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Home , Homeotopy, Homotopy category, Mapping class group, Torus

Mapping Class Groups

Nikolai V Ivanov

December

Contents

Introduction

of surfaces

The dimension

Classication of surfaces

and arcs on surfaces

Pants decomp ositions

Geometric structures on surfaces

Spaces of dieomorphisms and generalities

Spaces of dieomorphisms and embeddings surfaces

Gluing discs to a

The DehnNielsenBaer Theorems

Complexes of

Denitions

Connectivity of C S and the MorseCerf theory

The type of the complexes of curves

Hyp erb olic prop erties

Dehn twists generators and relations

Dehn twists

The DehnLickorish theorem

Finite presentations

Teichmuller spaces

Denitions

Length functions and FenchelNielsen ows

The top ology of Teichmuller spaces

Supp orted in part by the NSF Grants DMS and DMS

Corners and the mapping class groups

Ideal triangulations

Cohomological prop erties

of groups and the PoincareLefschetz duality

Classifying spaces and universal bundles

Mess

Virtual cohomological dimension

stability

The lowdimensional homology groups

Virtual

Torsion in mapping class groups torsion in their cohomology and

related topics

Thurstons theory and its applications

Classication of mapping classes

Measuring against circles and its applications

Action of the mapping classes on Thurstons compactication of

Teichmuller space

Some applications

Commuting elements and Dehn twists

of complexes of curves

The theorem ab out automorphisms

Intersection number prop erty

The case of a with holes

Complexes of curves and ideal triangulations

An application to subgroups of nite index

An application to Teichmuller spaces

Mapping class groups and arithmetic groups

Arithmetic groups

Abstract commensurators

The original approach to the nonarithmeticity

Rank of the mapping class groups

The MostowMargulis sup errigidity

Introduction

The Mo d of an orientable surface S is dened as the

S

group of isotopy classes of orientationpreserving dieomorphisms S S In

to b eing a central ob ject of the top ology of surfaces cf these

groups also play an imp ortant role in the theory of Teichmuller spaces and in

algebraic where they are known under the name Teichmuller modular

groups or simply modular groups Our notations are derived from the latter

terminology There are several closely related groups which also deserve the

name of the mapping class groups or Teichmuller mo dular groups First of

all one may consider the extended mapping class group Mo d of S dened

S

as the group of the isotopy classes of al l dieomorphisms S S The pure

mapping class group PMo d of S is dened as the group of isotopy classes of all

S

orientationpreserving dieomorphisms S S preserving setwise all b oundary

comp onents of S Finally one may consider the group M of all orientation

S

preserving dieomorphisms S S xed on the b oundary S considered up to

isotopies xed on the b oundary If S then dieomorphisms xed on S are

automatically orientationpreserving If S then of course M Mo d

S S

All these groups could b e also dened as the th homotopy groups of suitable

dieomorphisms groups of S For example Mo d Di S where Di S

S

is the group of all orientationpreserving dieomorphisms of S considered with

for example C top ology or any other reasonable top ology

The study of the mapping class groups was initiated in the ies by M

Dehn D D D and J Nielsen NiNi Although their work had some

common themes cf for example the DehnNilesenBaer theorems in in

general their approaches were fairly dierent M Dehn was interested in the

prop erties of the mapping class group as a whole addressing for example

such questions as the existence of a nite set of generators He developed and

exploited an imp ortant to ol for this purp ose the action of the mapping class

group Mo d on the collection of the isotopy classes of all circles on S He

S

called this collection the arithmetic eld of S On the contrary J Nielsen was

mainly interested in understanding the ne structure of the individual elements

of Mo d His metho ds draw heavily on hyperb olic geometry a to ol also favored

S

by M Dehn For quite a while the work of b oth M Dehn and J Nielsen was

apparently forgotten The ideas of M Dehn related to the arithmetic eld

of a surface found a natural continuation in the ideas of W Harvey Harv

Harv ab out the complex of curves of a surface which is nothing else but

the arithmetic eld made in a simplicial complex in a natural way see

A closely related ob ject was considered in an inuential pap er of A Hatcher

and W Thurston HatT The ideas of J Nielsen were partially rediscovered

extended and brought to an essentially complete form by W Thurston in his

theory of surface dieomorphisms Th FLP see Chapter Later on his

theory was applied also to the structure of the mapping class group and not

only to its individual elements

In the present notes we have adopted a p oint of view going back to that

of M Dehn Compared to the other thread going back to J Nielsen it is

more elementary and allows us to reach more quickly some really deep results

ab out mapping class groups Everyone interested in the mapping class groups

eventually should learn the other p oint of view also esp ecially Thurstons theory

of surface dieomorphisms We hardly do justice to it in a chapter devoted

to Thurstons theory and some of its applications If one takes M Dehns

terminology seriously our exp osition is slanted toward the arithmetic of our

sub ject which seems to b e only natural for an introduction The analogy with

the arithmetic groups see for a denition briey touched on in Chapter

ts nicely in this approach for example the main results of Chapter are in

fact analogues of some fundamental theorems ab out arithmetic groups giving

a deep er meaning to the word arithmetic in this context

Our exp osition is centered around three topics the DehnLickorish theorem

providing a nite set of generators of Mo d for closed S J Harers theorem

S

computing the socalled virtual cohomology dimension of Mo d and the au

S

thors theorem describing all automorphisms of W Harveys complex of curves

The rst two topics are presented with complete pro ofs and a detailed review

of the prerequisites for the third topic only an outline is given The pro ofs we

give for the theorems of DehnLickorish and Harer contain some new ideas and

may b e interesting even for exp erts The table of contents gives a go o d idea of

the topics included in these notes The limitations of time and space prevented

us from giving a more detailed treatment of many included topics and from dis

cussing other topics equally or may b e even more imp ortant As we already

mentioned our treatment of Thurstons theory is hardly adequate although we

hop e that it is still useful the same applies to the asp ects of the theory of the

Teichmuller spaces related to the mapping class groups Nothing is said ab out

relations with algebraic geometry and mathematical physics we refer to the

recent survey of R Hain and E Lo oijenga HL for this Two omitted topics are

very closely related to the ones included In order not to pass them in complete

silence we very briey discuss them here

The rst one is the study of the socalled Torelli group the of

Mo d consisting of the isotopy classes of dieomorphisms of S acting trivially

S

on H S Z The pioneering work in this eld is due to D Johnson In J he

proved that the Torelli group of S is nitely generated if the of S is at least

and S has no more than b oundary comp onent and in J J he computed

the rst homology group of the Torelli group under the same restrictions See

J for a survey of this work It was continued in particular in the algebro

geometric work of R Hain see H and esp ecially the remarkable pap er H

where R Hain proved the existence of a nite presentation for a Lie algebraic

analogue of the Torelli group Note that the question ab out the existence of a

nite presentation of the Torelli group itself remains op en

The second topic is the theory and applications of the characteristic classes of

surface bundles which are nothing other than the elements of the cohomology

group H Mo d constructed indep endently by D Mumford Mumf Sh

S

Morita Mor Mor and E Miller Mi Sh Morita Mor Mor and E

Miller Mi proved the nontriviality of these classes and Sh Morita Mor

Mor found a sp ectacular application of this nontriviality he proved that the

canonical homomorphism Di S Mo d admits no section In his further

S

pap ers see for example MorMor MorMor Sh Morita studied

these characteristic classes and applied them to the theory of the Torelli group

extending in some resp ects the ab ove mentioned work of D Johnson and to the

Casson invariant of homology and related problems of the top ology

of We refer to his own surveys MorMor for a discussion of

these results

I am very grateful to R Sher for the careful reading of this pap er

Topology of surfaces

In this section we attempt to outline the material which should b e at the foun

dation of the theory of the mapping class groups All this material is fairly

classical although some results for example the ones ab out spaces of dieo

and embeddings are not so widely known as they deserve Our

discussion is more broad than is needed for later applications in order to get

closer to a coherent picture still our treatment is far from b eing complete The

pro ofs in this section are not to b e taken to o seriously mostly they are only

outlines or just indications of the ideas involved in the real pro ofs

The dimension

In order to put the top ology of surfaces in a p ersp ective let us start with a

particular view of the top ology of manifolds in general

The central problem of the top ology of manifolds is the problem of classi

cation of manifolds It is usually considered to b e more of less solved for at

least simply connected manifolds of dimension Another ma jor problem

is the problem of isotopy classication of dieomorphisms Consideration of

dieomorphisms up to isotopy distinguishes top ology from other sub jects es

p ecially from the theory of smo oth dynamical systems which concentrates on

the classication of dieomorphisms up to conjugacy A satisfactory solution of

the isotopy classication problem for dieomorphisms is known only for some

sp ecial classes of manifolds The problem which naturally comes after the clas

sication of dieomorphisms is the problem of understanding of the homotopy

type of the dieomorphism group of a It is extremely dicult even

for such manifolds as spheres These problems actually form a natural series

Namely the classication of manifolds the isotopy classication of dieomor

phisms and the computation of homotopy groups Di M for i where

i

Di M is the group of all dieomorphisms M M with say C top ology

form a series of problems parallel and related to the study of K K

and K for i of the algebraic K theory We can add also to this row the

i

problem of the existence of a smo oth manifold within a given homotopy type

and the K

In the dimension the solution of the classication problem is well known

and is in fact a part of any introductory course in top ology The homotopy

groups Di M with i are known for all compact two dimensional

i

manifolds and moreover most of them are equal to We will discuss these

results in This leaves us in the dimension with only one of the ab ove

problemsthe problem of the classication of dieomorphisms Interpreted

broadly this problem naturally includes a search for an understanding of the

mapping class groups Di M which is the main fo cus of our survey But

b efore we concentrate on Di M we will review some other parts of the

top ology of surfaces partly b ecause they are needed to understand Di M

and partly for their own sake

Classication of surfaces

Throughout the pap er all surfaces are assumed to b e orientable and unless the

contrary is obvious from the context compact

There is no doubt that the reader is familiar with the classication of sur

faces Let us recall for the sake of completeness that an orientable connected

surface S is determined up to a dieomorphism by its genus g and the number

of the b oundary comp onents b Following Riemann the genus can b e dened

as the maximal number of disjoint circles on S such that these circles do not

separate S ie the complement in S of the union of these circles is connected

The genus g and the number of b oundary comp onents b are related to the Euler

characteristic S of S by the well known formula S g b A surface

of genus g with b b oundary comp onents is also known as a sphere with g hand les

and b holes the terminology is justied by a well known picture

The standard pro of of this classication theorem based on triangulations

and cutting and pasting arguments provides a cannonical mo del of a surface of

genus g with b b oundary comp onents Namely such a surface can b e obtained

from a g gon by a well known identication of sides and removing b disjoint

discs from the interior The identication of sides is describ ed by the word

a b a b a b a b and this leads to a presentation of S

g g

g g

S ha b a b d d j a b a b a b

g g b g g

a b d d i

b

g g

If b then one can eliminate one d from this presentation and conclude that

i

S is a free group One can obtain the same result in a more top ological

manner by observing using the ab ove g gon mo del for example that S has

a dimensional CW complex as a deformation retract But considering S

for surfaces with b oundary as just a free group is to a great extent misleading

b ecause in this way we lose all information ab out the b oundary It is much

b etter to keep the track of the b oundary by introducing the socalled peripheral

structure the set of b conjugacy classes of elements of S corresp onding

to lo ops going once round a b oundary comp onent It is also useful to consider

the oriented peripheral structure the set of b conjugacy classes of elements of

S corresp onding to lo ops going once round a b oundary comp onent in the

direction prescrib ed by a xed orientation of S which induces an orientation of

the b oundary S

Note that all generators in the ab ove presentation can b e obviously repre

sented by embedded lo ops ie by lo ops f S such that f x f y

only if x y or fx y g f g of course f f is the base p oint

Circles and arcs on surfaces

Circles and arcs on surfaces and the action of dieomorphisms on them are

some of the main to ols and ob jects of the top ology of surfaces By circles

we understand the embedded circles ie submanifolds dieomorphic to the

standard Similarly by arcs we understand submanifolds dieomorphic

to the interval Usually one restricts attention to prop erly embedded

arcs an arc I on a surface S is called properly embedded or simply proper if

I I S and I is transverse to S

It is convenient to single out circles and arcs considered to b e trivial We call

a circle C on a surface S trivial if C is either contractible in S or is homotopic

to some b oundary comp onent of S Similarly we call an arc I trivial if it can

b e deformed into the b oundary of S in such a way that its b oundary I stays in

S Also it is imp ortant to distinguish b etween separating and nonseparating

circles We call a circle C on S nonseparating if S n C is connected and separating

otherwise

As we noticed in all generators in the standard presentation of S

can b e represented by embedded lo ops The image of an embedded lo op is a

circle and we call an embedded lo op separating or nonseparating if this im

age is separating or resp ectively nonseparating The obvious lo ops repre

senting a b a b are nonseparating while the obvious lo ops represent

g g

ing d d are separating in fact their images can b e deformed into the

b

corresp onding b oundary comp onents But one can easily represent elements

a d a d by embedded nonseparating lo ops It follows that S can b e

b

generated by elements represented by nonseparating embedded lo ops This re

mark will b e used in our pro of of the DehnLickorish theorem see the pro of of

Theorem C

If C is a circle on S or more generally union of several disjoint circles we

can cut S along C and get a new surface which will b e denoted by S If C is a

C

nonseparating circle then S is connected If C is a separating circle then S

C C

consists of two comp onents Note that we can always reconstruct S from S by

C

gluing along the b oundary comp onents of S resulting from C More generally

C

one can cut S also along prop er arcs and more generally disjoint unions of

circles and prop er arcs If prop er arcs are involved the result of cutting is a

smo oth surface with corners

Supp ose now that C is a nonseparating circle Then S is a connected

C

surface having two more b oundary comp onents than S The additivity of the

Euler characteristic implies that S has the same Euler characteristic as S

C

Hence by the classication of surfaces S is determined up to a dieomorphism

C

by S and the fact that C is nonseparating

Lemma A If C C are two nonseparating circles on S then there is a

dieomorphism f S S such that f C C

0

are dieomor Pro of By the remarks preceding the Lemma S and S

C C

phic Let us x an orientation of S it induces an orientation of b oth S and

C

0

Since every orientable surface obviously admits an orientationreversing S

C

dieomorphism we can always choose an orientationpreserving dieomorphism

0

F S S Let C and C b e two b oundary comp onents of S result

C C C

ing from C and let C F C C F C Let I C C and

0

I C C b e two gluing dieomorphisms giving S back from S and S

C C

resp ectively If F agrees with these gluings ie if F I I F on C then F

gives rise by gluing to a dieomorphism f S S such that f C C Note

that b oth I and I are orientationreversing assuming that C C C C are

0

and F is orientationpreserving oriented as b oundary comp onents of S S

C C

on b oth C and C It follows that dieomorphisms F I I F C C

are b oth orientationreversing Since every two orientationreversing as also

every two orientationpreserving dieomorphisms of a circle are isotopic we

can nd a dieomorphism F C C isotopic to F j C and such that

F I I F j C By extending the isotopy b etween F j C and F to

an isotopy of F and without changing F j C we get a dieomorphism

0

which agrees with gluings As we noted ab ove this is su F S S

C C

cient to complete the pro of 2

If C is nonseparating then there is a circle C transversely intersecting C

at exactly one p oint To construct such a C take a small arc I intersecting C

transversely at one p oint in the interior of I and connect the two ends of I by

an arc in S n C In particular the homological intersection number C C is

This immediately implies that C is nontrivial

Lemma B A circle C contractible in S bounds a disc in S A circle C

homotopic to a component B of S bounds together with B an

in S In particular a circle is nontrivial if and only if it does not bound a disc

in S and does not bound an annulus in S together with a boundary component

Pro of First notice that a trivial circle C is always separating by the remarks

preceding the Lemma For such a C we can represent S as the union of two

surfaces S S intersecting only along their common b oundary comp onent C

If C is contractible in S then van Kamp ens theorem together with the basic

prop erties of the amalgamated pro ducts implies that C is contractible in one

of the surfaces S S say in S In view of any embedded lo op with image

C is freely homotopic to a lo op representing d or d for some generator d of

i i

i

S Because C is contractible in S this implies that d Obviously this

i

is p ossible only if the genus of S is and S has only one b oundary comp onent

namely C ie only if S is a disc In this case C b ounds a disc in S namely

S A similar argument applies if C is homotopic to a b oundary comp onent

We leave it to the reader 2

Pants decomp ositions

With few exceptions a surface has negative Euler characteristic by the classi

cation of surfaces The simplest such surface is a disc with two holes often

called a pair of pants or simply pants see Fig a As the next theo

rem shows pairs of pants are the elementary building blo cks of all surfaces of

negative Euler characteristic

i i i i

a b

Fig

Theorem A If S is a surface of negative Euler characteristic ie if S

is not a sphere a a disc or an annulus then there is a collection of

disjoint circles on S such that the result S of cutting S along their union C is

C

a disjoint union of pairs of pants In other words C decomposes S into pairs

of pants

Pro of Using the classication of surfaces one can easily exhibit the required

collection of circles in each case Let us outline a pro of indep endent of the

classication Cho ose a Morse function f S R on S such that f is equal

to on S f is nonnegative and the values of f at dierent critical p oints are

dierent Let a a a b e a sequence of noncritical values of f such

n

that every interval a a i n contains exactly one critical value of f

i i

and Im f a Consider a comp onent of the preimage f a a If it

n i i

do es not contain a critical p oint of f it is an annulus If it contains a critical

p oint of f of index or then it is a disc Finally if it contains a critical p oint

of f of index then it is easily seen to b e a disc with two holes So if we

cut S along the union of the comp onents of all preimages f a i the

i

resulting surface will consist of discs annuli and pairs of pants We can simplify

this decomp osition of S into discs annuli and pairs of pants in the following

manner Note that gluing a disc to a b oundary comp onent of a disc annulus

or a pair of pants results in a sphere a disc or an annulus resp ectively Also

gluing an annulus to a b oundary comp onent of any surface do es not change it

up to a dieomorphism Hence by rep eatedly gluing discs and annuli to other

pieces of the decomp osition we will eventually reach a decomp osition of S ei

ther into several pairs of pants or into two discs glued together in this case

S is a sphere or into an annulus glued to itself in this case S is a torus

or into just one disc or annulus To complete the pro of note that a gluing op

eration corresp onds to removing a circle from our collection of cutting circles 2

This theorem can b e used to give an alternative pro of of the classication of

surfaces Let us cut S along the union of several circles into a nite collection of

pairs of pants as in the theorem Then let us reassemble S from this collection

in the following manner Start with some pair of pants and glue to it the other

pairs of pants one at time and only along one b oundary comp onent at a time

as long as this is p ossible At every stage we will have as one can easily see by

induction a disc with several holes When no further such gluing is p ossible

the remaining gluings identify some pairs of b oundary comp onents of our disc

with holes If there are g such pairs and b b oundary comp onents not in any

such pair then the result of the gluing of these pairs is obviously a sphere with

g handles and b holes ie a surface of genus g with b b oundary comp onents

Sometimes it is useful to decomp ose pairs of pants further Namely three

nontrivial arcs shown in Fig b decomp ose a pair of pants into two top o

logical hexagons which are smo oth manifolds with corners and are homeo

morphic but not dieomorphic to a disc Combining this decomp osition with

Theorem A we can decomp ose S into such top ological hexagons

From the p ersp ective of dimensional top ology nontrivial circles and arcs

are analogues of incompressible surfaces and the ab ove decomp osition of S into

hexagons is an analogue of the Haken decomp osition of a into

top ological balls In fact since pairs of pants are simple enough one rarely

needs to use nontrivial arcs to decomp ose a surface into simpler pieces Note

that from the p oint of view of this analogy all surfaces with the exception of

discs and spheres are Haken

Geometric structures on surfaces

By a geometric structure on a surface S we understand simply a riemannian

metric on S having constant and geo desic b oundary By scaling the

metric if necessary we can assume that its curvature is equal to or

It turns out that every surface admits a geometric structure For surfaces of

nonnegative Euler characteristic this is very simple For a sphere it is sucient

to take the metric of the standard unit sphere in R For a disc it is sucient

to take the metric of a hemisphere of a standard sphere For an annulus one

may take a pro duct metric on S For a torus one may take the metric

on R Z induced from the standard Euclidian metric on R In particular

spheres and discs admit metrics of constant p ositive curvature and annuli

and tori admit metrics of curvature The really interesting case is the case of

surfaces of negative Euler characteristic They all admit hyperbolic structures

ie geometric structures with metric of curvature see Theorem B b elow

Geometric structures on surfaces can b e used on two levels On the rst

level the very existence of a geometric structure is useful For example xing

a geometric structure on a surface S allows us to represent isotopy classes of

circles by geo desics for hyperb olic structures such representatives are unique

Moreover the following lemma holds

Lemma A Suppose that S is endowed with a geometric structure Then

every nontrivial circle on S is isotopic to a circle on S If S is en

dowed with a hyperbolic structure then such a geodesic circle is unique If two

nontrivial circles are disjoint then any two geodesic circles isotopic to them are

either disjoint or equal

We refer to FLP Exp ose xxI and III for a pro of On a second level one

considers all p ossible geometric structures on S simultaneously It is convenient

