}D_{13}D_{12}D_{13}D_{12}D_{23}

D_{23}D_{13}D_{12}D_{12}D_{23}D_{13}

### .

_{M F}

_{0;5}

^{0}

_{ }

_{P}

_{4}

_{}

_{Z}

^{4}, and it admits a presentation with

Generators: fA_{1};A_{2};A_{3};A_{4};D_{12};D_{13};D_{14};D_{23};D_{24};D_{34}g where the curvesa_{i};d_{ij}
are represented in Figure 5.1.

Relations: A_{i}A_{j} A_{j}A_{i} for alli;j
A_{i}D_{jk} D_{jk}A_{i} for alli;j;k
D_{34}D_{12} D_{12}D_{34}

D_{14}D_{23}D_{23}D_{14}

D23D13D12D13D12D23D12D23D13

D24D14D12D14D12D24D12D24D14

D34D14D13 D14D13D34 D13D34D14

D34D24D23D24D23D34 D23D34D24

D24D^{ÿ1}_{34}D^{ÿ1}_{14}D34D14D13D^{ÿ1}_{34}D^{ÿ1}_{14}D34D14D13D24

Figure 5.1. Generators for the mapping class group ofF_{0;}^{0}_{4}andF_{0;5}^{0} .

In what follows, we will refer to these presentations as standard presentations for the sporadic surfaces of genus0.

As far as the sporadic surfaces of genus 1 are concerned,

### .

_{M F}

_{1;0}

^{0}

_{ }

_{hA;}

_{B}

_{j}

_{ABA}

_{}

_{BABi}

_{}

_{SL 2;}

_{Z}

_{(a;}

_{b}as in Figure 5.2);

### .

_{M F}

_{1;1}

^{0}

_{ }

_{SL 2;}Z, and admits the presentationM F

_{1;1}

^{0} hA;BjABABABi (a;bas in Figure 5.2);

### .

_{for}

_{M F}

_{1;}

^{0}

_{2}

_{}we have, with a

_{1}, a

_{2}, b, c

_{1}, c

_{2}as in Figure 5.2, the following presentation:

Generators: fA1;A2;B;C1;C2g Relations: A1A2A2A1

AiBAiBAiB i1;2
AiCjCjAi i;j1;2
CjBBCj j1;2
A_{1}^{2}A2B^{3}C1C2

A1A_{2}^{2}B^{3}C2C1

These presentations are particular cases of a general result of Gervais in [7] (see Section 7); therefore in what follows we will refer to them as toGervais presentations for the sporadic surfaces of genus1.

Since we will discuss in detail only the non-punctured case in the next sections, we avoid to describe here the mapping class group of the sporadic surfaces with punc- tures. We just remark that at least one simple ®nite presentation for each of them is well known, so our inductive argument works also in the punctured case. We refer the interested reader to [1, 6] for the case of genus 0, and to [20] for the case of genus 1.

6 The non-punctured case

Proposition 12. If Fg;rF_{g;r}^{0} is not sporadic, and g>0, then for every choice of the
representatives and of the maximal tree, it is possible to express all the symbols E
appearing in Theorem 3 as a product of elements in6_{v}_{A}_{V
X}ordStabs
v.Hence,the
presentation obtained applying Theorem3 to the action of the mapping class group on
the ordered complex of curves reduces to

M
F_{g;r} h 6

vAV
X^{ord}

S_{v}j 6

vAV
X^{ord}

R_{v}URg^{
3}URg^{
4}i;
6:1

1;r

where

g_{c;}_{i}g_{T;c}g^{ÿ1}_{T;a}g_{a;i}^{ÿ1}AStabs
f;

g_{a;}_{t}g_{T;}_{a}g^{ÿ1}_{T;b}g^{ÿ1}_{b;}_{i}AStabs
m;

g_{b;t}g_{T;b}g^{ÿ1}_{T;}_{c}g_{c;t}^{ÿ1}AStabs
p:

Therefore, it two of the edges ofTare in the maximal treeT, once we ``kill''Tusing
the relationsR^{
2}, the relation (6.2) gives an expression for the third edge as a product
of stabilizers of the representatives for the vertices. The same is true if in (6.2) two of
the symbols for the edges were already expressed as products of stabilizers.