to identify geometric structures on S which dier one from another in a trivial

way namely to identify two geometric structures if one is the pullback of the

other by a dieomorphism S S isotopic to the identity The resulting set of

equivalence classes is the Teichmuller space of S We explore this p oint of view

in Section

Theorem B If a surface has negative Euler characteristic then it admits

a hyperbolic structure

Pro of There are at least two ways to prove this result The rst one works

b est for closed surfaces S Represent S as a result of the standard identication

of sides of a g gon cf One can realize this g gon as a regular p olygon

on the hyperb olic with all angles equal to g g Note that

the hyperb olic plane has a canonical riemannian metric of constant negative

curvature in the upp er half plane mo del it is the metric dx dy y on

y g this metric induces a riemannian metric on S b ecause fx y R

the lengths of glued edges are equal and the sum of angles is equal to notice

that all vertices of our g gon are identied in the gluing pro cess Clearly this

metric on S has constant negative curvature Note that it is not necessary to

consider a regular g gon a weaker condition would suce For the details of

this construction we refer to St Sections and

Now one can use Lemma A to deal with the surfaces with b oundary

Given a surface with b oundary S we can endow the double dS of S with a

hyperb olic structure Recall that dS is the result of gluing S with a copy of

itself along the b oundary in particular dS S Lemma A implies that S

is isotopic in dS to a union of disjoint geo desic circles If we cut dS along this

union we get two surfaces dieomorphic to S each of them endowed with a

hyperb olic structure

Another construction works equally well b oth for closed surfaces and for

surfaces with b oundary and is based on Theorem A Let us prove rst that

a pair of pants P admits a hyperb olic structure such that all three b oundary

comp onents have the same length To this end we represent P as the union of

two top ological hexagons as in Fig b Let us realize these hexagons as two

isometric regular hexagons in the hyperb olic plane with all angles equal to

Then gluing P from these hexagons endows P with a hyperb olic structure

one should take as the gluing maps this is p ossible b ecause lengths

of edges are all equal the b oundary is geo desic b ecause angles of hexagons are

equal to Clearly the lengths of all b oundary comp onents of P are equal

to twice the length of an edge of our hexagons in particular the lengths of all

b oundary comp onents of P are equal Now given a surface S of negative Euler

characteristic we can decomp ose S into pairs of pants as in Theorem A Let

us endow all pairs of pants of this decomp osition with the hyperb olic structure

we just constructed Since the lengths of all b oundary comp onents of these

pairs of pants are equal one can take isometries as gluing maps and then these

hyperb olic structures on pieces of S induce a hyperb olic structure on S itself

For the details and a more general version of this approach we refer to Bu

Section and Chapter It can b e easily generalized to give a description of

all hyperb olic structures on S see Bu Chapters and and FLP Exp ose

and Section 2

Spaces of dieomorphisms and embeddings generali

ties

For an orientable manifold M we denote by Di M the group of all dieo

morphisms M M and by Di M the subgroup of all orientationpreserving

dieomorphisms Since we are interested mainly in the groups Di M we

chose for them a simpler notation For X M we denote by Di M x X

the subgroup of dieomorphisms f M M xed on X ie such that f x x

for all x X We denote by Di M x X the intersection Di M x X

Di M The most imp ortant case is that of X M we denote the groups

Di M x M and Di M x M by Di M x and Di M x re

sp ectively

If N is a submanifold of M we denote by Emb N M the set of all embed

dings N M which can b e extended to a dieomorphism M M If N has

co dimension in M the main case is that of a disc in the interior of M then

it makes sense to sp eak ab out orientationpreserving embeddings N M and

we denote by Emb N M the subset of all orientationpreserving embeddings

By Sub N M we denote the set of submanifolds of M dieomorphic to N by

a dieomorphism which extends to M ie the set of all images of embeddings

from Emb N M

As usual all dieomorphisms and embeddings are assumed to b e of the

class C and the ab ove groups of dieomorphisms and sets of embeddings

are considered together with their C top ology Sub M N is considered with

the top ology of a quotient space of Emb N M As is well known groups

of dieomorphisms with the C top ology are top ological groups Also for a

submanifold N of M the natural Di M Emb N M given by the

restriction of dieomorphisms to N is continuous

It is easy to see that if M is a surface then Mo d Di S Mo d

S

S

Di S and M Di S x

S

Theorem A Let N be submanifold of M such that N intersects M along

several components of N and N is transverse to M Then the natural map

Di M Emb N M is a Serre bration If N is of codimension in M

then the natural map Di M Emb N M is a Serre bration If N is of

codimension in M and is contained in the interior of M then the natural map

Di M x Emb N M is also a Serre bration

Pro of Recall that a continuous map is a Serre bration if the homotopy lifting

prop erty holds for cub es For Di M Emb N M the homotopy lifting

prop erty for dimensional cub es amounts to the extension of isotopies of N

in M to isotopies of the whole M This extension prop erty is well known and

easy to prove see for example Kos Theorem I I The homotopy lifting

prop erty for higher dimensional cub es is essentially a multiparameter version

of this isotopy extension prop erty and can b e proved by a direct generalization

of the standard pro of the usual isotopy extension prop erty Also the second

statement of the theorem obviously follows from the rst The last statement

of the theorem can b e proved in exactly the same manner as the rst one 2

Theorem B The natural map Emb N M Sub N M Emb N M

Sub N M given by the formula f f N are Serre brations

This theorem can b e proved using a variant of the ideas of the pro of of

Theorem A

The b ers of the brations Di M Emb N M and Di M

Emb N M are easy to identify They are Di M x N and Di M x N

resp ectively In addition if N is a submanifold of co dimension contained in

the interior of M then we can identify Di M x N and Di M x N with

Di M n int N x N and Di M n int N x N resp ectively The b ers of the

maps Emb N M Sub N M Emb N M Sub N M are obviously

Di N and Di N resp ectively

Now let us x a p oint m M and consider the space FM of all orientation

preserving T M T M x M where T M is as usual the

m x x

space to M at x There is an obvious pro jection FM M more

over FM is a principal GL n Rbundle over M where n dim M Let

d Di M FM b e the map assigning to a dieomorphism M M its

dierential at m If N is a submanifold of co dimension and m N then

there is a similar map d Emb N M FM

Theorem C The maps d Di M FM and d Emb N M FM

are Serre brations If N is a disc of codimension in the interior of M then

the second map is a weak homotopy equivalence

Pro of We may consider p oints of FM as framed p oints in M Then the ho

motopy lifting prop erty for d b ecomes a version of the multiparameter isotopy

extension prop erty for isotopies of p oints which is not much more dicult to

prove the usual isotopy extension prop erty The second statement of the theo

rem is essentially a multiparameter version of a well known theorem to the eect

that there is exactly one isotopy class of orientationpreserving embedings of a

co dimension disc in an orientable manifold See for example Kos Corollary

I I I Again it is easy to add parameters to the usual pro of 2

If dim M it convenient to x a riemannian metric on M and consider the

unit tangent bundle UT M instead of FM Let us x a vector v T M Then

m

the evaluation at v followed by the normalization denes a map FM UT M

which is a homotopy equivalence if dim M Also dene u Di M

UT M as the map assigning to a dieomorphism M M the normalized image

of v under its dierential at m Similarly dene u Emb N M UT M

Theorem D The maps u Di M UT M and u Emb N M

UT M are Serre brations If N is a disc of codimension in the interior of

M and dim M then the second map is a weak homotopy equivalence

Pro of The rst statement is similar to the rst statement of Theorem C

and in fact holds in all The second statement follows from the

second statement of Theorem C and the fact that the ab ove map FM

UT M is a homotopy equivalence 2

The results of this section go back to J Cerf Ce and for closed manifolds

R Palais P In fact they proved that our Serre brations are actually b er

bundles But we need only the Serre bration prop erty b ecause we are inter

ested only in the homotopy groups of spaces of dieomorphisms and embeddings

in fact only in and and this prop erty is much easier to prove

Spaces of dieomorphisms and embeddings surfaces

In this section we describ e the weak homotopy type of some spaces of dieo

morphisms of surfaces and embeddings of circles While all these results are

known to b e true for the usual homotopy type instead of the weak one the

results ab out the weak homotopy type are easier to prove in particular there

is no need to know that Serre brations from are b er bundles and only

they are needed for the applications we have in mind

We start with two preliminary results ab out dimensional manifolds

Lemma A Di D x is contractible where D is the dimensional

disc

Pro of We may assume that D Now if f is a

dieomorphism xed on f g then x tf x tx is a dieomorphism

for every t b ecause its derivative is always 2

Corollary B Di S is homotopy equivalent to S and moreover con

tains S as a deformation retract

Pro of Note that S is naturally contained in Di S as the group of rota

tions Fix x S and consider the Serre bration Di S Emb fxg S

Its base can b e obviously identied with S and S Di S denes a sec

tion Using this section it is easy to see that this Serre bration is actually a

b er bundle If we notice that its b er is homotopy equivalent to Di D x

the corollary follows 2

The rst basic result ab out the groups of dieomorphisms of surfaces is the

following theorem of S Smale Sm

Theorem C Di D x is contractible where D is the dimensional

disc

In that follows we understand by a weak deformation retract of a top ological

space X a subspace Y X such that the inclusion Y X is a weak homotopy

equivalence The following corollary is also due to S Smale Sm who actually

proved it with all adjectives weak omitted

Corollary D Di D is weakly homotopy equivalent to a circle Di S

is weakly homotopy equivalent to SO and moreover contains SO as a

weak deformation retract

Pro of Consider the Serre bration Di D Di D Its base is

homotopy equivalent to S by Corollary B and its b er is homotopy equiv

alent to a p oint by Theorem C The rst statement follows Now x

a disc D S and consider the bration Di S Emb D S By

Theorem D its base Emb D S is weakly homotopy equivalent to the

unit tangent bundle of S which can b e identied with SO The b er is

Di S x D which can b e identied with Di S n int D x D But

S n int D is a disc hence by Theorem C the b er is contractible It follows

that Di S is weakly homotopy equivalent to SO Moreover the bration

u Di S UT S SO from is a weak homotopy equivalence

The inclusion of SO in Di S as the subgroup of rotations denes a sec

tion of this bration Clearly the image of this section ie SO is a weak

deformation retract of Di S 2

Corollary E Let A be an annulus ie a manifold dieomorphic to S

Then Di A x has weakly contractible components and the group of

components Di A x is isomorphic to Z

Pro of Consider the bration Di D x Emb B D where D is a

disc and B is a disc in the interior of D Its total space is contractible by

Theorem C By Theorem D its base is weakly homotopy equivalent to the

unit tangent bundle of D and hence to S Its b er is Di D n int B x

which can b e identied with Di A x The corollary easily follows from

these facts and the homotopy sequence of our bration 2

Theorem F Let T S S be a torus The components of Di T are

weakly homotopy equivalent to T and in fact contain T as a weak deformation

retract

Note that since T is a T is naturally contained in Di T In fact

this subgroup T Di T is a weak deformation retract of the comp onent of

the identity of Di T

Theorem G If S is a surface of negative Euler characteristic then the

components of Di S x and of Di S are weakly contractible

Theorems F and G in fact with adjectives weak omitted are

due to C Earle and J Eells EE The pro ofs in EE are analytical and are

based on the theory of Beltrami equations The corresp onding results ab out

spaces of are due to ME Hamstrom Ham Ham her

pro ofs are purely top ological The corresp onding results in the piecewise linear

were obtained by G P Scott Sco Nowadays the easiest way to

obtain Theorems F and G is probably to consider them as dimensional

versions of dimensional results of A Hatcher Hat and the author I ab out

Haken manifolds cf end of Unfortunately such an approach is not written

down yet

Theorem H Let S be a surface of negative Euler characteristic and let

C be a nontrivial circle on S Then the components of Emb C S are weakly

homotopy equivalent to a circle and the components of Sub C S are weakly

homotopy equivalent to a point

This theorem can b e deduced from the ab ove results ab out spaces of dif

feomorphisms using brations from Alternatively if one proves Theorems

F and G along the lines of theorems of Hat and I ab out Haken

manifolds then this theorem is essentially the main step of the pro of

Finally we give an application of these theorems which will b e needed later

see

Theorem I Let R be a connected subsurface of a connected surface S of

negative Euler characteristic Suppose that the boundary of R is connected and

nontrivial in S Then the natural map M M given by the extension of

R S

dieomorphisms of R xed on R by the identity to dieomorphisms of S is

injective

Pro of Let T b e the closure of S n R Clearly T is connected The natural

map Di S x Emb T S is a Serre bration with b er Di R x The

map Di R x Di S x induced by the inclusion of the b er

in the total space is nothing other as our map M M In view of the homo

R S

topy sequence of this bration this map will b e injective if Emb T S

Note that since R is nontrivial T has negative Euler characteristic and hence

Di T Let us consider now the natural map Emb T S Sub T S

from Theorem B which is a Serre bration with b er Di T In view of the

homotopy sequence of this bration Emb T S if Sub T S

But a submanifold of S dieomorphic to T is determined by its b oundary

may b e up to two p ossibilities if R is dieomorphic to T It follows that

Sub T S Sub T S The latter group is in view of Theorem

H This completes the pro of 2

Gluing discs to a surface

If a surface has nonempty b oundary then we can construct a closed surface or

a surface with a smaller number of b oundary comp onents by gluing discs to

b oundary comp onents Very often this gluing pro cedure allows us to reduce

theorems ab out surfaces with p ossibly nonempty b oundary to the case of closed

surfaces or use the induction on the number of b oundary comp onents In this

section we discuss some basic constructions and results used in such arguments

Let R b e a surface with nonempty b oundary and let Q b e the result of

gluing discs D D to n dierent b oundary comp onents of R Given a

n

dieomorphism f R R preserving setwise all comp onents of R there exists

an extension f Q Q of f Such an extension is obviously not unique but it

is unique up to isotopy b ecause the spaces Di D x are connected Hence

i

extension of dieomorphisms from R to Q denes a map PMo d PMo d

R Q

Clearly is a homomorphism We are interested in particular in its kernel and

to understand it we need the notion of the pure braid groups of surfaces and

the prop erties of these groups discussed b elow

m m

For any surface S consider S fy y S y y for i j g

m i j

m

The pure PB S is dened as S where we tacitly assume

m

m

that some base p oint x x S have b een chosen

m

Let S S n fx x x x g note that S is a noncompact

i i m

i i

m

surface Let x b e the base p oint of S The map S S given by

i

i i

the formula x x x x x x induces a map S x

i i m i

i

m

S PB S

m

Theorem A The group PB S is generated by the images of the above

m

m

maps S x S PB S

i m

i

l l

Pro of Use induction on m and the natural brations S S given by

the formula y y y y 2

l l

We are mainly interested in the group PB Q We will assume that the

n

n

base p oint x x Q has b een chosen in such a way that x int D

n i i

i n Let Q b e the result of gluing only the disc D to R Clearly Q is

i i i

contained in Q and hence natural maps Q x PB Q are dened In

i i n

i

fact Q is a deformation retract of Q

i

i

Corollary B The group PB Q is generated by the images of the natural

n

maps Q x PB Q

i i n

Now we construct a homomorphism j PB Q PMo d Let

n R

n

PB Q represent by a lo op in Q Such a lo op may b e considered as

n

an isotopy of the dimensional submanifold fx x g of Q Let us extend

n

this isotopy to an isotopy ff Q Qg of the whole manifold Q in par

t t

ticular f id f x x for all i We can easily arrange that in addition

Q i i

f D D for all i It follows from Theorem C that the space of discs

i i

in a surface containing a given p oint in the interior is connected Then f

preserves R and preserves all b oundary comp onents of R We take as j the

isotopy class of the restriction of f on R One can easily check that j is cor

rectly dened using Theorem C again We leave it as an exercise to check

that j is a homomorphism If n then PB Q PB Q is nothing more

n

than the Q and j is a homomorphism Q PMo d

R

The rst main prop erty of j is contained in the following theorem

Theorem C The homomorphism is surjective and Ker Im j in other

words the sequence

PB Q PMo d PMo d

n R Q

is exact If Q has negative Euler characteristic then j is injective in other

words the sequence

PB Q PMo d PMo d

n R Q

is exact

Pro of Let us prove the surjectivity of PMo d PMo d rst Represent

R Q

an element of PMo d by a dieomorphism g Q Q of

Q

the spaces of orientationpreserving embeddings of discs into a surface easily

implies that g is isotopic to a dieomorphism g such that g D D i

i i

n Let b e the isotopy class of the restriction of g to R Clearly

This proves the surjectivity of

Now let Ker Let us represent by a dieomorphism f R R

and extend f to a dieomorphism f Q Q We may assume that f x

i

x i n Since Ker the dieomorphism f is isotopic to the

i

identity id Let ff Q Qg b e some isotopy b etween f f and

Q t

t

n

x denes a lo op in Q with the base x f id Then t f f

n Q

t t

p oint x x and hence an element Q PB Q Obviously

n n

n

j This proves that Ker Im j The opp osite inclusion follow directly

from the denitions

Supp ose now that the Euler characteristic of Q is negative Let Ker j

n

Let us represent by a lo op in Q and construct an isotopy ff Q Qg

t t

as in the denition of j In particular f D D for all i Since Ker j

i i

the restriction of f on R is isotopic to id It follows that after changing the

R

isotopy if necessary we may assume that the restriction of f on R is equal to

id Next in view of Theorem C we may assume that f id Now

R Q

ff Q Qg denes a lo op in Di Q By Theorem G this lo op is

t t

contractible Clearly this implies that the original lo op representing is con

tractible and hence 2

This Theorem is due to J Birman Bir See also Bir Theorem In

general Ker j is isomorphic to the quotient of the group PB Q by its center

n

which can b e easily identied see Bir Chapter

Now let Mo d Supp ose that some and then every dieomorphism

R

f R R representing preserves setwise the union of circles D

i

i n Then f can b e extended to a dieomorphism f Q Q Such

extension is unique up to isotopy cf the construction of We may assume

that f x x for some p ermutation f ng f ng Extension

i

i

f is unique up to isotopy even in the class of such dieomorphisms this follows

from the fact that discs are connected and the results of Such an extension

n n

f induces a map Q Q mapping the base p oint fx x g to

n

fx x g In particular f induces a homomorphism of the

n

fundamental groups Q Q It is easy to see that this

n n

homomorphism dep ends only on By a slight abuse of notations we will

denote it even if id by PB Q PB Q keeping in mind that the

n n

second pure braid group is dened using a dierent base p oint

Note that obviously maps the image of Q x in PB Q to the

i i n

image of Q x

i i

The homomorphisms and j are related by the following fundamental

formula

j j

which follows immediately from the constructions of and j

As the last application of the extension of dieomorphisms from R to Q con

sider the situation when only one disc D is added to R Then the extension of

dieomorphisms denes a map Mo d Di Q x x where Mo d is the

R R

subgroup of Mo d consisting of isotopy classes of dieomorphisms preserving

R

D It turns out that this map is an Again this follows from

The DehnNielsenBaer Theorems

For a group let Aut b e the group of automorphisms of and let Inn

b e the subgroup of inner automorphisms of A trivial check shows that Inn

is a subgroup of Aut The quotient group Aut Inn is denoted

by Out and called the outer automorphisms group of These groups natu

rally show up when one deals with homotopy selfequivalences of a connected

space X with X Namely any selfmap f X X denes a homo

f X X and if f is a homotopy equivalence then f is

an isomorphism But if we do not assume a base p oint to b e xed f is well

dened only up to an inner of X Hence a homotopy

equivalence f X X denes an element f Out Obviously f dep ends

only on the homotopy class of f

Now let X S b e a compact connected orientable surface In view of the

ab ove remarks there is a natural homomorphism Mo d Out S If S

S

has nonempty b oundary then the elements of the image of this homomorphism

obviously preserve the p eripheral structure of S from

Theorem A If S is closed and is not a sphere then the natural homomor

phism Mo d Out S is an isomorphism If S has nonempty boundary

S

and negative Euler characteristic ie S is not a disc or an annulus then the

natural homomorphism Mo d Out S is an isomorphism onto the sub

S

group of Out S consisting of elements preserving the peripheral structure

One can complement this theorem by a description of the subgroup of

Out S corresp onding to Mo d Supp ose rst that S is closed and is not a

S

sphere For a review of the cohomology of groups used b elow see Then

S is an EilenbergMacLane space and hence H S Z H S Z Z

The group Out S naturally acts on H S Z b ecause inner automor

phisms act trivially on cohomology groups Let Out S b e the subgroup of

Out S consisting of elements acting trivially on H S Z Clearly the

image of Mo d under the homomorphism Mo d Out S is contained in

S

S

Out S If S has nonempty b oundary and negative Euler characteristic

then the image of Mo d in Out S is obviously contained in the subgroup of

S

Out S consisting of elements preserving the oriented p eripheral structure

from

Theorem B If S is closed and is not a sphere then the natural homomor

phism Mo d Out S is an isomorphism If S has nonempty boundary

S

and negative Euler characteristic then the natural homomorphism Mo d

S

Out S is an isomorphism onto the subgroup of Out S consisting of

elements preserving the oriented peripheral structure

For closed S the surjectivity part of these theorems is due to M Dehn who

did not publish his pro of and to J Nielsen Ni who published a pro of partially

based on Dehns ideas Another pro of was suggested by H Seifert SeiThe

injectivity part is due to R Baer Ba The surjectivity part for surfaces with

b oundary was proved by W Magnus M and the injectivity part much later

by H Zieschang Zi Pro ofs and a detailed discussion of these results can

b e found in ZiVC see ZiVC Theorem surjectivity for closed surfaces

following Seifert Theorem surjectivity for surfaces with b oundary and

Theorem injectivity

In principle these results give a purely algebraic description of mapping class

groups and allow one to reduce every question ab out them to a purely algebraic

question Surprisingly this reduction turned out to b e not very successful

Probably the only algebraic prop erty of the mapping class groups which for a

long time could b e proved only in terms of the outer automorphisms groups is

the residually niteness

Theorem C The groups Mo d are residually nite ie for every nontriv

S

ial element f Mo d f there is a homomorphism h Mo d G onto a

S S

nite group G such that hf

This theorem is due to E Grossman Gros whose pro of was based on the

description of mapping class groups in terms of outer automorphisms groups and

fairly complicated combinatorial arguments A more conceptual

pro of still based on the description in terms of outer automorphisms groups was

given by H Bass and A Lub otzky BasL Now a simple direct pro of not based

on the results of DehnNielsenBaer is available see I or I Exercise

In one case namely if S is a closed torus the DehnNielsenBaer theorems

provide a complete and ecient description of Mo d and Mo d In this case

S

S

S is ab elian and moreover is isomorphic to Z and hence Out S

Aut S Aut Z GL Z This chain of isomorphisms clearly maps

the subgroup of Out S to SL Z It follows that Mo d GL Z and

S

Mo d SL Z for a closed torus S

S

Complexes of curves

Complexes of curves are probably the most fundamental geometric ob jects on

which the mapping class groups act but the Teichmuller spaces are certainly

the most imp ortant They were discovered by W Harvey Harv Harv

and have played an ever increasing role since then In this section we give the

denition and discuss the basic prop erties of them We prove that complexes

of curves are connected and do this in a way which can b e generalized to

prove much stronger homotopy prop erties of them After this we discuss the

homotopy type and the hyperb olic prop erties of them Later sections will amply

illustrate the usefulness of complexes of curves

Denitions

Complexes of curves are simplicial complexes in the sense of Sp Chapter or

Bro for example Thus a simplicial complex consists of a set of vertices and

a set of simplices Simplices are non empty nite sets of vertices sub ject only

to the following two conditions a non empty subset of a simplex is a simplex

every vertex b elongs to some simplex The dimension of a simplex is the number

of vertices in it minus One dimensional simplices play a sp ecial role and are

called edges

The vertices of the complex of curves C S of a compact orientable surface

S are the isotopy classes of simple closed curves as usual we call them circles

on S which are nontrivial in the sense of ie not contractible in S into

a p oint or into the b oundary S We denote the isotopy class of a circle C

by hC i A set of vertices f g is declared to b e a simplex if and only

n

if hC i hC i for some pairwise disjoint circles C C The

n n n

extended mapping class group Mo d acts on C S in an obvious way the

S

isotopy class f Mo d of a dieomorphism F S S maps hC i to hF C i

S

C S can b e dened in another way According to we can equip S with

a riemannian metric of constant curvature with geo desic b oundary Then we

can take as the set of vertices of C S the set of geo desic circles in S n S and

as simplices the sets of pairwise disjoint geo desic circles In view of Lemma

A this denition is equivalent to the previous one This denition makes

clear that a set of vertices is a simplex if and only if all its element subsets are

simplices ie any two its vertices are connected by an edge In the language of

the theory of buildings cf Bro for example this means that C S is a ag

complex Thus C S is in fact completely determined by its skeleton Higher

dimensional simplices do not contribute any new combinatorial information but

they make the of C S much more relevant and interesting

Sp eaking ab out the homotopy prop erties of C S we of course have in

mind the homotopy prop erties of its geometric realization cf Sp which is a

top ological space and even a CWcomplex

Obviously C S is an innite complex It is even locally innite when it

has p ositive dimension every vertex is connected by an edge with an innite

number of vertices But it is nitely dimensional its dimension is equal to

the maximal number of pairwise nonisotopic disjoint nontrivial circles on S

minus It follows that if S is a connected orientable surface of genus g with b

b oundary comp onents then the dimension of C S is equal to g b except

in the case g and b when the dimension is zero if this formula leads

to a negative number the complex of curves is empty

Connectivity of C S and the MorseCerf theory

Before discussing other homotopy prop erties of C S we will present a pro of

of the connectedness of C S This pro of can b e generalized to give the b est

p ossible connectivity result for closed surfaces S and surfaces S with one b ound

ary comp onent cf Theorem C the case of surfaces with more b oundary

comp onents can b e reduced to the case of closed surfaces in at least two dif

ferent ways cf It is based on some elementary facts from the theory of

singularities more precisely on a generalization of the to families

of functions due to J Cerf Ce Ce Namely we need the following result

Lemma A Any family of functions ff S Rg can be approxi

t t

mated say in the C topology by a family of functions fg g such that

t t

al l functions g belong to one of the fol lowing three classes

t

i Morse functions with al l critical values ie values at critical points

dierent

ii Morse functions with two equal critical values and al l other critical values

dierent from them and from each other

iii functions having al l critical values dierent and exactly one nonMorse

critical point such that in appropriate local coordinates x y around this critical

point the function has the form x y c for some constant c R

This lemma was originally proved in Ce Chapter I I and has b ecome well

known since then The pro of shows that one can also require that all but nitely

many functions g b elong to the rst class but we do not need this fact For

t

a general introduction to the relevant ideas of the theory of singularities one

can recommend a nice short b o ok by T Poston and I Stewart PoS cf PoS

Chapters

Theorem B Suppose that S is neither a sphere with holes nor a torus

with holes Then C S is connected

Pro of If S is a torus with two holes it is easy to see directly that C S is

connected We leave this as an exercise to the reader with the following hints

there are two types of nontrivial circles namely non separating circles and

circles separating S into a torus with one hole and a disc with two holes the

circles of the rst type are never disjoint but any two of them can b e made by

an isotopy disjoint from a circle of the second type

In the remaining part of the pro of we assume that S is neither a sphere with

holes nor a torus with holes

Let hD i hD i b e two vertices of C S We need to prove that

there is a sequence of vertices such that is connected by

n i

an edge with or is equal to for all i n

i i

To b egin with let us choose two smo oth functions f f S R such that

D resp ectively D is a comp onent of a level set f a of f resp ectively of

f and moreover f and f have no critical p oints on D and D resp ectively In

addition we may assume that b oth f and f are Morse functions having dierent

critical values Clearly we can connect f and f by a path ff S Rg

t

t in the space of smo oth functions f f f f

Let fg g b e some approximation of ff g provided by Lemma

t t t t

A Since g is C close to f some comp onent C of the level set g a is

a circle close and isotopic to D In particular this circle b elongs to the isotopy

class Similarly some comp onent C of some level set of g b elongs to the

isotopy class The following claim provides similar comp onents of level sets

for all functions g

t

Claim If a function g S R b elongs to one of the three classes of

Lemma A then some comp onent of some level set g r of g contains no

critical p oints and is a nontrivial circle

In order to prove this claim let us choose a critical p oint x of g which is

not a lo cal maximum or minimum such a critical value exists b ecause S is not

a sphere or a disc Let c g x and let L b e the comp onent of the level set

g c containing x Cho ose an such that c is the only critical value in

c c Then exactly one comp onent of the set g c c contains

L Let us denote this comp onent by L If one of the b oundary comp onents

of L is nontrivial then we are done Supp ose that all of them are trivial ie

b ound in S either a disc or an annulus together with a b oundary comp onent

of S

If one of these discs or annuli contains L then we replace g by a function g

equal to g outside this disc or annulus X and having only one Morse critical

p oint on it if it is a disc or no critical p oints if it is an annulus We may assume

that if g has a critical p oint in X ie if X is a disc then the corresp onding

critical value is dierent from all other critical values of g Note that if X is a

disc then g has at least two critical p oints on X the p oint x and a minimum

a b c

Fig

or a maximum and g has only one If X is an annulus then g has at least

one critical p oint on X namely the p oint x and g has none In b oth cases

g has fewer critical p oints than g If some comp onent of some level set of g

is nontrivial then the same is true for g itself b ecause all comp onents of level

sets of g contained in X are trivial So in this case our problem is reduced

to a similar problem for the function g which has fewer critical p oints than g

Since g obviously b elongs to one of three classes of Lemma A we can use

induction

If none of these discs or annuli contain L then S is equal to the union of

L with these discs or annuli It follows that the Euler characteristic S is

equal to L d where d is the number of the discs Clearly L is a

deformation retract of L and hence S L d But L if L

contains one Morse critical p oint L if L contains two Morse critical

p oints and L if L contains a nonMorse critical p oint in the last case

L do es not contain any other critical p oints and is a top ological circle with a

cusp at the critical p oint The various p ossibilities for L are presented in Fig

It follows that S in this case But any such S is either excluded

by the assumptions of the theorem or is a torus with two holes which is also

excluded now This completes the pro of of the claim

Now we return to the pro of of the theorem The ab ove claim implies that

for any t t the function g has a comp onent C of a level set g a

t t t

t

such that C do es not contain any critical p oint of g and is a nontrivial circle

t t

We may assume that C C and C C If u is suciently close to t say u

b elongs to a neighborho o d U of t then the level set g a of the function g

t t u

u

has a comp onent C close and isotopic to C b ecause g has no critical p oints on

tu t t

C The family of neighborho o ds fU g forms an op en cover of the interval

t t t

By using a well known Leb esgue lemma we can divide the interval

by several p oints x x x into intervals x x such that

n i i

We any interval x x i n is contained in a neighborho o d U

i i t

i

may assume that t and t

n

i We claim that is the required sequence of ver Let hC

n i t

i

tices By our choice and Let us prove that is connected

n i

by an edge with for i n Let i n and let x x

i i

y x z x and v t w t Note that C is isotopic to C

i i i i v v y

b ecause y x y U and C is isotopic to C b ecause y y z U

v w w y w

i hC i hC i But i hC i hC i and hC Hence hC

w w y v v y i t i t

i+1 i

C and C are comp onents of the level sets of the same function g Hence

v y w y y

they are either equal or disjoint and their isotopy classes and are either

i i

equal or connected by an edge This completes the pro of 2

This theorem for closed surfaces S was originally deduced by the author I

from the results of A Hatcher and W Thurston HatT which were also based

on the ideas of the MorseCerf theory Later on the metho d was streamlined

and extended to cover the higher connectivity of C S cf the discussion b e

low In the meantime J Harer Har Har proved that C S is homotopy

equivalent to a b ouquet of spheres all having the same dimension and com

puted this dimension giving in particular the b est p ossible estimate of the

higherconnectedness Cf

In Ce J Cerf also proved a version of Lemma A for parameter families

of functions Here some new types of functions app ear such as Morse functions

with three equal critical values or functions having critical p oints of the form

x y c Using this version of Lemma A one can mo dify the pro of

of Theorem B and prove the simply connectivity of C S in most cases

Instead of a sub division of the interval into subintervals one should use a

suciently ne of a disc

The MorseCerf theory is based on a complete classication of p ossible sin

gularities of functions in generic families Such a classication is unknown for

families with suciently many parameters But in fact one can carry out the

main arguments of the pro of of Theorem B without complete classication

One needs to prove only some restrictions on the p ossible complexity of sin

gularities of functions in generic families cf I Lemmas and This

program is carried out in I Sections and and leads to the following result

Theorem C If S is a closed surface respectively a surface boundary

component a surface with boundary components then the geometric re

alization of C S is S g connected respectively S

connected S connected where S is the Euler characteristic of S

This result is not the b est p ossible for surfaces with b oundary comp o

nents the b est is the S connectedness but the closed case is sucient

for the main applications cf for example and in fact the general case

can b e reduced to it

The homotopy type of the complexes of curves

The following theorem gives an almost complete description of the homotopy

type of C S

Theorem A The geometric realization of C S is homotopy equivalent to a

bouquet of spheres of dimension S if S is closed and of dimension S

if S has non empty boundary where S is the Euler characteristic of S

This theorem is due to J Harer Har His pro of is based on a connec

tion of the complexes of curves with ideal triangulations of Teichmuller spaces

cf and a combinatorial argument used to give an upp er estimate of the

homotopy dimension of C S ie of the minimal dimension of a CWcomplex

homotopy equivalent to it This upp er estimate in fact matches the connec

tivity results and together they imply that C S is homotopy equivalent to a

b ouquet of spheres Another pro of based on Theorem C a duality b etween

the prop erties of C S and Mo d cf and and Harers combinatorial

S

argument giving an upp er estimate of homotopy dimension was suggested by

the author I We refer to I for a detailed presentation of this pro of and a

discussion of its relations with Harers approach cf I Section

Both approaches use an induction on the number of b oundary comp onents

reducing in fact the general case to the closed one Harers inductive argument

Har is made on the geometric level of complexes of curves He shows that

making a hole in a surface with non empty b oundary increases the connectivity

of C S by making the rst hole is more complicated In I this inductive

argument is made on the algebraic level of the mapping class groups where

the exact sequences of together with theory of groups with duality BieE

provide a convenient to ol An improved version of this argument is used in the

pro of of Theorem B

The following table summarizing the dep endence of the homotopy dimension

hdim C S and the usual dimension dim C S of C S on the genus g and the

number b of b oundary comp onents of S is often quite useful

g b hdim C S dim C S

g b

g b b b

g b

g b b b

g b g g

g b g b g b

Corollary B Suppose that S is connected and has non empty boundary

Then the structure of the simplicial complex C S considered up to isomor

phism determines if the genus of S is and the whole topological type of S

if it turns out that the genus of S is

Pro of The table shows that hdim C S dim C S if and only if the genus

is or the genus is and the b oundary is empty The last line of the table

allows us to nd g and b if we know already that g and b 2

Hyp erb olic prop erties

Now we turn from the homotopy prop erties to the geometry of complexes of

curves Let C S b e the skeleton of C S ie the graph with the same

vertices and edges as C S As we noted in C S is a ag complex and

hence C S completely determines the whole structure of C S

Any graph or rather its geometric realization has a canonical structure of

a metric space which is constructed as follows any edge is made isometric to

the interval and then the distance b etween two p oints is dened to b e the

inmum of the lengths of paths connecting them Clearly the distance b etween

two vertices is equal to the length n of the shortest chain

n

such that is connected by an edge with compare the pro of of Theorem

i i

B This structure of a metric space turns any graph into a geodesic space

This means that any two p oints x y can b e connected by an isometric image of

an interval in the R Any such image is called a geodesic and is denoted

by xy despite the fact that it is usually not unique

A geo desic metric space X is called hyperbolic where is some real

number if for any three p oints x y z X any geo desic xz is contained in a

neighborho o d of the union xy y z A geo desic metric space is called hyperbolic

if it is hyperb olic for some These notions are due to M Gromov Gro

and serve as the starting p oint of his theory of hyperb olic spaces and groups

The main examples are provided by the classical hyperb olic spaces innite

trees ie graphs without cycles and Cayley graphs of fundamental groups of

closed negatively curved manifolds For trees one can take Note also

that any geo desic metric space of nite diameter D is trivially hyperb olic with

D

Theorem A If C S is connected then its geometric realization is a hy

perbolic space of innite diameter

This remarkable and completely unexp ected theorem was recently proved

by H Masur and Y Minsky MasM An overview of the pro of is provided

by Y Minsky Min The replacement of C S by C S is not signicant we

can turn the geometric realization of C S into a metric space by making every

simplex b e isometric to a regular simplex with edges of length Then the

resulting metric space will also b e hyperb olic This easily follows from Theorem

A and basic prop erties of hyperb olic spaces It simply seems that it is more

convenient to work with C S in this context

Even the fact that C S is of innite diameter is nontrivial and interesting

It means that for any natural number N there are two vertices of C S which

cannot b e connected by a chain of edges shorter than N This seems to b e

nearly obvious but apparently was not proved b efore MasM

The fact that C S is not lo cally nite presents a serious obstacle if one

attempts to use Theorem A along the lines of M Gromovs theory of hyper

b olic groups and spaces Gro In MasM H Masur and Y Minsky developed

some to ols to overcome this obstacle As a rst application to the mapping class

groups they proved the following theorem ab out pseudoAnosov elements for

the denition see of the mapping class groups

Theorem B Let us x a nite set of generators of Mo d and let us denote

S

by j j the minimal word length with respect to these generators If f f

W

are two conjugate pseudoAnosov elements of Mo d then f g f g for an

S

element g Mo d such that

S

j g j C j f j j f j

W W W

where the constant C depends only on S and the generating set

To put this theorem into prop er p ersp ective one should note that G Hemion

He proved that the conjugacy problem for the mapping class group is algo

rithmically solvable PseudoAnosov elements are in a denite sense typical

elements of Mo d and L Mosher Mos gave an explicit algorithm for deter

S

mining the conjugacy for pseudoAnosov elements In b oth He and Mos

no explicit b ounds on the time required by the algorithm was given Theorem

B obviously implies an explicit b ound for the conjugacy problem b ecause it

provides a b ound on the word length of a conjugating element

Dehn twists generators and relations

The main examples of nontrivial elements of mapping class groups and also the

main building blo cks in various constructions of other elements are the socalled

Dehn twists They were introduced for the rst time by M Dehn D and then

played a central role in his fundamental pap er D This section starts with a

denition and some basic prop erties of Dehn twists and then pro ceeds to one

of their main applications namely to their role as generators of mapping class

groups The material of this section is crucial for developing an intuitive feeling

of the mapping class groups

In this section S is always assumed to b e a compact oriented surface

Dehn twists

The goal of this section is to dene Dehn twists and to prove their basic prop

erties Dehn twists are by denition the isotopy classes of some sp ecial dif

feomorphisms called twist dieomorphisms We b egin with a discussion of the

latter

We start with a description of a standard twist dieomorphism of an annulus

Let A b e the annulus in R given by the inequality r in the standard

p olar co ordinates r in R Its b oundary A consists of two comp onents

A A dened by the equations r r resp ectively Let us x a

I T

Fig

smo oth function R R such that x for x x for

x and the derivative is Let us dene a dieomorphism

T A A by the formula T r r r in our p olar co ordinates r

We call T the standard twist dieomorphism of the annulus A It is easy to

see that up to an isotopy xed on the b oundary A the dieomorphism T

do es not dep end on the choice of Fig illustrates the action of T In

fact up to an isotopy xed on the b oundary T is the unique dieomorphism

xed on the b oundary and taking the radial arc I in Fig a into an arc

isotopic to the arc in Fig b Using the fact that Di I x I is connected

any two such dieomorphisms can b e made equal on I by an isotopy By an

additional small isotopy these two dieomorphisms can b e made equal on a

neighborho o d N of I A Consider a disc D contained in A with the b oundary

contained in N Since Di D x is connected by Theorem C any two

such dieomorphisms can b e made equal by an isotopy So Fig denes T

up to an isotopy xed on A

Now let e A S b e an orientationpreserving of A into our

surface S We may use e to transplant T from A to S as follows take the

dieomorphism e T e eA eA and extend it by the identity to

a dieomorphism T S S We call any such dieomorphism T a twist

e e

dieomorphism of S Clearly up to isotopy xed on S the dieomorphism T

e

dep ends only on the isotopy class of the embedding e The isotopy class of an

embedding e A S is determined by the isotopy class of the oriented image

ea of the oriented say counterclockwise axis a fr r g of the

annulus A and by which b oundary comp onent of A is mapp ed to the right side

of ea the notion of the right side is determined by the orientations of ea

and S So given the isotopy class of the unoriented image ea there are four

p ossible isotopy classes of e but only two of them are orientationpreserving In

fact if e is orientationpreserving then e has to map A to the right of ea

It is easy to see draw a picture that b oth orientationpreserving embeddings

e with the same isotopy class of ea lead to isotopic dieomorphisms T

e

In particular up to isotopy xed on S the dieomorphism T dep ends

e

only on the image C ea of the axis a of A and even only on its isotopy

class hC i By a slight abuse of the language we call this dieomorphism

a twist dieomorphism about C Clearly any twist dieomorphism preserves

orientation and is xed on the b oundary of S We call the isotopy class of

a twist dieomorphism ab out a circle C the about C Since this

isotopy class dep ends only on the isotopy class hC i of C we usually denote

it by t and call it also the Dehn twist about There is an ambiguity in these

notations and terminology namely one may consider only isotopies xed on

the b oundary and then t M or one may consider all isotopies and then

S

t PMo d The context usually resolves this ambiguity

S

The Dehn twists we dened are often called the left Dehn twists and the

elements of the form t are called the right Dehn twists Similarly one may

sp eak ab out the left and the right twist dieomorphisms This terminology is

justied rst by the fact that the right twist dieomorphisms and Dehn twists

can b e constructed in a manner completely similar to the construction of the left

ones but using orientationreversing embeddings A S instead of orientation

preserving ones we leave this as an exercise Second one can distinguish

b etween the right and the left twist dieomorphisms in the following way Let

e A S b e an embedding and T b e the corresp onding left or right twist

e

dieomorphism If J is an arc in S transversely intersecting C e A then

as we approach the annulus eA along T J and pass the circle C we will b e

e

turning left if T is a left twist dieomorphism and we will b e turning right if T