We say that a symbol Eisdeterminable(or simply that the corresponding edgee
isdeterminable) if, using recursively relations of typeR^{
4}, after ``killing'' the maximal
tree it is possible to expressEas a product of elements in6_{vA}_{V
X}ordSv.

Given an edgeeinX^{ord}, we calltopological inverse of ethe edgeebeing the same
element inX, but having the opposite orientation, i.e. directed fromt
eto i
e. It
is clear that, if there exists in X^{ord} a 2-cell with edges a;b;c, then there are also all
the 2-cells with the same vertices that one can build usinga;b;cand their topological
inverses (see Figure 3.2). Therefore, if two of the edgesa;b;care determinable, so is
the third one and the three topological inverses as well.

Sinceg>0, there exists a single non-separating vertex inX^{ord}that we denote byv_{0}.
Lemma 13. For each vAV
X, v0v_{0}, there always exists at least one edge e with
i
e v_{0} and t
e v. More precisely, the number of such edges is 1 if one of the
connected components of F_{v}has genus0,or if vv_{sym},and it is2otherwise.

Proof.The assertion follows immediately from the analysis of Subsection 4.1.

Lemma 14. For each vAV X, v0v0, at least one of the edges joining v0 and v is determinable.

Proof.LeteATbe an edge joiningv_{0}and a separating vertex, sayv_{1}. Then (at least)
one amongv_{0} andv_{1}is connected with another vertexv_{2} by an edgee^{0}belonging to
the maximal treeT. Ife^{0}joins the two separating vertices, then there always exists
a 2-cell with verticesv0;v1;v2, containing the two edgeseande^{0}: this follows from
the fact that any representative for e^{0} dividesF into three subsurfaces, and at least
one of them has genus >0. Therefore we may always ®nd a representative for the
non-separating vertex disjoint from the other two chosen representatives (that means
we always have a triangleTwith verticesv0,v1andv2); moreover, if there is only one

along the maximal tree touching all the other vertices), we get the assertion for every vAV X.

In the sequel, we sometimes writee^{
n}_{i;}_{j} to denote an edge inX^{ord}with initial vertex
v_{i}and terminal vertexv_{j}, emphasizing, by means of the progressive numbern, the fact
that in general such an edge is not unique.

Lemma 15. Let e^{
0}_{0;}_{0} be the loop of X^{ord} based in v0 and such that F_{s
e}^{
0}

0;0 is connected
(and therefore homeomorphic to Fgÿ2;r4);then e^{
0}_{0;}_{0}is determinable.

Proof. Let us suppose there exists an edge eAT with one end in v0 and such that
there exists a 2-cell with edgese^{
0}_{0;0},e,e. In such a situation, the symbol corresponding
toe^{
0}_{0;}_{0}is determinable by the relationR^{
4}corresponding to that 2-cell.

If, on the contrary, such an edge does not exist, this means that gd2, and the
only eAT with one end in v_{0} is the edge such that F_{s
e} has a connected com-
ponent homeomorphic to the pair of pants F_{0;3} and the other one toF_{gÿ1;r1}, with
gÿ1;r1>
1;1. In this case, the other end ofe, that we denote by v_{1}, is con-
nected to another vertexv_{2}by an edgee^{0}AT, and there exists a triangle with vertices
v_{0},v_{1},v_{2} and edgese,e^{0} ande^{00}, where e^{00} is an edge with endsv_{0} andv_{2}. Therefore
the edgee^{00}is determinable, and, since there exists a 2-cell with edgese^{
0}_{0;0},e^{00},e^{00}, so
is the loope^{
0}_{0;0}.

Lemma 16.The following edges are determinable:

(i) all the loops based in v_{0};
(ii) all the edges with one end in v0;
(iii) all the loops based in vi,for every i;

(iv) all the edges e^{
n}_{i;}_{j} with ends in vi and vj,i0j and i;j00.