e e

is a right twist dieomorphism Of course the meaning of turning left or right

is derived from the orientation of S A useful exercise is to convince yourself

that we will turn to the same direction if we approach eA from the other side

crossing C e A Curiously enough M Dehn himself D D did not

distinguished b etween left and right twists in the notations or otherwise but

always describ ed explicitly which twist he deals with

Let us consider now twist dieomorphisms ab out trivial circles If ea

b ounds a disc D in S we may assume that D contains A and not A

replacing if necessary e by e i where i A A is dened by the formula

ir r clearly ea e ia but e and e i b elong to dierent

isotopy classes Then we can extend e to an embedding e D S of the disc

D fr r g such that e D D where D fr r g

D D g of D dened by We can use e to transplant the isotopy fT

u

u

the formula T r r r u r from D to S in the same

u

manner as we transplanted T from A to S Clearly the isotopy fT g is

u

u

xed on the b oundary D A and connects the extension T of T to D by

note that r r Hence our trans the identity with T id

D

2

planted isotopy connects T with the identity dieomorphism of S and is xed

e

on S Therefore any twist dieomorphism ab out a circle b ounding a disc in S

is isotopic to the identity by an isotopy xed on S A similar argument shows

that any twist dieomorphism ab out a circle C b ounding an annulus together

with a b oundary comp onent of S or equal to a such comp onent is isotopic to

the identity this time by an isotopy moving this b oundary comp onent

It follows that in PMo d any Dehn twist ab out a trivial circle ie a circle

S

b ounding a disc or homotopic to a b oundary comp onent is trivial ie equal

to But in M only the Dehn twists ab out circles b ounding a disc are trivial

S

and the Dehn twists ab out circles homotopic to a b oundary comp onent are

nontrivial of course the last statement requires a pro of

Now we prove some basic prop erties of Dehn twists We start with a rene

ment of the second statement of Corollary E

Lemma A The group Di A x A is an innite gener

ated by the isotopy class of the standard twist dieomorphism T

Pro of Consider the bration Di D x Emb D D from Theorem

A recall that D fr r ig i Since Di D x is

i

contractible by Theorem C and since Emb D D is weakly homotopy

equivalent to the unit tangent bundle of D by Theorem C and hence to

a circle the homotopy sequence of our bration implies that Di A x A

is isomorphic to Z Obviously Emb D D is generated by the homotopy

class of the lo op uniformly rotating D once around its center To compute the

image of this generator in Di A x A we need to extend this rotation

to an isotopy of D xed on D This was done in fact in our discussion of

twist dieomorphisms ab out trivial circles ab ove The extended isotopy ends

in a dieomorphism equal to the identity on D and to the standard twist

dieomorphism on A The lemma follows 2

Corollary B Let C be circle on S and let H H S S be two orienta

tionpreserving dieomorphisms leaving C invariant and preserving the orien

tation of C If the results of cutting H and H along C are isotopic with

free boundary then up to isotopy H diers from H by a power of a twist

dieomorphism about C

Pro of Changing H and H if necessary by an isotopy we may assume that

b oth H and H are equal to the identity on C b ecause Di C is connected

Obviously b oth H and H preserve the sides of C and hence H and H may

b e assumed to b e equal to the identity on some annulus X having C as one

of the b oundary comp onents If we remove the interior of X from S we get a

surface S dieomorphic to the result S of cutting S along C Obviously the

X C

dieomorphisms H H S S induced by H H are isotopic after a

X X X

X

natural identication of S with S to the results of cutting H H along C

X C

Hence H H are isotopic By extending this isotopy to S we get an isotopy

X

X

b etween H and some dieomorphism H which is equal to H outside X and

hence is equal to H G where G is some dieomorphism supp orted in X

Since X is an annulus an application of the lemma completes the pro of 2

The supp ort of a dieomorphism f is the closure of the set fx f x xg

Lemma C If f Mo d and is the isotopy class of a circle on S then

S

f t f t

f

Pro of If e A S is an embedding and F S S is a dieomorphism

then clearly F T F T The lemma follows 2

e F e

Corollary D Any two Dehn twists about nonseparating circles on S are

conjugate

Pro of It is sucient to apply Lemma A 2

a b

c d

Fig

Lemma E Let C and D be two circles on S transversely intersecting at

one point and let respectively be their isotopy classes

i t t

ii t t t and t t t can be represented by a dieomorphism taking

C to C and reversing the orientation of C or what is the same interchanging

the sides of C

Pro of First notice that for any two pairs of circles transversely intersecting

at one p oint there is a dieomorphism of S taking one pair to the other So it

is sucient to prove the lemma for one such pair for example for the pair C D

in Fig a the orientations of circles play a role only in the pro of of ii

i The image E of C under a twist dieomorphism ab out D is illustrated

in Fig b together with a circle isotopic to C The image of E under a

twist dieomorphism ab out this last circle is illustrated in Fig c Clearly

the circle in Fig c is isotopic to the circle in Fig d which is nothing

other than D This proves i

ii In order to prove that t t t it is sucient to apply i twice In

order to prove the remaining part of ii one needs to trace the orientations of

our circles The image of C oriented as in Fig a under the comp osition

of two twist dieomorphisms from the pro of of i is illustrated in Fig d

This image is isotopic to D with the orientation given by turning counterclock

wise the orientation of C at the unique intersection p oint of C and D If we

apply these two dieomorphism once more but in the other order so we get

an oriented representative of t t t we get C with the orientation given by

turning the last orientation of D again counterclockwise Clearly this orienta

tion of C is opp osite to the original one This proves ii 2

I learned Lemma E from J Birmans survey Bir but the assertion i

is denitely due to M Dehn D cf D x a as are also Lemma C and

Corollary D Now we prove several fundamental relations b etween Dehn

twists

Lemma F Let C and D be two circles on S and let respectively be

their isotopy classes

i If C D then t t t t

ii If C and D intersect transversely at exactly one point then the fol lowing

Artin relation holds

t t t t t t

Pro of i If C D then obviously t and t can b e represented by

twist dieomorphisms with disjoint supp orts Such dieomorphisms obviously

commute

ii t t t t t t is obviously equivalent to t t t t t t By Lemma

C the latter equation is equivalent to

1

t t

t

t

Hence it is sucient to prove that t t or what is the same

t t But this is the content of Lemma Eii 2

Lemma G Let C and D be two circles on S transversely intersecting at

one point and let respectively be their isotopy classes Let N be a regular

neighborhood of the union C D this means that N cut along C D is an

annulus with corners and let B N hB i Then

t t t

V H isotopy

a b c d

e f g h

Fig

Pro of Instead of N we may consider the standard torus with one hole re

sulting from the identication of the opp osite sides of a square having a small

hole around its center As C and D we may take circles represented by a verti

cal and a horizontal segment in this square resp ectively Let V and H b e the

corresp onding twist dieomorphisms

Consider the horizontal arc h in Fig a Fig ad compute H

V h up to isotopy If we apply V to the arc in Fig d we get the same

arc So the arc in Fig d is isotopic to V H V h Outside the big

circle in Fig e the arc in Fig d lo oks exactly as the arc in Fig a

turned Such a turn preserves orientation and hence interchanges V and H

more precisely conjugates one to the other So by turning Fig ad

we can nd the image of the arc in Fig d under H V H ie we can

nd H V h H V H V H V h The result is the arc in Fig

f This arc coincides with h outside the big circle in Fig g This

allows us to nd the image of the arc in Fig f under H V or what is

the same H V h Clearly the result is isotopic to the image of h under a

twist dieomorphism ab out the b oundary the small circle in our pictures

Next we would like to nd H V v where v is a vertical arc similar to

h cf Fig h Using a turn as ab ove we see that it is sucient to nd

V H h Note that the rst H in V H acts on h trivially so V H h

V H V h V H V h But V H V h H V h b ecause

in the computation of H V h the last H acts trivially like the last V in

the computation of V H V h Hence V H h H V h and hence

V H h is equal to the image of h under a twist dieomorphism ab out the

b oundary It follows that the image H V v is equal to the image of v under

a twist dieomorphism ab out the b oundary

It remains to notice that if we cut our torus with a hole along h and v we

get a disc with corners and hence a dieomorphism of our torus xed on the

b oundary is determined up to an isotopy xed on the b oundary by its action

on h and v recall that Di D x D is connected for any disc D 2

This lemma is also due in a slightly dierent form to M Dehn D cf

D x c

C

2

C C

23 12

C

0

C C

1 3

C

13

Fig

Lemma H Let S be a sphere with four holes Let the boundary compo

be C C and for i j let C denote a circle encircling nents of S

ij

C and C as in Fig Suppose that S is embedded in S Let t M be

i j i S

the Dehn twist about C i and let t M be the Dehn twist about

i ij S

C i j Then the fol lowing lantern relation holds in M and hence

ij S

in Mo d

S

t t t t t t t

Pro of Connect C with C C C by three disjoint arcs I I I Clearly

if we cut S along these three arcs we get a disc with corners Hence as in

the pro of of Lemma G a dieomorphism of S xed on S is determined

up to an isotopy xed on S by its action on I I I Therefore in order to

prove our relation it is sucient to compute the action of some representatives

of b oth sides on the arcs I I I Of course we will take comp ositions of twist

dieomorphisms as our representatives and then their action on the arcs can

b e computed in a straightforward manner as in the pro of of Lemma E One

just needs to draw several pictures and we leave this task to the reader 2

The relation of Lemma H was discovered by M Dehn D cf D x

g and much later rediscovered and p opularized under the name lantern

relation by D Johnson J

Finally we will prove a lemma b orrowed from Bir cf Bir the discussion

after Corollary It is crucial for the induction step in some arguments using

the induction on the number of b oundary comp onents Cf Section

Lemma I Suppose that S Let R be the result of gluing a disc D to

one of the boundary components of S Let R x be an element repre

sented by an embedded loop b Without any loss of generality we may assume

that the base point x is contained in the interior of D and the loop b intersects

D in a single arc containing x Then we can choose an annulus B containing D

and b in its interior and having boundary components isotopic to b Let C be

the component of B lying at the left side of b and let C be the component of

B lying at the right side of b where the meaning of left and right is determined

by the direction of the loop b and the orientation of S Then

t j t

1

2

where j R x PMo d is the homomorphism from and hC i

S

hC i

Pro of j can b e obtained in the following manner cf Consider the

lo op b as an isotopy of the base p oint x and extend this isotopy to an isotopy

fF R R g F id We may assume that the ending dieomorphism

t t R

F of this isotopy is xed on D Then it induces a dieomorphism F S S

and j is equal to the isotopy class of F Obviously the isotopy fF g

t t

may b e chosen to b e xed outside of the annulus B Moreover it can b e chosen

is such a way that F is equal to the comp osition of a left twist dieomorphism

supp orted in an annulus contained in B with the b oundary C and a right

twist dieomorphism supp orted in an annulus disjoint from the rst one and

also contained in B with the b oundary C One can see this by drawing a couple

of simple pictures or alternatively by dening fF g by explicit formulas

t t

compare our discussion of twist dieomorphisms ab out trivial circles We

leave this as an exercise to the reader the pictures can b e found in I Section

Clearly this description of F implies the lemma 2

The DehnLickorish theorem

The main result of this section is Theorem D known as the DehnLickorish

theorem It provides an explicit nite set of Dehn twists generating Mo d for

S

closed S We also discuss some corollaries of this result Let us start with a

couple of simple lemmas

Lemma A The Dehn twists about the longitude and the meridian of a

closed torus S generate Mo d

S

Pro of By the discussion after Theorem A Mo d is isomorphic to SL Z

S

This isomorphism of course dep ends on the choice of an isomorphism S

Z If the latter isomorphism takes the elements represented by the longitude

and the meridian into the standard basis of Z then as it is easy to see the

Dehn twists ab out the longitude and the meridian corresp ond to matrices

and

On the other hand it is well known that these matrices generate SL Z The

lemma follows 2

Lemma B Suppose that S is a closed surface of genus Let be

two vertices of C S represented by nonseparating circles on S Then there

is a sequence of vertices of C S such that any two

n

consecutive vertices are connected by an edge in C S and moreover if

i i

they are represented by two disjoint circles C C then the union C C

i i i i

does not separate S

Pro of Since C S is connected there is a sequence of

m

vertices of C S such that any two consecutive vertices are connected by

i i

an edge in C S It may happ en that some is the isotopy class of a circle D

i i

separating S into two parts S S In this case can b e represented

i i

by two circles D D resp ectively disjoint from D If D and D are

i i i i i

contained in dierent parts of S then D D and we can delete

i i i

from our sequence Supp ose now that b oth D and D are contained in the

i i

same part of S say in S Since S is closed and D is nontrivial the genus of

i

S is and hence S contains some nonseparating circle D In this case we

i

replace by hD i After rep eating this pro cedure several times we will get a

i

i

new sequence such that all are the isotopy classes of

m i

nonseparating circles and any two consecutive vertices are connected

i i

by an edge

Now let i m and let D D b e two disjoint circles representing

i i

resp ectively Supp ose that D D separates S into two parts S S

i i i i

Since S is closed b oth these parts have genus we assume that

i i

It follows that S as also S contains some nonseparating circle D Clearly

i

b oth D D and D D do not separate S Let us replace the part

i i i i

i i

of our sequence by hD i hD i hD i After rep eating this

i i i i

i i

pro cedure several times we will get the required sequence 2

Theorem C If S is a compact surface of genus then PMo d is gen

S

erated by Dehn twists about nonseparating circles

Pro of We use the double induction on the genus and the number of b ound

ary comp onents The induction starts with the closed torus In this case the

theorem follows from Lemma A

Supp ose that S Let R b e the result of gluing a disc to one of the

b oundary comp onents of S By the inductive assumption PMo d is generated

R

by twists ab out nonseparating circles Obviously every circle C on R is isotopic

0

where PMo d PMo d is to a circle C contained in S and t t

S R C C

the homomorphism from Moreover if C is nonseparating then C is also

nonseparating It follows that in order to prove the theorem for S it is sucient

to prove that ker is generated by Dehn twists ab out nonseparating circles

The latter follows from Lemma I and the fact that R is generated by

the homotopy classes of embedded nonseparating lo ops This completes the

induction step of the induction on the number of b oundary comp onents

Next supp ose that S is closed Let f Mo d Let us choose a nonsepa

S

rating circle C on S Consider two vertices hC i f of C S and

connect them by a sequence as in Lemma B For

m

every i m we can represent by disjoint circles C C

i i i i

Since C C do es not separate S we can choose a circle D transversely

i i i

intersecting each of C and C at exactly one p oint Let hD i By

i i i i

Hence g and t and t t Lemma Ei t

i i i i

i i i i+1

i m t t t g f where g is the pro duct of elements t

i i+1 i i

Since g f we can represent g f by a dieomorphism H S S pre

g Lemma t t serving C If H interchanges the sides of C we replace g by t

1 1

1

Eii implies that for the new g the element g f can b e represented by a

dieomorphism H S S preserving C and preserving the sides of C Now

let us cut S and H along C We will get a dieomorphism H S S pre

C C C

serving b oth b oundary comp onents of S Hence the isotopy class h of H

C C C

Since the genus of S is less than the genus of S h can b elongs to PMo d

C C S

C

b e presented by the inductive assumption as a pro duct of Dehn twists ab out

nonseparating circles on S Every circle on S is at the same time a circle

C C

naturally dene Dehn twists in on S and hence these Dehn twists in PMo d

S

C

PMo d The pro duct of these Dehn twists in PMo d can b e represented by a

S S

dieomorphism H S S preserving C and isotopic to H after cutting along

C

C It follows that H diers from H up to isotopy by some p ower of a Dehn

twist ab out C cf Corollary B Hence g f is equal to a pro duct of Dehn

twists ab out nonseparating circles Since g is also such a pro duct by its con

struction we conclude that f is a pro duct of Dehn twists ab out nonseparating

circles This completes the induction step of the induction on genus and hence

the pro of 2

This theorem is due to M Dehn D His work was forgotten for a while

and the theorem was rediscovered by W B R Lickorish Li To a certain

extent our pro of follows the pro of of J Birman Bir cf Bir Theorem

who simplied the pro of of Lickorish The use of the connectedness of C S

C

2

C

1

C

3

A B A B A

1 1 2 2 3

Fig

leads to further radical simplications and is in fact closer in spirit to Dehns

original approach than to the one of Lickorish The main to ol of M Dehn D

was the action of Mo d on the set of the isotopy classes of systems of circles

S

What was lacking in D and is crucial to our approach is an appropriate

structure on this set namely the structure of a simplicial complex introduced

much later by W Harvey Harv

Now we are ready to prove the DehnLickorish theorem

Theorem D If S is a closed surface of genus g then Mo d is generated

S

by Dehn twists about the g circles A A B B C C pre

g g g

sented in Fig

Pro of By Lemma A the theorem holds for g Using induction on g

we may assume that the theorem holds for all surfaces of genus g

Let us denote by the isotopy classes of the

g g g

circles A A B B C C resp ectively Let us denote by G

g g g S

the subgroup of Mo d generated by Dehn twists ab out these g circles So

S

we have to prove that G Mo d In view of Theorem C it is sucient

S S

to prove that G contains all Dehn twists ab out nonseparating circles By its

S

denition the subgroup G contains some such twists Hence by Lemma C

S

it is sucient to prove that G acts transitively on the set of isotopy classes

S

of nonseparating circles Moreover since C is nonseparating it is sucient to

prove that for every nonseparating circle C there is an element f G such

S

that f where hC i

Now we claim that it is sucient to prove the last statement only for circles

C such that C C and C C do es not separate S In fact supp ose that

our statement is proved for such sp ecial circles C and consider an arbitrary

nonseparating circle C Let us connect with hC i by a sequence

as in Lemma B By our assumption there is g G

n S

such that g g Note that is connected with g

C

2

C

3

A B A

2 2 3

Fig

by a shorter sequence g g g g

n

with the prop erties of Lemma B Using induction we may assume that

g h for some h G Then g h and g h G This

S S

proves our claim

So let C b e a circle such C C and C C do es not separate S and

t t t let hC i By Lemma Ei we have t

1 1 1 1

It follows that g for some t and t t t

2 2 2 1

g G namely for the pro duct of the ab ove eight Dehn twists Therefore it

S

is sucient to prove that f for some f G Note that b oth C and

S

C are disjoint from C and b oth C C and C C do not separate S

Let R b e the result of cutting S along C and let Q b e the result of glu

ing two discs D D to the two b oundary comp onents of R Clearly b oth

C and C are contained in Q and even in R and do not separate Q Con

sider the subgroup G of Mo d generated by Dehn twists ab out the circles

Q Q

A A B B C C clearly all these circles are contained in

g g g

Q Cf Fig Since the genus of Q is equal to g we have G Mo d

Q Q

by the inductive assumption

Since G Mo d we can nd a dieomorphism F Q Q such that

Q Q

F C is isotopic to C in Q and the isotopy class of F b elongs to G Since G

Q Q

is generated by the Dehn twists ab out the circles A A B B C

g g

C we may assume that F is equal to a comp osition of twist dieomorphisms

g

supp orted in some annuli with b oundary comp onents isotopic to these circles

Clearly we may assume these annuli to b e disjoint from the discs D D added

to R In particular we may assume that F is xed on D and D Such a dieo

morphism F can b e obtained by cutting a unique dieomorphism F S S

id xed on C along C and gluing in the identity dieomorphisms id

D D

2 1

Obviously F is equal to a comp osition of twist dieomorphisms supp orted in

the same annuli as F In particular its isotopy class f b elongs to G

S

If F C is not only isotopic but equal to C then F C is also equal to

D

Fig

C and hence f This completes the pro of in the case F C C

since f G In the rest of the pro of we deal with the complications arising

Q

from the fact that in general we only know that F C is isotopic to C in Q

Let fK Q Qg b e an isotopy moving F C into C so that K

t t

id K F C C During this isotopy the discs D D may b e moved

Q

Note that D and D are disjoint from F C b ecause F is xed on D and D

and hence K D and K D are disjoint from C Since C do es not separate

Q we can move the discs K D K D back to their original p osition D

D without moving C More precisely there is an isotopy fL Q Qg

t t

such that L id L is xed on C for all t and L K j D id

Q t D

1

Let H L K Clearly H F C C H is isotopic L K j D id

D

2

Like F the dieomorphism H can b e H j D id to id and H j D id

D Q D

2 1

obtained by cutting a unique dieomorphism H S S xed on C along

Let h b e the isotopy id C and gluing in the identity dieomorphisms id

D D

2 1

class of H

Since H F C C we have H F C C and hence h f

Recall that f G and hence h f G if h G Therefore in order

S S S

to complete the pro of it is sucient to show that h G In view of the

S

prop erties of H stated in the last paragraph it is sucient in fact to prove the

following claim

Claim Let H S S b e a dieomorphism xed on C preserving the

sides of C and such that the result H of cutting H along C and gluing in the

is isotopic to id Then the isotopy class id identity dieomorphisms id

Q D D

2 1

h of H b elongs to G

S

Let us now prove this claim First by Lemma Eii some representative

maps C to C and interchanges the sides t t I of the isotopy class i t

1 1

1

of C Let I R R b e the result of cutting I along C and let i Mo d

R

is equal to t t b e its isotopy class Next by Lemma G t t

0 1 1

where is the isotopy class of the circle D in Fig In particular t

can b e represented by a twist dieomorphism T ab out D We may assume

that T is supp orted in an annulus disjoint from C Let T R R b e

the result of cutting T along C and let t Mo d b e its isotopy class Let

R

G b e the subgroup of Mo d generated by i t and the Dehn twists ab out

R

A A B B C C

g g g

R R b e the result of cutting H along C recall that R Let H

C

G ie Mo d b e its isotopy class Supp ose that h S and let h

R C

C C

can b e represented as a pro duct of i t and the Dehn twists ab out that h

C

A A B B C C on R Then h diers from the corre

g g g

sp onding pro duct of i t and the Dehn twists ab out A A B B

g g

ab out C C considered now on S by some p ower of the Dehn twist t

g

1

b elong to G it follows that in t and t t t t C Since i t

S

1 1 1 1

1

this case h G Therefore it is sucient to prove that h G By the

S

C

assumption h b elongs to the kernel of the natural map PMo d PMo d

R Q

C

from So it is sucient to prove that ker G

The kernel ker is equal to the image of the canonical homomorphism

j PB Q Mo d from and hence is generated by the images of the

R

fundamental groups Q Q where Q is the result of gluing only one

i

disc D to R and i

i

Consider rst the element a of Q represented by the lo op in Fig a

In view of Lemma I its image in Mo d is equal to t t where are the

R

isotopy classes of circles B C from Fig b Clearly B C are isotopic in

G Let us t R to B C resp ectively and hence this image is equal to t

1

2

consider next the element b of Q represented by the lo op in Fig c a

lo op going round the hole In view of Lemma I its image in Mo d is equal

R

to t t where are the isotopy classes of circles D E from Fig d

Clearly D is isotopic to D and E is a trivial circle homotopic to a b oundary

t t and hence t t t comp onent It follows that t t

0 0

Therefore the image of b also b elongs to G

Recall that f j f j f for any f Mo d B Q cf

R

It follows that j f G if f G j G In particular if the image

of some element c Q b elongs to G and f G then the image of f c

also b elongs to G if f is not in PMo d ie if f interchanges the holes then

R

f c Q but actually we do not need this case Now Fig ag

shows how to pro duce new elements of Q from the element a by applying

consecutively various Dehn twists from G to a Clearly together with b these

elements generate Q It follows that the image of Q in Mo d is

R

contained in G

It remains to show the same for the image of Q This can b e done

by similar pictures But it is easier to note that conjugation by the element

i interchanging the holes of R maps the image of Q into the image of

Q It follows that since i G the image of Q is also contained in G

And since the images of Q Q generate ker by Corollary B and

a

a

C

B

b

b

c

E

d

D

Fig

a

t

2

b

t

2

c

isotopy

d

t

2

Fig ad

e

t

2

f

isotopy

g

etc

Fig eg

Theorem C we conclude that ker G As we already said this completes

the pro of of the claim and hence of the theorem 2

M Dehn D was the rst to prove that the groups Mo d are nitely gen

S

erated and to provide an explicit nite set of generators for them His set of

generators consisted of g g Dehn twists for a closed surface S of genus

g and of Dehn twists for a closed surface of genus Much later and

indep endent of Dehns work Theorem D was proved by W B R Lickorish

Li Li An exp osition of the work of Lickorish was given by J Birman

Bir see Bir Chapter Again to a certain extent our pro of follows the

simplied pro of of J Birman Bir cf Bir Theorem and Corollary

and the use of C S leads to further radical simplications and is closer in spirit