Proof. (i) Given any loope^{
i}_{0;0} based in v0, di¨erent from e^{
0}_{0;}_{0}, ifgd2 there always
exists a triangle with edgese^{
i}_{0;0},e^{
0}_{0;0},e^{
0}_{0;0}, and thene^{
i}_{0;}_{0}is determinable. In caseg1,
we observe that there always exists a triangle with edgese^{
i}_{0;}_{0},e,e, a representative of
which is depicted in the left-hand side of Figure 6.1; ifeAT, we are done, if not,
then the two edges e^{0},e^{00}, whose representatives are shown again in Figure 6.1, are
inT. If this is the case,eis determinable by the trianglee,e^{0},e^{00}(right-hand side of
Figure 6.1), and therefore the loope^{
i}_{0;0} is determinable.

(ii) The assertion follows immediately from Lemma 14, Lemma 15, and the obser-
vation that, if vis such that there are two di¨erent edges with initial vertex v_{0} and
terminal vertexv, then there exists a triangle whose edges are these two edges and the
loope^{
0}_{0;0}.

(iii) Given any loop based in v, there always exists a triangle whose edges are
this loop and two edges with one end invand the other one inv_{0}, hence we have the
assertion.

(iv) Just notice that there always exists a triangle with edges e^{
n}_{i;}_{j} and two edges
with one end inv0and the other one respectively inviandvj.

This concludes the proof of Proposition 12.

Moreover, we have the following result:

Theorem 17.There exists a choice of the representatives and of the maximal tree such
that (6.1)expressesM_{g;r} as the free product of the isotropy subgroups of the chosen
representatives for the vertices,amalgamated along the subgroupsStabs
e,as e varies
in E
X_{g;r}.

Proof. Let us ®x a representative s v0 for the non-separating vertex. It is always possible to choose, for every separating vertex vi, a representative s vi such that s v0Vs vi q.

Hence, we may takeTto be the union, for all the separating verticesv_{i}AV
X_{g;}_{r},
of the edges ewithi
e v0 andt
e vi, such that it is possible to choose s
e
hs
v0;s
vii.

Concerning all the other edges, notice that we may always choose the representa-
tivess
ein such a way that at least one of the ends ofs
eis the chosen representative
for its class, and we may always choose for the topological inverseeof the edgeethe
representatives
e ht
s
e;i
s
ei, so thatg_{e;}_{i}g_{e;t} andg_{e;t} g_{e;i}.

By Proposition 12, every edgeeBTis determinable; with the choices we made, it turns out that actually the symbolEassociated to an edgeeis determined as

E1 if i s e s i eandt s e s t e;

Eg^{ÿ1}_{e;}_{t} if i
s
e s
i
eandt
s
e0s
t
e;

Eg_{e;i} if i
s
e0s
i
eandt
s
e s
t
e;

6:3

where theg_{e;i}andg_{e;t} may be taken to be products of stabilizers of suitable vertices.

Figure 6.1. Determinability of a loope^{
i}_{0;0}inF1;r.

in Stabt
s
ewith the copy of the same subgroup in Stabi
s
e, after, by means
of suitable conjugations, expressing the elements of the intersection as products of
elements in6_{vA}_{V
X}_{g;}_{r}_{}Stabs
v.

The proofs we gave of Proposition 12 and Theorem 17 strongly depend on the
existence of a non-separating vertex, v_{0}; hence, the argument obviously does not
work for the case of genus 0.

Actually, the structure ofX_{0;r} is quite di¨erent from that of X_{g;}_{r} when gd1; in
order to describe it, we enumerate the boundary components ofF_{0;r}, denoting them
byq_{1};q_{2};. . .;q_{r}.

The vertices ofX_{0;r} arev_{I}, with multi-indexI
i_{1};i_{2};. . .;i_{s}such that
1ci_{1}<i_{2}< <i_{s}cr;

s2;. . .;r=2 wherexdenotes the integer part of x
and; if ris even; thev_{I} withaIr=2 havei_{1}1;

where v_{i}_{1}_{;...;i}_{s} denotes theM_{0;r}-equivalence class of a curves
v_{i}_{1}_{;...;}_{i}_{s}separating F_{0;r}
into two connected components, both of genus 0, such that one of them has boundary
componentsqi1;. . .;qis ands
vi1;...;is.