to Dehns original approach We also added some details corresp onding to the

pro of of Corollary from Bir omitted in Bir

Corollary E Mo d is nitely generated for any compact orientable sur

S

face S

Pro of Since PMo d is of nite index in Mo d it is sucient to prove that

S S

the groups PMo d are nitely generated Let us glue discs to the b oundary

S

comp onents of S one at a time Using the second of Theorem

C with n recall that PB Q Q for any surface Q the fact

that the fundamental groups of compact surfaces are nitely generated and the

induction on the number of b oundary comp onents we can reduce our assertion

to the case of closed surfaces S But for closed S Theorem D provides an

explicit nite system of generators 2

Corollary F If S is a closed surface of genus g then Mo d is generated by

S

Dehn twists about the g circles A A B B C C presented

g g

in Fig

Pro of This result is due to S Humphries Hum and we refer to Hum for a

sequence of pictures showing how to transform C into a circle isotopic to C

i i

by a sequence of twist dieomorphisms ab out A B C A and B

i i i i i

In view of Lemma C this implies that one can express the Dehn twist ab out

C in terms of Dehn twists ab out C A B C A and B It follows

i i i i i i i

that we can consecutively eliminate Dehn twists ab out C C C from

g g

our list of generators 2

This result is complemented by a theorem of S Humphries Hum to the

eect that g is actually the minimal number of Dehn twist generators of

Mo d if S is a closed surface of genus g If we do not require the generators

S

to b e Dehn twists then one needs in fact only two generators Namely the

following theorem holds

Theorem G If S is a closed surface of genus g then Mo d is generated

S

by the two elements

t t t t t t t

g g 1 2 2 1 1 1

t t

g 1

g

where are respectively the isotopy classes of the cir

g g g

cles A B A C C C presented in Fig

g g g

This nice result is due to B Wajnryb W He deduced it from the ab ove

Corollary F by some ingenious computations His pro of also covers the case

of surfaces with one b oundary comp onent if we made a hole in the surface in Fig

at the right end of it then the Dehn twists ab out the same circles may serve

as generators The p oint of this theorem is in providing the minimal number

of generators clearly Mo d is not cyclic and hence cannot b e generated by one

S

element As B Wajnryb explains in W it is quite easy to nd a generating

set consisting of three elements and already W B R Lickorish Li noticed

that four elements generate Mo d While the prop erty of b eing generated by

S

two elements has no immediate imp ortant consequences it traditionally attracts

the attention of group theorists

Finite presentations

After we have proved that the groups Mo d are nitely generated it is only

S

natural to ask if they are nitely presented and to lo ok for a nite presentation

if they are In the spirit of our pro of of the DehnLickorish theorem ie

Theorem D we will base our approach to this question up on the action of

Mo d on C S Such an approach is made p ossible by the following general

S

theorem

Theorem A Suppose that a group G acts on a CWcomples X permuting

its cells and suppose that

i X is simply connected in particular X is connected

ii the isotropy group of every vertex of X is nitely presented

iii the isotropy group of every edge of X is nitely generated

iv the number of orbits of cells of dimension is nite

Then G is nitely presented

An elegant pro of of this theorem based on the BassSerre theory Ser is

given by K Brown Bro cf Bro Theorem But it can also b e easily proved

directly In the sp ecial case when G acts transitively on the set of vertices such

a pro of is contained in the rep ort of F Laudenbach La ab out the pap er of A

Hatcher and W Thurston HatT cf La x Implicitly this sp ecial case was

proved and used already by A Hatcher and W Thurston HatT Laudenbachs

pro of can b e easily extended to the general case K Brown Bro gives a long

list of precursors of Theorem A but the pap ers HatT La apparently were

unknown to him It is worth adding to his list also the results of H Behr Be

Section and of J L Kozsul Kosz Chapter I I I x While H Behr Be

states his sucient condition for the nite presentability in terms of actions

on discrete metric spaces J L Koszul Kosz reworked it in the language of

actions on graphs and obtained in fact a necessary and sucient condition

y

Actually as it is clear from his denition of the homotopy of lo ops J L Koszul

Kosz works not with graphs but with ag complexes cf Compared with

Kosz the main novelty of Theorem A lies in the fact that lo cally innite

complexes are admitted According to W Harvey Harv the fact that the

complexes of curves are not lo cally nite also was an at least a psychological

obstacle to overcome

Note that all pro ofs of Theorem A allow in principle to construct a

nite presentation of G starting from nite presentations of the isotropy groups

of vertices nite sets of generators of the isotropy groups of edges the com

binatorial structure of X and of course the action of G on X This leads

usually to a fairly complicated presentation which one may try to simplify For

Mo d such an approach was successfully realized by B Wajnryb in his work

S

W discussed b elow

The following two lemmas admit direct and elementary pro ofs But we

prefer to deduce them from Theorem A and the well known fact that for

any nitely presented group there is a nite CWcomplex having this group as

its fundamental group Our pro of of the rst lemma was suggested by Kosz

xI I I Remark

Lemma B Let H G F be an exact sequence of groups If

H and F are nitely presented then G is also nitely presented

Pro of Let Y b e a nite CWcomplex with Y F and let X b e the

universal cover of Y So X is simply connected and F freely acts on X This

action induces an action of G on X Since F acts on X freely the isotropy

groups of vertices and edges under the action of G are all isomorphic to the

group H which is nitely presented Since Y is nite the number of orbits of

cells under the action of either F or G is nite Hence we may apply Theorem

A and conclude that G is nitely presented 2

Lemma C Let H be a subgroup of nite index of a group G Then H is

nitely presented if and only if G is nitely presented

Pro of Supp ose rst that G is nitely presented Let Y b e a nite CW

complex with Y G and let X b e the universal cover of Y Then H acts

y

in Kosz p on the second line a a a a should b e replaced by

n

0

i i+1

a a a a

n

0

i i+2

freely on X and the number of the orbits of the cells is nite b ecause Y XG

is nite and the index G H is nite Theorem A now implies that H is

nitely presented

Supp ose now that H is nitely presented Let

H g H g

g G

This intersection is actually nite b ecause G H and hence H is of nite

index in b oth G and H By the already proved part of the lemma H is nitely

presented Since H is obviously normal in G we have an exact sequence of

groups H G GH Since GH is nite it is nitely presented

Now Lemma B implies that G is nitely presented 2

Theorem D Mo d is nitely presented for any compact orientable sur

S

face S

Pro of In view of Lemma C it is sucient to prove that the groups PMo d

S

are nitely presented In order to deal with PMo d we will use double induc

S

tion on the genus and the number of b oundary comp onents as in the pro of of

Theorem C

Supp ose that S Let R b e the result of gluing a disc to one of the

b oundary comp onents of S By the inductive assumption PMo d is nitely

R

presented The kernel of the natural homomorphism PMo d PMo d from

S R

is nitely presented by Theorem C this kernel is isomorphic to R

with few exceptions these exceptions are easy to deal with directly and hence

Lemma C implies that PMo d is also nitely presented This completes the

S

induction step of the induction on the number of b oundary comp onents

Next supp ose that S is closed If S is a sphere then PMo d If S is

S

a torus then PMo d is isomorphic to SL Z It is well known that the latter

S

group is nitely presented cf Ser for example So we may assume that

the genus of S is In this case we are going to apply Theorem A to

the action of PMo d on the geometric realization of C S We need to verify

S

the assumptions of Theorem A Theorem C implies i cf also the

discussion preceding Theorem C Let us now check iii Supp ose that

hC i hC i are connected by an edge of C S and let G b e the isotropy group

of the geometric realization of this edge We may assume that C C

Let G b e the subgroup of PMo d consisting of the isotopy classes of dieo

S

morphisms F S S such that F C C F C C and F preserves

the sides of b oth circles C C Clearly G is a subgroup of nite index in G

in fact the index is Such dieomorphisms F can b e cut along C C

and this leads to a canonical homomorphism G PMo d where Q is the

Q

result of cutting S along C C Clearly this homomorphism is surjective its

kernel is generated by the Dehn twists ab out C and C the latter assertion

follows from Corollary B applied twice and these Dehn twists commute

by Lemma Fi Since S is closed the genus of the comp onents of Q is

strictly less than the genus of S So the inductive assumption implies that

PMo d is nitely presented By Lemma B G is also nitely presented Fi

Q

nally by Lemma C the isotropy group G is nitely presented So we proved

an assertion even stronger than iii The pro of of ii is similar and simpler

In order to check iv notice that the number of orbits of cells of dimension

is nothing more than the number of arrangements of disjoint circles

on S considered up to a dieomorphism Clearly this number is nite So we

can apply Theorem A and conclude that PMo d is nitely presented This

S

completes the induction step of the induction on genus and hence the pro of 2

Theorem D was originally proved by J McCo ol McC by metho ds of

the combinatorial group theory His approach allows in principle computation

of an explicit nite presentation of Mo d for a given S but this turned out to

S

b e to o dicult to b e done for any S Later on A Hatcher and W Thurston

HatT suggested a geometric approach to the nite presentability and to a

construction of an explicit nite presentation The ab ove pro of was suggested

by the author in I It has some features in common with HatT such as

the use of families of functions in order to prove the simply connectivity of an

appropriate complex cf the pro of of Theorems B and C ab ove and the

use of Theorem A only implicit in HatT In some sense this pro of can b e

considered as an ultimate simplication of the pro of of Hatcher and Thurston

at least for the time b eing One more approach to this theorem is indicated

in

The complex used in HatT is very complicated and A Hatcher and W

Thurston did not achieve their goal of writing down an explicit nite presenta

tion that do es not diminishes the signicance of their exceptionally b eautiful

pap er Their complex was somewhat simplied by J Harer Har and using

this simplied complex B Wajnryb W succeeded in writing down an explicit

nite presentation for closed surfaces and for surfaces with one b oundary comp o

nent The pro of involves some extremely complicated computations The reader

should b e warned that W contains some mistakes even in the statements of

the main theorems corrected much later in BW Recently B Wajnryb W

provided a new exp osition of his work In addition W contains a new pro of

of the simply connectivity of the HatcherThurston complex

Let us briey describ e the Wajnryb presentation for closed surfaces As

generators he uses the Humphries generators of Corollary F Relations are

the following First for any pair of disjoint circles from the list of Corollary

F the Dehn twists ab out them are related by the commutation relation

of Lemma Fi Second for any pair of circles from this list intersecting

transversely at one p oint the Dehn twists ab out them are related by the

Artin relation of Lemma Fii There are three additional relations In

order to describ e them let us as usual denote the isotopy classes of the circles

C A B A C

g

Fig

A A B B C C in Fig by

g g g g g

resp ectively

g

In order to describ e the rst relation we need a circle D on S intersecting

transversely A at exactly one p oint not intersecting any other circle in Fig

and not isotopic to C up to isotopy D is characterized by these prop erties

it is situated similarly to C but is contained in the lower part of S Let

as one may check t t t t t t hD i Then u where u t

2 1 1 1 1 2

1

t t t t u It turns out that t ut straightforwardly Hence t

2 2 1 1 1 2 2

and u we can express this relation in terms of and using our formulas for t

2

the Humphries generators This is the rst additional relation

The next relation is essentially the lantern relation of Lemma H ex

pressed in terms of the Humphries generators To write it down B Wajnryb

chose an embedding of S in S such that the b oundary circles C C C C of

are mapp ed to the circles C B B C of Fig Then he expressed the S

Dehn twists ab out the images of the circles C C C and the Dehn twist

ab out C in terms of the Humphries generators After this he wrote the lantern

relation in terms of these expressions This is the second additional relation

The last relation is concerned with the socalled hyperelliptic involution of

S It is an orientationpreserving dieomorphism H S S of order ie

H H id Up to isotopy it is characterized by the following prop erties

S

it maps each of the g circles C A B A B B A C to itself

g g g

and reverses the orientation of all these circles and hence interchanges the

comp onents of the complement in S of the union of these g circles It can

b e represented by a rotation around the horizontal axis of in Fig

The isotopy class h of H is also called the hyperelliptic involution Note

that as it is describ ed here the hyperelliptic involution dep ends on the choice of

a collection of circles dieomorphic to the one in Fig and dierent choices

lead to dierent but conjugate hyperelliptic involutions For the purp oses of

writing down the Wajnryb presentation we need only one such collection of

by h t circles Since H C C and H is orientationpreserving ht

g g

g g

h The third additional relation t Lemma C or what is the same ht

g g

where f t is essentially this commutation relation It turns out that h t

g g

if and Clearly h commutes with t t t t t t t t t f t

g g g 1 1 1 1 1 g 1 g

1

in terms of the Humphries generators only if f do es Wajnryb expressed t

g

and then using this expression and the ab ove formula for f he wrote the

f in terms of the Humphries generators This t commutation relation f t

g g

is his third and the last additional relation

The Wajnryb presentation for surfaces with one b oundary comp onent is

obtained from the Wajnryb presentation for closed surfaces by omitting the

third additional relation If we made a hole in the surface on Fig at the

right end of it the Dehn twist ab out the same circles serve as generators

The ab ove description of the Wajnryb presentations in geometrical terms is

essentially b orrowed from J Birmans survey Bir Note that the relation

in Bir is incorrect as stated One should replace in it the element G by the

ab ove expression for the hyperelliptic involution h

Now let S b e a closed surface of genus In this case the second additional

relation follows from the other relations The rst nite presentation of Mo d

S

in this case was discovered by P Bergau and J Mennicke BeM Their pro of

turned out to b e incomplete and their presentation was justied only ab out

ten years later by J Birman and H Hilden BirH Instead of the rst two

additional relations of the Wajnryb presentation this presentation contains the

following two relations

t t t t t

2 2 1 1 1

t t t t t t t t t

2 2 1 1 1 1 2 2

1

the second one expresses the fact that the hyperelliptic involution has order

A sp ecial feature of this case is that the hyperelliptic involution h commutes

with all generators of Theorem D or Corollary F known already to

Dehn D as is clear from the ab ove description of h and hence b elongs to the

center of Mo d In fact it generates the center in this case On the contrary if

S

and in fact the center is the genus is then h do es not commutes with t

2

trivial Cf

For surfaces S of genus ie a spheres with holes Mo d is closely related

S

to the classical Artin braid groups This allows to write down a presentation of

Mo d in this case For the details see Bir Chapter

S

Recently S Gervais Ge used the Wajnryb presentation and induction on

the number of the b oundary comp onents in order to derive an explicit and very

symmetric presentation of Mo d for surfaces S with an arbitrary number of

S

b oundary comp onents

Finally let us mention a related work of S Gervais Ge where he derived

some innite presentations of mapping class groups similar to the Wajnryb

presentation but having as generators all Dehn twists or all Dehn twists ab out

nonseparating circles His presentation with all Dehn twists as generators was

recently simplied by F Luo Luo

Teichmuller spaces

Teichmuller spaces provide another class of fundamental geometric ob jects on

which mapping class groups act Our approach to Teichmuller spaces is closer in

spirit to that of R Fricke FrK than to the approach of O Teichmuller himself

In particular we dene the Teichmuller spaces in terms of metrics of constant

curvature and not in terms of Riemann surfaces as Teichmuller did In fact the

work of Fricke precedes the work of Teichmuller by ab out forty years Probably

Fricke was also the rst to dene the mapping class groups

In our discussion we stress the connection of Teichmuller spaces with com

plexes of curves and esp ecially the results of W Harvey Harv Harv cf

and other combinatorial ob jects namely the socalled ideal triangulations

cf As the rst application we will indicate another approach to the

nite presentability of the mapping class groups cf Corollary B In the

next section we will present a much deep er application of Teichmuller spaces to

the mapping class groups namely the computation of the virtual cohomological

dimension of the mapping class groups cf Theorems A B and C

Denitions

Let us assume that our compact orientable surface S admits a riemannian

metric of constant curvature with geo desic b oundary As is well known see

such metrics exist if and only if S has negative Euler characteristic Let

us x some number l and consider only metrics such that all comp onents

of the b oundary S have length l The choice of l is largely irrelevant but at

a later p oint it will b e convenient to have l small Let us denote by H the

S

set of such metrics We can endow H with some convenient top ology say the

S

C top ology by considering riemannian metrics as tensor elds Since dieo

morphisms act on tensor elds by the pushforward the group Di S acts

on H The quotient space H Di S by the subgroup of dieomorphisms

S S

isotopic to the identity is called the Teichmuller space of S and is denoted by T

S

Clearly the quotient group Mo d Di S Di S acts on T and the quo

S S

tient space T Mo d is equal to H Di S which is called the

S S S

of hyperbolic structures on S and is denoted by M Hence T Mo d M

S S S S

Sometimes cf for example it is more convenient to consider another

version of Teichmuller spaces namely the one related to socalled punctured

surfaces By a punctured surface R we understand a closed orientable surface

together with a nite subset of it The p oints of this nite subset are called

punctures and we denote by R the result of deleting all punctures from the

surface Let R b e a punctured surface such that R admits a complete rie

mannian metric of constant curvature Again it is well known that such

metrics exist if and only if R has negative Euler characteristic We denote

by H the set of such metrics This set also can b e endowed with a natural

R

top ology This time the group Di R acts on H and the quotient space

R

H Di R by the subgroup of dieomorphisms isotopic to the identity is

S

called the Teichmuller space of R and is denoted by T Naturally the group

R

Mo d Di R Di R is called the mapping class group of R It nat

R

urally acts on T and the quotient space T Mo d is equal to H Di R

R R S R

which is called the moduli space of hyperbolic structures on R and is denoted by

M Clearly T Mo d M

R R R R

Note that there is no real dierence b etween mapping class groups of compact

and punctured surfaces If S is a compact surface then we can collapse each

comp onent of S into a p oint one for each b oundary comp onent and get a

closed surface Images of b oundary comp onents form a nite subset of this

surface and the resulting surface together with this nite subset is a punctured

surface Let us denote it by R Clearly S n S R This leads to a natural

map Di S Di R The homomorphism Mo d Mo d induced by it is

S R

easily seen to b e an isomorphism

Length functions and FenchelNielsen ows

Let b e a vertex of C S ie the isotopy class of a nontrivial circle For any

metric h H the isotopy class contains exactly one geo desic with resp ect to

S

the metric h Let us denote by l h the length of this geo desic Clearly this

length dep ends only on the image of h in T Hence we can consider l as a

S

function T R We call this function the length function corresp onding

S

to For a simplex f g of C S we may consider the map

m

m

xg Clearly it x l given by the formula L x fl L T R

S

m 1

m

comes from a similar map L H R The map L is esp ecially imp ortant

S

when is a simplex of maximal dimension In this case it turns out to b e a

m m

principal bundle over R having as the structure group the additive group R

m

Let us describ e the corresp onding action of R on T Let h H b e a

S S

metric representing a p oint x T and let C b e the geo desic circle with resp ect