Regarding the edges ofX0;r, it is easy to verify the following claims:

Lemma 18.(i)There are no loops;

(ii) for each pair of vertices vI and vJ there exists at most one edge with initial vertex vI and terminal vertex vJ;

(iii) there exists an edge connecting v_{I} and v_{J} if and only if4caIaJcrÿ1and,
ifaIcaJ,either IPJ or IVJq.If such an edge exists,we denote it by e_{I;J}.
Finally, the triangles ofX_{0;r}are theT_{I;}_{J;K} v_{I};v_{J};v_{K}such that 6caIaJaK
crand, ifaIcaJcaK, we have one of the following situations:

### .

_{I}

_{V}

_{J}

_{}

_{q,}

_{J}

_{V}

_{K}

_{}

_{q,}

_{I}

_{V}

_{K}

_{}

_{q;}

### .

_{J}

_{V}

_{K}

_{}

_{q,}

_{I}

_{P}

_{K;}

### .

_{I}

_{V}

_{K}

_{}

_{q,}

_{J}

_{P}

_{K;}

### .

_{I}

_{V}

_{J}

_{}

_{q,}

_{J}

_{P}

_{K,}

_{I}

_{P}

_{K;}

### .

_{I}

_{P}

_{J}

_{P}

_{K:}

Let us choose representatives for the vertices as in Figure 6.2. Hence, it is pos-
sible to choose representatives for the edges such thats
eI;J fs
vI;s
vJgif either
aI<aJ andIPJ, orIVJ q and the i_{h}'s do not alternate with the j_{k}'s with
respect to the cyclic ordering off1;. . .;rg. In the other cases, we may chooses
e_{I;}_{J}

fs
v_{I};~v_{J}g, where I is the multi-index with greater cardinality, or, ifaI aJ, I
contains the lower index (where~v_{J} is the representative forv_{J} depicted in the right-
hand side of Figure 6.2).

We restrict from now on to the case ofF_{0;}_{6}, the ®rst non-sporadic surface of genus
0, that, in spite of its simplicity, is paradigmatic of the caseF_{0;r}.

We choose the maximal treeTas follows: we put inTthe ten edges having one end inv1;2:

e_{
1;2;}_{
3;4}; e_{
1;}_{2;
3;6}; e_{
1;2;}_{
4;6}; e_{
1;2;
1;}_{2;3}; e_{
1;2;
1;}_{2;5}

e_{
1;2;}_{
3;5}; e_{
1;}_{2;
4;5}; e_{
1;2;}_{
5;6}; e_{
1;2;
1;}_{2;4}; e_{
1;2;
1;}_{2;6}:

From what we said before,v1;2 appears in 15 triangles, and clearly all of them have two edges inT; hence, the 15 other edges are determined, more precisely they are:

e_{
3;}_{4;
5;6}; e_{
1;2;}_{3;
4;5}; e_{
1;2;}_{4;
3;5}; e_{
1;2;}_{5;
3;}_{4}; e_{
1;2;}_{6;
3;}_{4};
e_{
3;}_{5;
4;6}; e_{
1;2;}_{3;
4;6}; e_{
1;2;}_{4;
3;6}; e_{
1;2;}_{5;
3;}_{6}; e_{
1;2;}_{6;
3;}_{5};
e_{
4;}_{5;
3;6}; e_{
1;2;}_{3;
5;6}; e_{
1;2;}_{4;
5;6}; e_{
1;2;}_{5;
4;}_{6}; e_{
1;2;}_{6;
4;}_{5}:

6:4

Each of these edges appears in two more triangles, whose third vertex is still not
reached by the maximal tree: for example, e_{
3;4;}_{
5;6} appears inT
3;4;
5;6;
1;3;4 and
T_{
3;4;
5;}_{6;
1;}_{5;6}. Hence, if we put in T the edges e_{
1;}_{3;4;
3;}_{4} and e_{
1;}_{5;6;
5;}_{6}, we
determine the edges e
1;3;4;
5;6 ande
1;5;6;
3;4. Applying the same argument to all
the edges listed in (6.4), we complete the maximal tree adding the 14 edges

e_{
1;}_{i;}_{j;
i;}_{j} 3ci< jc6
e_{
1;2;}_{j;}_{
1;}_{j} 3cjc6
e_{
1;2;}_{j;}_{
2;}_{j} 3cjc6

Figure 6.2. Representatives for the vertices ofX0;r.