S i

to h in the isotopy class i m If we cut S along this geo desic C

i i

we get a new surface with a metric of curvature and geo desic b oundary

which has two new b oundary comp onents resulting from C In order to restore

i

S from this new surface we have to glue back these two b oundary comp onents

Clearly these b oundary comp onents should b e glued by an but this

do es not dene the gluing uniquely Two isometric gluings may dier by a

rotation It seems at rst sight that the set of all p ossible isometric gluings is

parameterized by a circle In fact since we are interested in the resulting p oint

of T and not only of M ie we are interested in the resulting metric up to

S S

isotopy ie up to the action of Di S and not only up to isometry the set

of all p ossible gluings is parameterized by the universal cover of this circle ie

by a copy of R More precisely there are at least two natural identications

of this universal cover with R the one corresp onding to the measuring of the

amount of rotation by the angle and the other corresp onding to the measuring

of the amount of rotation by the length one can get from the rst to the second

h Another ambiguity comes from the need to by multiplication by l

i

orient the circle C in order to get the identication but none of these is of

i

any imp ortance for us now To summarize we may cut the original surface and

then glue it back by a gluing diering from the one leading back to the original

surface by a parameter r R This denes a map R T T and it is easy

S S

to see that this map is an action of the additive group R This action is usually

called a FenchelNielsen ow Of course this action dep ends on the vertex

i

For any h H the corresp onding circles C i m are disjoint b ecause

S i

is a simplex and this easily implies that the actions corresp onding to dierent

m

vertices of commute Hence they dene an action of R

The fact that this action turns L into a principal bundle amounts essen

tially to the following In the notation of the previous paragraph if the simplex

is maximal and we cut S along all circles C i m we will get a collection

i

of discs with two holes A metric of the constant curvature with geo desic

b oundary on a disc with two holes is uniquely determined up to isotopy by

the lengths of its b oundary comp onents Hence the cut surface is determined

by L x Clearly if two surfaces with metrics lead to the same cut surface

m

then they dier only by the gluing ie by the action of R It follows that

m

the orbits of R are exactly the b ers of the map L One needs also to check

m

that the action of R on T is free and also the existence of lo cal sections We

S

refer to FLP Exp ose for this

The top ology of Teichmuller spaces

As we just saw the Teichmuller space T is the total space of a principal bundle

S

m m m

over R with the b er R Since R is contractible it follows that T is

S

m m m

homeomorphic to R R and hence to R The number m is the maximal

number of pairwise disjoint pairwise non isotopic nontrivial circles on S hence

m g b where g is the genus and b is the number of b oundary comp onents

g b

of S We conclude that T is homeomorphic to R

S

In addition one can introduce a smo oth structure on T in a number of

S

ways all leading to the same smo oth structure This smo oth structure can b e

characterized by the fact that all length functions L are smo oth In fact a

n

nite number of them is sucient to embed T in a R as a

S

smo oth submanifold The straightforward approach requires g b length

functions cf FLP Exp oses and But for a closed S it is sucient to

use only dim T length functions And if we consider another version of the

S

Teichmuller space T in which the lengths of the b oundary comp onents are not

S

xed as in then for nonclosed S it is sucient to use only dim T length

S

functions See P Schmutz Sch and the references there for these results

No matter what denition of the smo oth structure is used it is easy to

see that the principal bundle from is smo oth It follows that T is not

S

only homeomorphic but also dieomorphic to a Euclidean space no results

ab out top ological manifolds need to b e used Also the FenchelNielsen ows

are smo oth Moreover we can replace smooth by real analytic everywhere

For closed S the Techmuller space T even has a canonical complex analytic

S

structure but it plays no role in the sort of questions we are interested in

applications to the mapping class groups

The discussion in and in this section applies equally well to Teichmuller

spaces of punctured surfaces In particular Teichmuller space T of a punctured

R

g b

surface R is dieomorphic to R where g is the genus of R and b is the

number of punctures unless R is a torus without punctures in which case T

R

is dieomorphic to R

Corners and the mapping class groups

As we noticed in Mo d acts on T This action is not free b ecause Mo d

S S S

has torsion and T is homeomorphic to a Euclidean space any element of nite

S

order has a xed p oint in T say by the Smith theory see for example FLP

S

Exp ose end of x IV But it is as close to b eing free as p ossible any torsion

free subgroup of Mo d acts freely on T the argument go es as follows if the

S S

isotopy class of a dieomorphism F xes a p oint of T represented by a metric

S

h H then F is isotopic to an isometry F of the metric h it is well known

S

that the group of isometries of a metric of constant curvature on S is nite

It is well known that Mo d contains torsion free subgroups even of nite index

S

In fact the kernel of the natural homomorphism Mo d Aut H S ZmZ

S

dened by the action of dieomorphisms on homology is such a subgroup for any

m This result is a combination of a theorem of JP Serre cf Theorem

A and a classical result of J Nielsen cf Theorem A ab out realization

of nite cyclic subgroups of Mo d by subgroups of Di S see Corollary B

S

In addition this action is prop erly discontinuous Hence for torsion free the

quotient space T is a K space and of course a manifold It turns out

S

that this manifold is never compact but one can construct a compact manifold

from it if is of nite index in Mo d either by adding some b oundary at innity

S

or by deleting a neighborho o d of innity Moreover one can do this in a fairly

canonical way

Before we describ e these constructions we note that the resulting compact

manifolds with b oundary will not b e smo oth manifolds in the usual sense but

smo oth manifolds with corners Recall that a smooth manifold with corners

n m

is simply a smo oth manifold mo deled on the pro ducts R R for all n m

and not only for n as manifolds with b oundary The simplest type of such

manifolds is presented by the subsets of usual smo oth manifolds lo cally describ ed

by inequalities of the form x x where x x x is

n n nm

a chart

Let us x some small and let us assume only for convenience that

the b oundary length l xed in is equal to Let T b e the subset of

S

T dened by the inequalities l x for all isotopy classes of nontrivial

S

circles Clearly T is invariant under the action of Mo d on T The basic

S S S

prop erties of T are summarized in the following theorem

S

Theorem A If is suciently smal l then the fol lowing holds

i The action of Mo d on T is properly discontinuous

S S

ii Any torsion free subgroup of Mo d acts on T freely and its quotient

S S

space T is a smooth manifold with corners

S

iii The quotient space T Mo d is compact

S S

iv T is contractible

S

v The boundary T is homotopy equivalent to the geometric realization

S

of the complex of curves C S

Pro of First the should b e chosen so small that the following holds if h is a

metric of constant curvature on S and C C are two geo desic circles with

resp ect to h on S of length then C C are either equal or disjoint The

existence of such an is well known cf A Chapter I I Corollary of Lemma

in Section it also follows from Bu Corollary The prop erty i and

the rst part of the prop erty ii immediately follow from the corresp onding

prop erties of T itself Prop erty iii follows from the socalled Mumford com

S

pactness theorem Mumf see Bu Theorem or A Chapter I I Theorem

in Section for a closely related result The pro of of iv is the most technical

part of the pro of of the theorem we refer to I Theorem for a pro of Next

we turn to the prop erty v For any vertex of C S let us consider the subset

T fx T j l x g of T This subset turns out to b e closely

S S S

where S is the result of cutting S along a representative of related to T

S

and one can use this together with iv for S in order to prove that T

S

is contractible It is easy to see that the b oundary of T is equal to the union

S

of subsets T By the choice of any collection T of such subsets

S S i

has a nonempty intersection if and only if the vertices can b e represented by

i

disjoint circles Hence C S is the nerve of a cover of T by contractible

S

subsets The prop erty v follows Finally the second part of the prop erty ii

follows from the fact that T itself is a manifold with corners This follows

S

in turn from the fact that in a neighborho o d of a p oint x T the subset

S

and T is dened by a nite number of the inequalities of the form l

S

i

moreover the corresp onding form a simplex and hence can b e included as

i

co ordinate functions in a chart by

A more detailed outline of the pro of is contained in I Section with

complete details provided in I An outline of another pro of is contained in

Har Section 2

Corollary B Mo d is nitely presented

S

Pro of Let b e a torsion free subgroup of nite index in Mo d It follows

S

from i and iv that T is a K space Because is of nite index

S

in Mo d it follows from iii that T is compact In view of ii T

S S S

is a smo oth manifold with corners The last prop erty implies that T is

S

homotopy equivalent to a nite CWcomplex moreover it admits a nite tri

angulation Hence there exists a nite CWcomplex which is a K space

It follows that is nitely presented By Lemma C Mo d is also nitely

S

presented 2

The idea b ehind Theorem A is due to W Harvey Harv Harv In

Harv W Harvey announced a construction of a manifold with corners X

S

containing the Teichmuller space T as its interior and such that the prop erties

S

iv of Theorem A hold with T replaced by X He was motivated by

S S

the analogy rst noticed also by him b etween the mapping class groups and

arithmetic groups cf and by the work of A Borel and JP Serre BorS

Adding the corners to T in this manner turned out to b e a more delicate matter

S

than excising a part from T as in Theorem A A pro of of the existence of

S

such an X was provided in I together with a geometric interpretation of

S

the p oints in the complement X n T X

S S S

Ideal triangulations

Let R b e a punctured surface in the sense of with a non empty set of

punctures An ideal triangulation of R is a collection of embedded arcs with

disjoint interiors connecting the punctures of R such that if we cut R along

all these arcs we will get a collection of triangles with sides resulting from

these arcs The word ideal is justied by the standard understanding that

the punctures are p oints at the innity In other words we ideally triangulate

R rather than R Note also that two endp oints of an arc are p ermitted to

coincide and a triangle of an ideal triangulation is not determined by the set of

its vertices which are punctures in general

The ideal triangulations considered up to isotopy are exactly the top di

mensional simplices of a simplicial complex AR which we now dene The

vertices of AR are the isotopy classes of nontrivial arcs connecting punctures

of R Here an arc is called trivial if two its endp oints coincide and the resulting

circle b ounds a disc in R which do es not contain other punctures The isotopies

are p ermitted naturally only in the class of arcs connecting punctures so they

are xed on the endp oints The simplices are dened in a manner similar to the

simplices of complexes of curves a collection of vertices forms a simplex if these

vertices can b e represented by arcs with disjoint interiors Like complexes of

curves AR is a ag complex cf As in the case of complexes of curves

this can b e established by using complete metrics of constant curvature this

time on R and geo desic arcs Clearly Mo d acts on AR

R

We also need a sub complex A R of AR It has the same vertices as

AR A collection of vertices forms a simplex of A R if those vertices can b e

represented by arcs with disjoint interiors such that the result of cutting R along

these arcs has at least one comp onent which is not a disc meeting the set of

punctures only in its b oundary In other words simplices of AR which are not

simplices of A R corresp ond to the isotopy classes of ideal cell decompositions

of R in an obvious sense

b

Theorem A Let be the op en standard b dimensional simplex

b

where b is the number of punctures of R Let us endow with the trivial

b

action of Mo d and the product T with the product action Then

R R

b

T is Mo d equivariantly homeomorphic to the geometric realization of

R R

AR with the geometric realization of A R removed

Note that the implied in this theorem is not canonical

dierent pro ofs lead to apparently dierent homeomorphisms

The early history of this theorem is not well do cumented in the literature

According to Har Chapter the idea of this triangulation is due to W

Thurston his approach was based on hyperb olic geometry The rst published

account is contained in Har where an alternative pro of due to D Mumford

and based on work of K Streb el on quadratic dierentials on Riemann sur

faces cf Str Str was outlined Pro ofs based on hyperb olic geometry

were published by R Penner Pe and by B Bowditch and D B A Epstein

BowE Penners approach seems to b e more remote from the original ideas of

W Thurston in that the Lorentz mo del of the hyperb olic plane plays a crucial

role in it His ideas were extended to some degree to the higher dimensions

EpP and to closed surfaces NP and in Pe R Penner applied them to the

socalled universal Techmuller space Unfortunately all this falls outside the

scop e of the present pap er

The complexes AR and A R are together with complexes of curves the

basic combinatorial ob jects of the top ology of surfaces To a large extent they

are simpler than complexes of curves Theorem A b eing the main reason

but they are not dened for closed surfaces Among the main applications of the

complexes AR and Theorem A are the computation of the virtual Euler

characteristic of the mapping class group by indep endently J Harer and D

Zagier HarZ and R Penner Pe Pe cf computations and estimates of

the WeilPetersson of Teichmuller spaces by R Penner Pe Theorem

A ab out automorphisms of complexes of curves cf the pro of of E

Wittens conjecture Wi by M Kontsevich Ko and the theorem of L Mosher

Mos Mos to the eect that the mapping class groups are automatic in the

sense of Ep For an exp osition of the work of M Kontsevich we refer to Lo o

Lo o and for an introduction to L Moshers results we refer to his own survey

Mos

As an easy and useful corollary of Theorem A we p oint out the following

result

Corollary B Every simplex of AR is a face of a top dimensional sim

plex Every codimension simplex of AR is a face of or top dimensional

simplices Any two top dimensional simplices of AR can be connected

by a chain of simplices such that any two consecutive

m

simplices have a common codimension face

i i

A much more elementary pro of of this Corollary was provided by A Hatcher

Hat cf Hat Corollary

Cohomological prop erties

This section is devoted to cohomological prop erties of mapping class groups It

is centered around the computation of the socalled virtual cohomological di

mension of mapping class groups see As preparation for this computation

we present in a very short and idiosyncratic introduction to the cohomology

of groups This introduction is heavily slanted toward phenomena crucial for an

understanding of b oth mapping class groups and arithmetic groups we will dis

cuss the latter in and known as the BieriEckmann duality cf BieE The

latter is actually a manifestation of the PoincareLefschetz duality as we try to

explain in After this introduction we prove in J Harers theorem

Har Har computing the virtual cohomological dimension of the mapping

class groups In we review mostly without pro ofs some related coho

mological prop erties of mapping class groups Finally in we review some

results ab out cohomology of the mapping class groups with nite co ecients

Cohomology of groups and the PoincareLefschetz du

ality

The goal of this section is to present an introduction to cohomology of groups

and a fragment of the theory of R Bieri and B Eckmann BieE see esp ecially

Theorem H

Let b e a and let X b e a K space with a xed isomor

phism X which will b e used to identify these groups The homology

or cohomology of the group can b e dened as the homology or cohomology

of the space X In order to capture the essential prop erties of one has to

consider also homology and cohomology with socalled twisted co ecients Let

M b e a mo dule in other words M is a mo dule over the group Z

Then the homology H M and cohomology H M of with twisted

co ecients M can b e dened as follows

For a space Y let us denote by C Y its singular complex or if Y is a CW

complex its complex of cellular chains Let X X b e the universal covering

of X The fundamental group X acts naturally on X turning C X

into a complex of mo dules By denition

H M H X M H C X M

where means the tensor pro duct over the group ring Z If M Z with

the trivial structure of mo dule ie m m for all m M then

C X M C X Z C X

and we recover the usual homology of and X In fact if two singular simplices

or cells c c in X are pro jected to the same simplex or cell in X then c c

for some and hence c m c m c m c m for any m M

Cohomology is dened in a similar way

H M H X M H Hom C X M

where Hom C X M C X M is the group of equivariant M valued

co chains ie the group of co chains f such that f c f c for any

and any chain c As ab ove if M Z with the trivial structure of mo dule we

recover the usual cohomology of and X

If we choose a lift to X of any singular simplex or cell in X then any

equivariant co chain will b e dened by its values on these lifts It follows that

Hom C X M and C X M are isomorphic as graded ab elian groups

But the dierentials are dierent in the rst complex the dierential is twisted

Now let us dene the cohomological dimension cd of as the supremum

n

which can b e of the numbers n such that H M for some mo dule

M

Theorem A If X is a closed of dimension n then

cd n

m

Pro of First note that H X M for m n by same reasons as for trivial

n

co ecients Thus we only need to nd a mo dule M such that H X M

If X is orientable then we can take M Z with the trivial structure of

mo dule otherwise we should take the orientation Z It is isomorphic

or

to Z as an ab elian group but acts on it by multiplication by if

reverses the orientation of X and by otherwise 2

This pro of at least partially motivates the need for twisted co ecients in the

denition of cd

As usual a short exact sequence of co ecients

M M M

leads to a long exact sequence of cohomology groups

n n n n

H M H M H M H M

It is the key to ol in the pro of of the following lemma

Lemma B If cd then cd is equal to the supremum of numbers n

n

such that H M for some free Zmodule M

n

Pro of Let n cd Cho ose M such that H M and choose a short

exact sequence

M F M

with F free Then we have an exact sequence

n n n

H F H M H M

n n

where H M and H M b ecause n cd It follows

n

that H F 2

n

Lemma C If X is a nite CWcomplex then H F for al l free

n

Zmodules F if and only if H Z

Pro of The only if part is trivial In order to prove the if part supp ose

n m

that H Z Then for any nitely generated free mo dule F Z

n n m n m m

we have H F H Z H Z

Now any free mo dule is equal to the of its nitely generated

free submo dules As we will explain in a moment the fact that X is a nite

n n

CWcomplex implies that the functor H H X commutes with the

direct limits Hence the vanishing of the cohomology with co ecients in nitely

generated free mo dules implies the vanishing of the cohomology with co ecients

in an arbitrary free mo dule This completes the pro of mo dulo the claim ab out

commuting with the direct limits

It is well known that for any ring R we are interested in the case R

Z and any pro jective R mo dule Q the functor Q commutes with the

R

direct limits In addition if P is a nitely generated pro jective R mo dule then

Hom P M P M for any R mo dule where P Hom P R It follows

R R

that for a nitely generated pro jective R mo dule P the functor Hom P

R

commutes with the direct limits

Returning to our situation notice that if X is a nite CWcomplex and

we consider cellular chains then C X are nitely generated free and hence

k

pro jective Zmo dules Thus the result of the previous paragraph implies that

the functor Hom C X commutes with the direct limits It remains to

k

note that taking the cohomology groups of a complex always commutes with

the direct limits 2

Corollary D If X is a nite CWcomplex then cd is equal to the maxi

n

mum of numbers n such that H Z

Pro of Since X is a nite CWcomplex dim X If we use cellular chains

k n

then C X for k dim X and hence H M H X M for

k

k dim X It follows that cd It remains to apply Lemmas B and

C 2

n

The last corollary shows the imp ortance of the groups H Z Our

next goal is to provide a geometric interpretation of these cohomology groups

Lemma E For any module M the group Hom M Z is naturally

isomorphic to the group Hom M Z of al l homomorphisms of abelian groups

c

f M Z such that for any m M the image f m is equal to for almost

al l ie for al l with the exception of nitely many of them

Pro of Any homomorphism F M Z of mo dules can b e written in

the form

X

F m f m

for some homomorphisms f M Z Clearly for any m the image f m

is equal to for almost all by the denition of Z Since F is a

homomorphism of mo dules we have F m F m for any m M

It follows that

X X

f m f m

and hence f m f m for all and m M In particular

f m f m for all m M where is the unit element

of It follows that F is determined by f Moreover b ecause for any m

1 1

the image f m f m f m is equal to for almost all the

homomorphism f b elongs to Hom M Z

c

Conversely for a homomorphism f M Z b elonging to Hom M Z we

c

can dene a homomorphism F M Z of mo dules by the formula

X

F m f m

Thus F f establishes a natural bijection Hom M Z Hom M Z

c

2

Lemma F If X is a nite CWcomplex then H X Z is naturally iso

X of X morphic to the cohomology group with compact support H

c

Pro of By Lemma E

H Z H X Z H Hom C X Z

H Hom C X Z

c

If we use the cellular chains then Hom C X Z consists of co chains which

c

are equal to on almost all cells in the preimage of any given cell in X Because

X has only a nite number of cells these are exactly the co chains equal to on

almost all cells in X ie the co chains with compact supp ort It follows that

the last cohomology group is nothing other than the cohomology groups with

compact supp ort of X 2

Corollary G If X is a nite CWcomplex then cd is equal to the maxi

n

mum number n such that H X Z 2

c

Now it is the time to make a crucial step and apply the PoincareLefschetz

duality

Theorem H Suppose that X is a compact topological manifold of dimen

sion d with boundary may be empty and simultaneously is a nite CW

complex Let m be the minimum number such that H X where H

m

as usual denotes the reduced homology Then cd d m If there is no

such m then cd

n

Pro of By the PoincareLefschetz duality H X H X X Be

dn

c

cause X is a K space X is contractible and hence H X X

dn

H X It remains to apply Corollary G note that m d n

dn

if and only if n d m 2

The assumptions of this theorem are satised for example when X is a

smo oth manifold with corners as in the applications we have in mind and

hence can b e triangulated

Classifying spaces and universal bundles

Here we discuss some auxiliary results ab out classifying spaces and universal

bundles which will b e needed in and

Let G b e a top ological group acting on a top ological space X Let EG BG

b e the Milnor universal principal Gbundle In particular EG BG is a

lo cally trivial bundle and its total space EG is contractible We will denote by

EX BG the asso ciated bundle with the b er X Recall that EX EG X

G

A choice of a base p oint x X denes a map G X given by the formula

g g x and a map EG EX given by the formula y y x

G

Lemma A If G X is a Serre bration then the map EG EX is also

a Serre bration

Pro of Lo cally over any suciently small op en set U BG the bundle

EG BG is isomorphic to the trivial bundle U G U the asso ciated

bundle is isomorphic to U X U and the map EG EX is the map

U G U X induced by G X If G X is a Serre bration then

obviously U G U X is a Serre bration also It follows that EG EX is

a Serre bration lo cally over EX By a well known theorem see for example

Bre Theorem VI I this implies that EG EX is a Serre bration 2

Lemma B Let H be the stabilizer of the base point x in G Suppose that

the above map G X is a Serre bration Then the part

X EX BG

of the homotopy sequence of the bund le EX BG can be canonically identied

with the part

X H G

of the homotopy sequence of the bration G X

Pro of Let BG G b e the b oundary map from the homotopy se

quence of the bundle EG BG Since EG is contractible it is an isomorphism

By Lemma A the map EG EX is a Serre bration obviously its b er is

H Let EX H b e the b oundary map from the homotopy sequence

of this bration Again since EG is contractible it is an isomorphism Direct

check shows that the following diagram

- -

X EX BG

? ? ?