Now, the argument applied again to each edge in (6.5) gives the remaining 36 edges. Moreover, it is easy to check that, independently from the choices of the representatives for the triangles, the expressions found for the symbols associated to the edges are:

E1 if i s e s i eandt s e s t e;

Eg_{e;}_{i} if i
s
e0s
i
eandt
s
e s
t
e;

Eg_{e;t}^{ÿ1} if i
s
e s
i
eandt
s
e0s
t
e:

Hence, the relation of typeR^{
3} associated to the edgeeidenti®es the copy of the
intersection Stabt
s
eVStabi
s
e(that is Stabs
e) in Stabt
s
ewith the copy of
the same intersection in Stabi
s
e, while the relations of typeR^{
4} that we did not
use to determine the edges are ``coherent'', hence they disappear once we substitute
the values obtained for theE's.

The generalization to the case F_{0;}_{r};r>6 is straightforward, which proves the
following result, analogous to Theorem 17:

Theorem 19.There exists a choice of the representatives and of the maximal tree such that,for each non-sporadic surface of genus0,we may expressM0;ras the free product of the isotropy subgroups of the s v, vAV X0;r, amalgamated along the subgroups Stabs ewith eAE X0;r.

Remark 20.It is unknown to the author if the result of Theorem 19 is actually inde- pendent from the choice of the maximal tree.

7 Recovering known presentations

Let us consider, on the surface F_{g;}_{r}, gd1, rd0, the curves of Figure 7.1. A triple
i;j;kAf1;. . .;2grÿ2g^{3}is said to begoodwhen

i i;j;kBf l;l;l jlAf1;. . .;2grÿ2gg;

ii icjckor jckciorkcicj:

Recall that we denote each curve by a lowercase letter, and the Dehn twist along the same curve by the corresponding capital letter. Gervais, in [7], proved the fol- lowing result:

Theorem 21.For all
g;rAN^{}N, the mapping class groupMg;r admits a presen-
tation with

Generators: B;B1;. . .;Bgÿ1;A1;. . .;A2grÿ2;
Ci;j_{1ci;}jc2grÿ2;i0j

Relations: handles C_{2i;2i1}C_{2iÿ1;2i} for all i, 1cicgÿ1,

braids for all X, Y among the generators, XY YX if the asso- ciated curves are disjoint and XYXYXY if the associated curves intersect transversally in a single point,

stars Ci;jCj;kCk;i
AiAjAkB^{3} for all good triples
i;j;k,
where we de®ne Cl;l1.

To prove this result, Gervais started from Wajnryb's presentation described in [31], which in turn was obtained exploiting the action ofM Fon the Hatcher±Thurston complex ([14]). We prove in this section that it is possible to get the Gervais presen- tation for the mapping class group of any non-sporadic surfaceFg;rusing the ordered complex of curves in place of the Hatcher±Thurston complex, provided we take as starting point of the inductive process the Gervais presentationfor the sporadic sub- surfaces of genus 1, and the standard presentation for the sporadic subsurfaces of genus 0 (see Section 5).

Let us consider a non-sporadic surfaceFg;r, of genus greater than 0; by Theorem 17,
its mapping class group is the free product of the isotropy subgroups of a (suitably
chosen) representative for each class inV
X_{g;r}^{ord}, amalgamated along the subgroups
Stabs
e, foreAE
X^{ord}.

Hence, reasoning recursively, we just need to knowsomepresentation for the spo-
radic surfacesF_{0;}_{r}^{0},r^{0}3;4;5 andF_{1;r}^{0},r^{0}1;2.

Theorem 22. Starting from the Gervais presentation forM1;1 andM1;2, and the stan-
dard presentation for M_{0;}_{3}, M_{0;4},M_{0;5},we get the Gervais presentation for any non-
sporadic F_{g;}_{r}of genus g>0.

Proof.The assertion follows immediately from the following lemma:

Figure 7.1. Gervais generators forM Fg;r.