- -

H G X

is commutative and hence provides the required identication 2

If G acts on another top ological space X and a Gequivariant map X X

is given then there is a natural map EX EX commuting with the pro jec

tions EX BG EX BG If X X is a weak homotopy equivalence

then EX EX is also a weak homotopy equivalence and we can identify

the homotopy sequences of brations EX BG and EX BG

Mess subgroups

In this section we discuss we discuss some subgroups of the mapping class groups

discovered by G Mess Me These subgroups will b e our main to ol for estimat

ing the virtual cohomological dimension of the mapping class groups from below

cf the pro of of Theorem A

Let S b e a closed orientable surface of genus g and let Mo d Mo d

g g S

g

Let S b e an orientable surface of genus g with b oundary comp onent and let

g

1

M M This section is devoted to some subgroups B B of Mo d M

g g

S

g g g

g

resp ectively where g They were introduced by G Mess Me and we call

them Mess subgroups Although the construction of these involves some choices

it is essentially canonical

The construction of the Mess subgroups has a recursive character We start

with any subgroup B of Mo d generated by Dehn twists ab out any three pair

wise disjoint pairwise nonisotopic circles on S Thus B is isomorphic to Z

Supp ose that B g is already dened

g

We may assume that S is obtained from S by gluing a disc D to the

g

g

b oundary of S The extension of dieomorphisms of S xed on S by the

g g g

identity across the disc D denes a homomorphism M Mo d as in Let

g

g

B b e the preimage of B under this homomorphism

g

g

In order to dene B consider some embedding S S and identify

g g

g

S with its image The extension of dieomorphisms of S xed on S by

g g g

the identity across the complement of S in S denes a homomorphism

g

g

i M Mo d Note that the closure of the complement of S in S is a

g g

g g

torus with one hole Let us choose some nontrivial circle in this torus with one

hole and consider the Dehn twist t Mo d ab out this circle Let T b e the

g

innite cyclic group generated by t We dene B as the group generated by

g

iB and T

g

This completes the construction of the Mess subgroups A couple of remarks

are in order First recall see that the restriction of dieomorphisms of S

g

on D denes a bration Di S Emb D S As explained in its

g g

x Since Di S is simply connected b er can b e identied with Di S

g

g

by Theorem G and Emb D S is connected the homotopy sequence of

g

this bration ends with

Emb D S Di S x Di S

g g

g

ie with

Emb D S M Mo d

g g

g

where the homomorphism M Mo d is exactly the one used ab ove Hence

g

g

we have a short exact sequence

Emb D S B B

g g

g

Since Emb D S is weakly homotopy equivalent to the unit tangent bundle

g

UT S by Theorem C we can also write this short exact sequence as

g

UT S B B

g g

g

Note also that the homomorphism i M Mo d is injective by Theorem

g

g

I and that the twist t obviously commutes with the image of i It follows

that B is isomorphic to B Z Now we come to the main prop erty of the

g

g

Mess subgroups cf Me Prop osition

Theorem A There exists a closed topological manifold of dimension g

which is a K B space Similarly there exists a closed topological manifold

g

of dimension g which is a K B space These topological manifolds can

g

be chosen to be simultaneously CWcomplexes

Pro of The idea is to start with the dimensional torus which can b e taken

as a K B space and to construct a K B space as the total space of a

g

bundle with the b er UT S over an already constructed K B space and

g g

then get a K B space by multiplying this total space by the circle S

g

We would like this bundle to realize the sequence UT S B B as

g g

g

the part of its homotopy sequence Note that UT S is a K space

g

b ecause it is the total space of a circle bundle over S which is a K space

g

Supp ose that a K B space K is already constructed

g g

Let G Di S X Emb D S with the obvious action of Di S

g g g

and let X b e the unit tangent bundle UT S with the following action of

g

Di S a dieomorphism acts on a unit vector by its dierential and the sub

g

sequent normalization Let X X b e the map u Emb D S UT S

g g

from It is obviously Gequivariant By Theorem C it is a weak homo

topy equivalence Hence the induced map EX EX is also a weak homotopy

equivalence It follows that we can naturally identify the homotopy sequences

of bundles EX BG and EX BG

Now since comp onents of Di S are weakly contractible by Theorem

g

G the BG B Di S is a K Mo d space Hence the

g g

inclusion B Mo d leads to a map K BG from our K B space K to

g g g g g

K with the b er X UT S BG B Di S Consider the bundle K

g g g

g

induced from the bundle EX BG by this map Since b oth UT S and K

g g

are K spaces the homotopy sequence of the induced bundle shows that

is also a K space In addition this homotopy sequence ends with the K

g

short exact sequence

UT S K K

g g

g

This short exact sequence naturally maps to the end

UT S EX BG

g

recall X UT S of the homotopy sequence of the bundle EX BG

g

is obviously isomorphic to the preimage of K B Moreover K

g g

g

under the map EX BG By the previous paragraph this short

exact sequence is naturally isomorphic to the end

X EX BG

of the homotopy sequence of the bundle EX BG Now consider the bration

G Di S X Emb D S Clearly H Di S x is the stabilizer

g g

g

of the embedding D S and hence is the b er of this bration By Lemma

g

B the last short exact sequence is naturally isomorphic to the end

X H G

of the homotopy sequence of the bration G X But this short exact sequence

is nothing other than our sequence

Emb D S M Mo d

g g

g

It follows that K is isomorphic to the preimage of B under the map

g

g

i M Mo d ie to B Hence we can take K as our K B space and

g

g g g g

then put K K S

g

g

This completes our construction of K B and K B spaces Since

g

g

they are constructed by consecutively taking the total spaces of lo cally trivial

bundles with manifold b ers and multiplying by S and starting with the

dimensional torus these spaces are obviously top ological manifolds With

some additional care one can turn these spaces into CWcomplexes or even

smo oth manifolds and then triangulate them This proves all statements of

the theorem with the exception of the correctness of the dimension values

Clearly dim K dim K dim K dim K and dim K

g g

g g

g This completes the pro of 2 Therefore dim K g and dim K

g

g

Corollary B cd B g cd B g

g

g

Pro of Combine Theorem A with Theorem A 2

Virtual cohomological dimension

One of the main prop erties of the cohomological dimension is the inequality

cd cd which holds for We refer to Bro for a pro of and

only note that it uses in a crucial way the fact that all twisted co ecients

are allowed cf Bro Chapter VI I I Prop osition It is well known that

cd A for a nite cyclic group A it follows that any group with torsion

has innite cohomological dimension In order to get an interesting invariant

for groups with torsion we mo dify the denition in the way suggested by JP

Serre Ser

If is virtual ly torsion free ie contains a subgroup of nite index which is

torsion free then by a fundamental theorem of JP Serre Ser cf also Bro

Chapter VI I I Theorem or BeK Section all torsion free subgroups of

nite index of have the same cohomological dimension The common value of

the cohomology dimension of such subgroups is called the virtual cohomological

dimension of and is denoted by vcd

Since Mo d always has torsion cd Mo d and the interesting invariant

S S

is vcd Mo d The results of and allow us to compute it We start with

S

closed surfaces

Theorem A If S is a closed surface of genus g then vcd Mo d

S

g If S is a sphere or a torus then vcd Mo d or respectively

S

Pro of If S is a sphere then Mo d and the result is trivial If S is a torus

S

then Mo d is isomorphic to SL Z The last group as is well known contains

S

a nitely generated free group as a subgroup of nite index It is easy to see

that the cohomological dimension of a free group is Hence vcd Mo d if

S

S is a torus

In the rest of the pro of we assume that S is a closed surface of genus g

Consider a subgroup of nite index in Mo d acting freely on the Teichmuller

S

space T such subgroups do exist by It is sucient to show that cd

S

g for any such we do not need to know in advance that such a group

is torsion free since cd implies this

Consider the intersection B of with the Mess subgroup B It

g g

is obviously of nite index in B Hence we can take as a K space a

g

nite sheeted of the K B space provided by Theorem A

g

Clearly this K space is a of dimension g By Theorem

A cd g It follows that cd g

In order to prove the opp osite inequality consider the manifold with corners

T from and put X T Since the action of on T is free X

S S S

is also a manifold with corners and since T is contractible cf we

S

can take T as the universal cover X of X By the b oundary T

S S

is homotopy equivalent to the geometric realization of the complex of curves

C S Hence Theorem C implies that X T is g connected

S

Now we apply Theorem H Since X is g connected the number m

from this theorem is g such an m exist b ecause we already saw that

cd Finally d dim X dim T dim T g and hence

S S

Theorem H implies that cd g g g This

completes the pro of 2

Theorem B If S is a surface of genus g with b boundary compo

nents then vcd Mo d g b

S

Pro of Let S b e a surface of genus g with b b oundary comp onents and let

g b

Mo d Mo d In order to compute vcd Mo d we will construct a sub

g b S g b

g b

group of nite index in Mo d and a K space Since the dimension

g b g b g b

of this space will b e and hence cd will b e the subgroup will

g b g b

b e automatically torsion free and therefore we will have vcd Mo d cd

g b g b

The cohomological dimension cd will b e computed with the help of Theorem

g b

H using our K spaces

g b

The construction of subgroups has a recursive character We start with

g b

any subgroup of nite index in Mo d acting freely on the Teichmuller space

g g

is the manifold with corners from where T Let L T T

g S g S S

g 0 g 0 g 0

L will b e our K space Supp ose that a subgroup of nite

g g g b

index in Mo d and a K space L are already constructed

g b g b g b

Let us x a p oint x in the interior of S Let fxg it is a dimensional

g b

submanifold of S Clearly we can identify Emb S with the interior

g b g b

int S of S recall that the embeddings in Emb S can b e extended to

g b g b g b

dieomorphisms of S see

g b

Now consider the group Mo d Di S x In view of the last

g b

g b

paragraph of the group Mo d is isomorphic to the subgroup of the group

g b

Mo d consisting of the isotopy classes of dieomorphisms S S

g b g b g b

preserving setwise one xed b oundary comp onent In particular Mo d is

g b

isomorphic to a subgroup of nite index actually of index b of Mo d

g b

Let G Di S and let X int S with the obvious action of G The

g b g b

map G X given by g g x is nothing but the evaluation of dieomorphisms

at x it can b e identied with the map Di S Emb S from In

g b g b

particular Theorem A implies that G X is a Serre bration The stabi

lizer H of the p oint x X ie the b er of G X is equal to Di S x

g b

The homotopy sequence of this bration ends with

int S Di S x Di S

g b g b g b

ie with

Int S Mo d Mo d

g b g b

g b

By Lemma B this short exact sequence can b e naturally identied with

X EX BG

where X int S

g b

Next we would like to replace the op en surface int S by the compact sur

g b

face S Let X b e the surface S with the natural action of G Di S

g b g b g b

The inclusion X X is obviously Gequivariant and is a homotopy equiva

lence In view of the remarks at the end of we can identify the last exact

sequence with

X EX BG

where X S We can write this short exact sequence more explicitly as

g b

S ES B Di S

g b g b g b

Now as in the pro of of Theorem A we use the fact that B Di S is

g b

a K Mo d space and hence the inclusion Mo d leads to a map

g b g b g b

L Di S from our K space L to B Di S Consider the

g b g b g b g b g b

bundle L L with the b er S induced from the bundle EX

g b g b g b

ES BG B Di S by this map Since b oth S and L are K

g b g b g b g b

spaces the homotopy sequence of this bundle shows that L is also a K

g b

space In addition this homotopy sequence ends with the short exact sequence

S L L

g b g b g b

The latter naturally maps to the last short exact sequence of the previous para

graph and a trivial diagram chase shows that L is isomorphic to the

g b

preimage of under the map ES B Di S In view of the

g b g b g b

ab ove this preimage is naturally isomorphic to the preimage of under the

g b

map Mo d Mo d We dene to b e equal to this preimage obvi

g b g b

g b

ously it is of nite index in Mo d and take L as our K space

g b g b g b

This completes the construction of our subgroups and K spaces

g b g b

L Since the latter are constructed by consecutively taking the total spaces of

g b

lo cally trivial bundles with surface in general with b oundary b ers starting

with a manifold with corners the spaces L are top ological manifolds with

g b

b oundary and we also can turn them into CWcomplexes compare the pro of of

Theorem A Obviously dim L dim L for all b

g b g b

In order to apply Theorem H we need to analyze the universal covers

L of manifolds L and their b oundaries L If the universal cover L

g b

g b g b g b

is already constructed we can construct L in two steps First take the

g b

bundle M L with b er S induced from the bundle L L by

g b g b g b g b

g b

the map L L Since the base L of this bundle is contractible this

g b

g b g b

bundle is actually trivial and hence M is homeomorphic to S L As the

g b g b

g b

where L second step we take the universal cover of M This leads to S

g b

g b g b

is homeomorphic to is the universal cover of the surface S Thus L S

g b

g b g b

S L and hence the b oundary L is homeomorphic to

g b g b g b

S L S L

g b g b g b g b

Now there are two dierent cases to consider b and b

In the rst case S has no b oundary and L is homeomorphic to S

g g g

L Since S is contractible it follows that L is homotopy equivalent

g g g

to L

g

is non empty and actually consists of a countable In the second case S

g b

is innite number of comp onents homeomorphic to the real line Since S

g b

contractible it follows that the pair S S is homotopy equivalent to the

g b g b

pair C Z Z where Z is a discrete space consisting of a countable number of

p oints and CZ is the cone of Z The last pair in turn is homotopy equivalent

to the pair R Z It follows that L is homotopy equivalent to

g b

Z L R L

g b g b

Since the universal covers L are contractible the last space is homotopy

g b

equivalent to a b ouquet of an innite number of susp ensions L

g b

Now let db dim L and let mb b e the minimum number m such

g b

we assume that the genus g is xed As we saw ab ove that H L

m

g b

db db for all b The results of the two previous paragraphs imply

that m m and mb mb for b Combining these remarks

with Theorem H we see that cd cd and cd cd

g g g b g b

for b Since cd vcd Mo d g by Theorem A the theorem

g g

follows 2

Theorem C If S is a sphere with b holes then vcd Mo d for b

S

and vcd Mo d b for b If S is a torus with b holes then vcd Mo d

S S

for b and vcd Mo d b for b

S

Pro of The pro of is similar to the pro of of the previous theorem Since for a

sphere S with holes or a torus S without holes the comp onents of Di S

are not contractible and hence B Di S is not a K Mo d space the in

S

ductive argument should start with a sphere with holes and a torus with

hole We leave the details to the reader 2

The rst nontrivial estimate vcd Mo d g of vcd Mo d for a closed

S S

surface of genus g was proved in I I the trivial estimate is vcd Mo d

S

dim T g It was deduced from the simply connectivity of C S which

S

in turn was deduced from the simply connectivity of the HatcherThurston

complex HatT Theorems AC are due to J Harer Har Har

Another pro of was provided by the author I The pap er I contains also a

computation of the virtual cohomological dimension of the mapping class groups

of nonorientable surfaces Both pap ers Har and I actually prove a stronger

result than Theorems AC Namely they prove that Mo d contains a

S

subgroup of nite index which is a group with duality of some dimension in the

sense of BieriEckmann BieE and compute this dimension The dimension of

a group with duality is built into its denition like the dimension of a smo oth

manifold or a Poincare space it is always and turns out to b e equal to the

cohomological dimension The last prop erty allows the use of this result for the

computation of vcd Mo d We refer to I Section for an exp ository account

S

of this stronger result and to Bro Chapter VI I I or BeK for more details

ab out groups with duality The pro of of the lower estimate for the cohomological

dimension in our pro of of Theorem A follows the ideas of G Mess Me while

the pro of of the upp er estimate which turned out to b e equal to the lower one

follows the standard approach of I I Har Har I The ab ove pro of

of Theorem B seems to b e new Both of these pro ofs avoid the full p ower

of the theory of groups with duality a fragment of this theory was included in

and also do not use the complete description of the homotopy type of C S

they use only Theorem C and only this for closed surfaces

Homology stability

In its simplest form the homology stability for the mapping class groups asserts

that the homology group H Mo d with trivial co ecients Z do es not dep end

n S

on S provided S is closed and the genus g of S is suciently large compared to

n The implied restriction on g is usually called the domain of stability The

rst result of this sort with the domain of stability g n was proved

by J Harer Har Har Such results are similar to and motivated by the

classical homology stability theorems in the algebraic K theory The simplest

of those asserts that the homology group H SL Z do es not dep end on g

n g

provided g is suciently large compared to n The more advanced versions of

the homology stability theorems in algebraic K theory cover some other natural

series of groups like the sp ecial linear groups over rings or the symplectic groups

and usually provide an explicit domain of stability the role of g is played by

some natural parameter A common feature of the pro ofs of the homology

stability theorems in algebraic K theory is the need to include into the picture

a wider class of groups than the original innite series this is required by the

way the induction is arranged in the pro ofs The situation for the mapping class

groups is similar one needs to consider the groups M for nonclosed surfaces

S

S in addition to the groups Mo d for closed S The groups M have a natural

S S

advantage over our usual version Mo d namely the existence of a natural map

S

M M when R is a subsurface of S This map is given by the extension

R S

of dieomorphisms of R xed on R by the identity to dieomorphisms of S

In fact we can completely switch to the groups M in this context b ecause

S

M Mo d for closed S and the homology stability do es not hold for the

S S

groups Mo d for nonclosed surfaces S for the second homology group this

S

follows from Theorem C

Theorem A Let R be a connected subsurface of a connected surface S Let

g be the genus of R The map H M H M induced by the natural

R m R m S

map M M is an epimorphism if

R S

g n

R

and is an isomorphism if

g n

R

If in addition S is not closed then the map H M H M is an

m R m S

epimorphism if

g n

R

and is an isomorphism if

g n

R

The pro of of Theorem A is fairly technical In the outline it follows the

Maazenvan der Kallen vdKal approach to the homology stability theorems in

algebraic K theory On the geometric side one of the basic to ols is the action of

the groups M on some versions of the complexes of curves C S The higher

S

connectivity of these versions plays a crucial role On the algebraic side the

central role is played by some sp ectral sequences asso ciated with an action of a

group on a simplicial complex and a carefully arranged induction on the genus

and the number of b oundary comp onents of S For an introduction to the ideas

of the pro of of Theorem A we refer to I Section and to I Section

The details omitted in I are provided in I The case of closed S requires

an additional argument and is dealt with in I

Corollary B If S is a closed surface of genus n then up to iso

morphism the homology group H Mo d does not depend on S

n S

Pro of First recall that Mo d M for closed S

S S

If S S are two closed surfaces of dierent genera there is no natural homo

0

In order to compare the homology morphism b etween the groups M M

S S

0

we can choose two dieomorphic subsurfaces R R of S S re of M M

S S

0 0

If M sp ectively and apply Theorem A to the maps M M M

S R S R

we choose subsurfaces R R such that their genus is equal to the lesser of the

genera of S S the corollary will follow immediately 2

The domain of stability of Theorem A and Corollary B is substantially

b etter than the J Harers original domain of stability g n from Har

For nonclosed surfaces the domain of stability g n of Theorem A is

exactly the same as the b est known domain of stability for H SL Z On the

n g

other hand if we consider homology with rational as opp osed to the integer

co ecients then the domain of stability can b e further improved A similar

phenomenon is well known in the algebraic K theory

Theorem C Let R be a connected subsurface of a connected surface S with

non empty boundary Let g be the genus of R The map H M Q

R m R

H M Q induced by the natural map M M is an isomorphism if

m S R S

n

g

R

For odd n this map is surjective if

n

g

R

This theorem is due to J Harer Har who also proved that the domain

of stability of this theorem cannot b e improved The question ab out the exact

domain of stability for integral co ecients remains op en

Finally we mention two other stability theorems In Har J Harer proved

a homology stability theorem for subgroups of the mapping class groups con

sisting of the isotopy classes of dieomorphisms preserving some spin structure

on the surface these subgroups obviously have nite index in the corresp onding

mapping class groups The main result of I is a homology stability theorem

for socalled twisted co ecients ie for co ecients in a nontrivial mo dule

Actually more imp ortant than the fact that the co ecients are twisted is that

the natural examples of twisted co ecients substantially dep end on the surface

in question ie are not constant As the simplest and typical example one

may consider H M H S where M acts on H S in the natural way In

n S S

this sp ecial case we have the following result

Theorem D Let R be a connected subsurface of a connected surface S

Let g be the genus of R Suppose that both R and S have exactly boundary

R

component Then the map H M H R H M H S induced by the

m R m S

natural maps M M H R H S is an epimorphism if

R S

g n

R

and is an isomorphism if

g n

R

This is a sp ecial case of Corollary from I the coecient system

S H S is of degree in the sense of I E Lo oijenga Lo o provided

a new pro of of a sp ecial case of the stability theorem for twisted co ecients of

I Namely he considers only co ecients which factor through the natural

representation M S p Z where g is the genus of S and which are rational

S g

vector spaces In this situation he extends the results of I to closed sur

faces and more imp ortantly explicitly computes the stable homology ie the

homology in the domain of stability with such twisted co ecients in terms of

the stable homology with trivial co ecients Q Lo oijengas metho ds are largely

algebrogeometric but he uses also the homology stability theorem for constant

co ecients

Surprisingly homology stability with twisted co ecients do es not hold for

closed surfaces even for the rst homology group as the following theorem

implies

Theorem E If S is a closed surface of genus g then H M H S

S

Zg Z

This result is due to Sh Morita Mor cf Mor Corollary

The lowdimensional homology groups

We start by computing the rst homology group of PMo d

S

Lemma A H PMo d depends only on genus of S

S

Pro of It is sucient to show that H PMo d do es not change if we make

S

a hole in S So let R b e the result of removing from S the interior of a disc

embedded in S Consider the exact sequence

S PMo d PMo d

R S

from If a S is the homotopy class of an embedded lo op in S then the

image of a in PMo d has the form t t for some circles in R Moreover if

R

a is the homotopy class of an embedded non separating lo op then b oth circles

C

12

C

0

C

23

C

3

C

1

C

2

C

0

C

13

C

2

C

3

C

1

Fig

and are non separating see Lemma I It follows that t is conjugate

to t and hence their images in H PMo d are equal Therefore the image

R

of a in H PMo d is Since S is generated by the homotopy classes of

R

embedded non separating lo ops if S has non empty b oundary then some of

the standard generators are separating but they can easily b e replaced by two

non separating generators the image of S in H PMo d is Using the

R

ab ove exact sequence we see that H PMo d is isomorphic to H PMo d 2

R S

Theorem B Let S be a surface of genus g The rst homology group

H PMo d is equal to ZZ if g to ZZ if g and to if g

S

Pro of If S is a closed surface of genus ie a torus then PMo d is

S

isomorphic to SL Z and the result is well known to algebraists at least

If S is a closed surface of genus then probably the b est way to compute

H PMo d is to use the presentation of Mo d due to BergauMennicke BeM

S S

and BirmanHilden BirH cf end of Given the presentation the compu

tation of H PMo d is routine and can b e left to the reader This proves the

S

theorem for closed surfaces of genus and and then Lemma A implies it

for all surfaces of genus and

In the generic case g we actually do not need Lemma A In this case

we can embed the sphere with holes S from Lemma H in S in such a way

that all circles C i and C i j will b e non separating in

i ij

S see Fig For such an embedding all Dehn twists t t from the lantern

i ij

relation are conjugate and hence represent the same element of H PMo d

S

The lantern relation t t t t t t t implies that this element is equal to

On the other hand by Theorem C PMo d is generated by Dehn twists

S

ab out non separating circles All such Dehn twists are conjugate to t and

hence represent in H Mo d It follows that H PMo d 2

S S

The path to this nice argument for the genus case which is due to

J Harer Har was fairly long First D Mumford Mumf proved that

H Mo d is cyclic of order dividing for closed surfaces S of genus

S

Then J Birman Bir computed H Mo d for closed surfaces S of genus

S

and and proved that for genus the order of this group is or Fi

nally J Powell Pow proved that this order is actually by means of some

complicated computations

Theorem C Let S be a surface of genus g with b boundary components If

b

g then H PMo d Z

S

This result is due to J Harer Har One should warn the reader that

Har contains some mistakes even in the statement of the main theorem it

is asserted in Har that H PMo d Z Zg Z for closed surfaces

S

S of genus g But the general agreement seems to b e that Theorem C is

correct The main to ol in Har is an action of PMo d on a mo dication of the

S

HatcherThurston complex HatT and the sp ectral sequences asso ciated with

this action Later on Harer Har suggested an alternative metho d to compute

the second homology group with rational co ecients and also extended this

new metho d to the third homology group The approach of Har is close in

spirit to the pro ofs of the homology stability theorems and uses the actions

of the mapping class groups on complexes similar to those used to prove the

stability theorems The sp ectral sequences asso ciated with these actions relate

the homology groups of a mapping class group with the homology groups of the

stabilizers of the simplices which are in fact mapping class groups of smaller

surfaces This allows use of an inductive pro cess for the computations Let us

state the main result of Har

Theorem D Let S be a surface of genus g

H PMo d Q is equal to if g and to Q if g

S

H PMo d Q if g

S

Using the same approach in Har J Harer computed H Q for sub

groups of the mapping class groups consisting of the isotopy classes of dieo

morphisms preserving some spin structure on the surface and in Har he also

computed the fourth homology group of PMo d More precisely the main result

S

of Har is the following theorem

Theorem E Let S be a surface of genus g with b boundary components

H PMo d Q Q if g b or if g

S

The computations involved in the pro of of this theorem are extremely com

plicated Finally in HL R Hain and E Lo oijenga rep ort that recently J Harer

has extended his computations to H PMo d Q This homology group turns

S

out to b e for suciently high genus

Virtual Euler characteristic

In the spirit of Serres approach Ser the usual Euler characteristic ch

P

i

rank H is not the right invariant for discrete groups with torsion

i

even when it is well dened One should consider instead the socalled virtual

Euler characteristic If contains a subgroup of nite index admitting

a nite CWcomplex as a K space this implies that is torsion free

then by denition

ch

It turns out that this number do es not dep end on the choice of In general

and in the most interesting cases is not an integer but only a rational

number

For the virtual Euler characteristic there is an analogue of the usual formula

computing the Euler characteristic of a CWcomplex by counting the number of

cells of various dimensions Supp ose that acts on a CWcomplex T preserving

its CWstructure Supp ose that in addition T is contractible some subgroup

of nite index acts freely on T and hence T is a K space the

number of orbits of cells is nite and the isotropy groups of cells are nite

Then

X

dim

where is the isotropy group of the cell is its order and runs

over some set of representatives of the orbits of cells If acts on T freely this

formula reduces to the usual formula for the Euler characteristic of the K

space T For more details ab out the virtual Euler characteristic we refer to

Ser and to Bro Section IX

Theorem A If S has one boundary component then Mo d B g

S g

g where g is the genus of S B is the g th Bernoul li number and

g

is the Riemann function

Pro of Let R b e a oncepunctured surface of the same genus as S By

Mo d is isomorphic to Mo d Hence we may consider Mo d instead of Mo d

S R R S

We will use the ideal triangulation of the Teichmuller space T introduced

R

in Let us consider the rst barycentric sub division of the complex AR

and take the union R of the barycentric stars of those simplices of AR

which are not simplices of A R This union has a natural structure of a

CWcomplex with the barycentric stars as cells One may say that R is the

complex dual to the ideal triangulation of T from It is not hard to see that

R

R is a Mo d equivariant deformation retract of T In particular R is

R R

contractible Cells of R are in onetoone corresp ondence with simplices of

AR which are not simplices of A R ie with ideal cell decomp ositions of

R Under this corresp ondence the orbits of cells corresp ond to the topological

types of ideal cell decomp ositions Since the number of these top ological types is

obviously nite the number of orbits is also nite The isotropy group of a cell

corresp onding to an ideal cell decomp osition is clearly equal to the group of

isotopy classes of dieomorphisms preserving this ideal cell decomp osition and

hence is nite On the contrary the isotropy groups of simplices in A R are

innite As in the case of Teichmuller spaces T of surfaces with b oundary see

S

there are subgroups of nite index in Mo d acting freely on T and hence

R R

on R It follows that we can use the CWcomplex R with its natural

action of Mo d in order to compute Mo d by the ab ove formula

R R

The application of this formula to this situation is not a simple matter at all

The problem is essentially a combinatorial one since an ideal cell decomp osi

tion of R is an essentially combinatorial ob ject and the corresp onding isotropy

group is the group of symmetries of this combinatorial ob ject But there is

no simple way to list all ideal cell decomp ositions up to homeomorphism of

course and to nd the orders of their groups of symmetries By some breath

taking combinatorial arguments lo oking like sort of a miracle for an uninitiated

top ologist or algebraic geometer one can compute exactly the combination of

these orders leading to Mo d We refer to the original pap ers of J Harer

R

and D Zagier HarZ and R Penner Pe for the details While in HarZ the

needed technique integration over the space of hermitian matrices app ears

as a deus ex machine R Penner Pe motivates it by some ideas from quan

tum physics A simplied pro of was provided by M Kontsevich Ko see Ko

App endix D More recently D Zagier Z provided a considerably simpler pro of

of the key combinatorial fact needed for the calculation of Mo d This new

R

pro of is based on the theory of characters of symmetric groups and avoids the

integration over the space of hermitian matrices Despite this dramatic simpli

cation the original pro ofs retain in the authors view considerable interest

2

Corollary B If S is a closed surface of genus g then Mo d

S

B g g

g

Pro of The corollary immediately follows from the theorem if we use the short

exact sequence

Mo d S Mo d

S S

1

where S is the result of making one hole in S cf the fact that virtual

Euler characteristic is multiplicative in short exact sequences cf Bro Chap

ter IX Prop osition and the easy fact that S is equal to the Euler

characteristic of S ie to g 2

Finally we note that HarZ also contains some information ab out the usual

P

i

Euler characteristic dim H Mo d Q

i S

Torsion in mapping class groups torsion in their co

homology and related topics

Mapping class groups contain many torsion elements which can b e exploited

to detect torsion in their cohomology Clearly any nite group acting on

a surface S by orientationpreserving dieomorphisms may b e considered as a

nite subgroup of Di S and hence leads to a nite subgroup of Mo d In

S

fact the latter subgroup of Mo d is isomorphic to as immediately follows

S

from the next theorem

Theorem A Let S be an orientable surface of negative Euler characteristic

If is a nite subgroup of Di S then the canonical homomorphism

Aut H S ZmZ

dened by the action of dieomorphisms on homology is injective for any m

Note that the homomorphism Aut H S ZmZ obviously factors

through Mo d This theorem is essentially due to JP Serre Ser for an

S

elementary pro of see I Theorem and Supplement

According to a famous theorem of S Kerckho Ke Ke every nite

subgroup of Mo d can b e obtained as the image of a nite subgroup of Di S

S

see Theorem B In practice nite subgroups of Mo d are always obtained

S

from an action of a nite group on S ie as images of nite subgroups of

Di S Kerckho s theorem tells us that we will not miss any nite subgroup

by restricting ourselves to this metho d Note that every nite group admits

an embedding into some mapping class group b ecause as is well known every

nite group o ccurs as a group of symmetries of a Moreover

L Greenberg Gre proved that every nite group is isomorphic to the group of

all automorphisms of some Riemann surface

Finite subgroups of Mo d are used to detect nontrivial cohomology classes

S

of Mo d by restricting cohomology classes to appropriate nite subgroups Of

S

course one can detect in this way only torsion cohomology classes b ecause

cohomology groups of nite groups always consist of torsion classes The rst

result of this sort was obtained by R Charney and R Lee ChL It was

improved by H Glover and G Mislin GlM GlM and by R Charney and

R Lee ChL It is convenient to state these results in terms of the stable

cohomology groups in the sense of homology stability discussed in Note

that there are cohomology stability theorems exactly similar to the homology

stability theorems of In fact stability theorems for cohomology follow from

m

the corresp onding stability theorems for homology In particular H M

S

do es not dep ends on S if the genus of S is suciently large compared to m Let

m m

us denote by H M the group H M for surfaces S of suciently large

S

genus Similar notation will b e used for untwisted co ecients other than Z

k

Theorem B For al l k the stable cohomology group H M contains an

element of order equal to the denominator of the rational number B k where

k

B is the k th Bernoul li number

k

This theorem is due to H Glover and G Mislin GlM GlM The num

b ers app earing in this theorem are the same as in the computation of the virtual

Euler characteristic cf Theorem A This remarkable fact apparently has

no known explanation In order to construct elements of H M Glover

and Mislin consider closed surfaces S and use the canonical representation

M Aut H S C dened by the action of dieomorphisms on homol

S

ogy This leads to at complex vector bundle with the b er H S C over

K M and the Chern classes of this bundle can b e considered as elements

S

of H M Since the bundle is at they are torsion elements One needs

S

to know that these Chern classes are nonzero and moreover have suciently

high order This is checked by restricting pulling back them to sp ecially con

structed nite cyclic subgroups of M The elements alluded to in the theorem

S

are some combinations of these Chern classes

The cyclic subgroups used to detect the order of cohomology classes from

Theorem B are constructed only for surfaces of fairly high genus compared

to the dimension of the classes The cohomology stability theorems imply

that these cohomology classes survive till a much lower genus It happ ens that

sometimes these cyclic groups cannot act nontrivially on surfaces of such a low

genus and hence cannot b e subgroups of the mapping class groups of these low

genus surfaces In this way one gets examples of ptorsion cohomology classes

of mapping class groups having no ptorsion elements where p is some prime

number Such torsion elements in cohomology are somewhat unusual and are

called exotic torsion elements The existence of exotic torsion elements was

observed in GlM

Theorem C The stable cohomology H M Z with coecients Z

contains as a direct summand the cohomology H Im J Z of the space Im J

with the same coecients

This theorem is due to R Charney and R Lee ChL The space Im J is a

well known character from homotopy theory It is known that its cohomology

groups consist of torsion elements replacing the co ecients Z by Z just

kills the torsion R Charney and R Lee use some cyclic subgroups in a

manner similar to that of H Glover and G Mislin GlM but the subgroups of

Charney and Lee are smaller and simpler to construct together with some deep

ideas from homotopy theory and algebraic K theory the metho ds of H Glover

and G Mislin are more elementary Another dierence is that R Charney

and R Lee use the canonical homomorphisms M Aut H S ZpZ their

S

images are contained in the symplectic groups over the eld F ZpZ and

p

a lot is known ab out the cohomology of these groups from the work of Quillen

and others instead of the homomorphisms M Aut H S C used by H

S

Glover and G Mislin Theorem C can b e used to pro duce some torsion

classes in addition to those constructed by H Glover and G Mislin GlM

b e One can easily realize H M as the cohomology of some space Let S

g

a surface of genus g with one b oundary comp onent We may assume that S

g

is contained in S and hence S n int S is a torus with two holes for all

g g g

g Then we have a sequence of inclusions S S S

g g

inducing inclusions of the corresp onding mapping class groups M

1

M where M M M Let M b e the direct limit ie

S

n g g

n

the union of these groups Clearly the cohomology H M of this group

is exactly what was denoted by H M ab ove By the denition H M is

the cohomology H K M of the space K M A natural problem is to

rene Theorem C and pro duce a splitting on the level of spaces not just

cohomology groups In fact it is natural to replace K M by the simply

K M having the same cohomology groups as K M

Here denotes Quillens plusconstruction which is welldened for K M

b ecause H K M as a corollary of Theorem B Charney and Cohen

ChC proved that it is p ossible to nd such a splitting in the stable homotopy

category Note that the dierence b etween K M and K M disapp ears

in the stable

Theorem D Im J Z ie the space Im J localized away from is

a stable retract of K M

The pro of uses some noncyclic nite groups they are iterated wreath

k

pro ducts of cyclic groups But R Charney and F Cohen do not embed them

in any M Instead of this they construct maps K K M in

S k

fact only over nite skeletons using the homology stability theorems for map

ping class groups in order to split natural maps of the form K M

S

K M where S is a surface with one b oundary comp onent and R is the

R

result of gluing a disc to this b oundary comp onent

Recently U Tillmann Til proved that the splitting of Theorem C

indeed can b e realized on the level of spaces without passing to the stable

homotopy category Namely she proved the following theorem

Theorem E There is a space Y such that K M is homotopy equiva

lent to Im J Z Y

The pro of is based on Theorem D and the following fundamental theorem

of U Tillmann Til

Theorem F K M has the homotopy type of an innite

This theorem is obviously of great interest indep endent of its applications

to Theorem E The pro of is based on a construction of suitable categories

using the results of a previous pap er of U Tillman Til which lead to an

acceptable input for an innite lo op space machine Then a new group comple

tion theorem and Harers homology stability theorem see are used to iden

tify the resulting innite lo op space with K M Tillmanns innite lo op

space structure on K M extends the lo op space structure on K M

constructed previously by more elementary means by E Miller Mi and CF

Boedigheimer Bo But it is not clear yet if it extends the double lo op space

structure of Mi and Bo See also CoT for related results ab out the homology

op erations in H M

of a closed surface Now we turn to the mapping class group Mo d Mo d

S

2

S of genus R Lee and S Weintraub LeW and F Cohen Co proved that

H Mo d Z is a torsion group having nontrivial ptorsion for prime p only

for p or F Cohen Co Co showed that the torsion is completely

captured by a cyclic subgroup of order More precisely the following theorem

holds

Theorem G There is a homomorphism ZZ Mo d inducing an iso

morphism on primary components in homology and on homology with coe

cients ZZ In particular H Mo d Z is equal to ZZ plus some and

n

torsion groups for odd n and is a sum of and torsion groups for even n

The cohomology of Mo d with co ecients ZZ ZZ and ZZ was es

sentially completely describ ed by F Cohen and D Benson Co Ben BeCo

For the co ecients ZZ and ZZ even the ring structure of the cohomology

was determined while for the co ecients ZZ the ranks of the cohomology

groups were computed but the question of which of two p ossible ring structures

is correct was left op en These pap ers also contain some information ab out the

integral cohomology of Mo d the structure of which was further elucidated by

F Cohen Co We state only the result for the co ecients Z which is

esp ecially simple

Theorem H The ring H Mo d Z is isomorphic to

Zx y z x y z z

where x y z have degrees respectively

The pro of invokes a connection b etween Mo d and the mapping class group

6

Mo d Mo d of a sphere with six holes S due to Birman and Hilden BirH

S

0

Namely there is a short exact sequence

ZZ Mo d Mo d

with the image of ZZ generated by the hyperelliptic involution cf end of

for the latter In addition to this a close relation b etween Mo d and

the Artin braid group on six strings is exploited This geometric input leads

to cohomological computations by means of some p owerful homotopytheoretic

techniques For further results obtained by this approach we refer to Co

Co where some noncyclic nite subgroups of M namely quaternion and

S

dihedral groups in M play an essential role

S

For further information ab out many of the topics discussed in this section

and for other related results we refer to an excellent exp ository pap er Mis

by G Mislin In particular it contains a discussion of results of H Glover G

Mislin and Y Xia GlMX GlMX and of related results of Y Xia XX

see also XX and PriX for further results

Thurstons theory and its applications

This section is devoted to an overview of Thurstons theory of surface dieomor

phisms and some of its applications For more details we refer to Thurstons

up dated announcement of the main results Th and for a detailed exp osition

to the pro ceedings of the Orsay seminar FLP In the case of closed surfaces a

nice introduction from the p oint of view of hyperb olic geometry is provided by

A Casson and S Bleiler CB See also A Hatcher Hat

Classication of mapping classes

An element f Mo d is called reducible if it xes some simplex of C S but

S

p erhaps p ermutes its vertices and irreducible otherwise As usual we say that

n

an element f is of nite order if f for some n If an element f is

not of nite order and is irreducible then it is called pseudoAnosov It turns

out that elements in all three classes reducible nite order pseudoAnosov

can b e represented by dieomorphisms having some sp ecial prop erties These

sp ecial representatives play a crucial role in the understanding of corresp onding

mapping classes

First of all for a reducible element f xing a simplex of C S let us

consider the union C of some disjoint circles representing the vertices of We

will call such a union a realization of In this case there is a dieomorphism

F S S representing the isotopy class f and leaving C invariant F C C

Such a dieomorphism F induces a dieomorphism F S S of the

C C C

result S of cutting S along C The comp onents of S are in a denite sense

C C

simpler than S they have larger Euler characteristic and this allows the use

this pro cedure in inductive arguments One minor diculty comes from the fact

that S has in general several comp onents and F may p ermute them in a

C C

nontrivial way The standard way to deal with this is to replace F by some

C

p ower of it not p ermuting the comp onents Another more serious problem

comes from the fact that F may b e isotopic to the identity and then we

C

lose all the information after the cutting This happ ens exactly when f is a

pro duct of Dehn twists around the comp onents of C This case needs to b e

dealt with separately Anyhow any reducible mapping class can b e represented

by a reducible dieomorphism ie a dieomorphism preserving the union of

several pairwise disjoint and nonisotopic nontrivial circles

While the results outlined in the previous paragraph are essentially elemen

tary the next case namely the case of elements of nite order is much deep er

Sp ecial representatives of such elements are provided by the following theorem

of J Nielsen Ni

Theorem A If f Mo d is an element of nite order n then f can

S

represented by a dieomorphism F S S of the same order n Moreover

one can choose F to be an isometry of a metric of constant curvature on S with

geodesic boundary

See FLP Exp ose x for a pro of of this theorem in the context of

Thurstons theory and an outline of another pro of due to W Fenchel Fe As

a simple application of this theorem consider the kernel m of the natural

S

homomorphism

Mo d Aut H S ZmZ

S

dened by the action of dieomorphisms on homology Clearly m has nite

S

index in Mo d

S

Corollary B If m then m is torsion free

S

Pro of If the Euler characteristic of S is negative it is sucient to combine

Nielsens Theorem A with Serres Theorem A The remaining cases are

elementary and left to the reader 2

Finally the remaining elements which we called pseudoAnosov also can b e

represented by some very sp ecial dieomorphisms namely by pseudoAnosov

dieomorphisms To b e honest they are only homeomorphisms but their

nonsmo othness is very mild and harmless We will not give their denition

here which generalizes the twodimensional case of the denition of Anosov

dieomorphisms introduced by D V Anosov An and playing a central role in

the theory of dynamical systems see FLP or Th The existence of pseudo

Anosov dieomorphisms is one of the main achievements of Thurstons theory

of the dieomorphisms of surfaces

While the existence of nontrivial reducible mapping classes or mapping

classes of nite order is quite obvious it is not so for pseudoAnosov map

ping classes Let us indicate a couple of ways to get such mapping classes We

need the following notion we say that a set A of vertices of C S ie a set

of isotopy classes of nontrivial circles on S l ls S if for every vertex there is

a vertex A such that any two circles representing have a non empty

intersection For nite sets A this is equivalent to the following prop erty for

any collection of circles representing all isotopy classes from A all comp onents

of the complement of the union of these circles in S are either op en discs or

halfop en annuli and each of the latter contains a comp onent of S

Theorem C If are two isotopy classes of nontrivial circles such that

m

n

f g l ls S then al l elements t t for m n are pseudoAnosov

See FLP Exp ose Remark after theorem I I I Theorem I I I itself

shows that under the assumptions of Theorem C many other combinations

of t t are also pseudoAnosov These results were generalized by D Long

Lo and then by R Penner Pe to the situations when more than two Dehn

twists are involved Further examples were provided by M Bauer Bau The

following remarkable result due to A Fathi Fathi while not giving completely

explicit examples of pseudoAnosov elements shows that they are in a very

strong sense generic

n

Theorem D Let f Mo d and be a vertex of C S If the set ff

S

n

n Zg l ls S then al l elements t f are pseudoAnosov except for at most

consecutive values of n

n

For example if f g lls S then ft n Zg also lls S The pap er

Fathi includes some other results of this sort but the following question re

mains op en is it p ossible to replace the Dehn twist t in this theorem by an

arbitrary comp osition of Dehn twists ab out several disjoint circles p ossibly at

the cost of replacing the number by some may b e undetermined but universal

number The results of A Fathi were preceded by a theorem of D Long and

H Morton LoM in which only the niteness of the number of exceptions ie

n

number of n such that t f is not pseudoAnosov was asserted instead of the

existence of a universal constant by Theorem D

When one deals with individual mapping classes one can usually reduce the

problem to the case of irreducible mapping classes by the metho d of cutting

outlined ab ove When one deals with the whole mapping class group or a

general subgroup of it usually there is no way to avoid reducible elements But

one can often restrict attention to following in some sense simplest type of

reducible elements Let us call a mapping class f Mo d pure if it can b e

S

represented by a dieomorphism F S S xing p ointwise some union C of

disjoint and pairwise nonisotopic nontrivial circles on S and such that F do es

not p ermute the comp onents of S n C and induces on each comp onent of the

cut surface S a dieomorphism isotopic either to a pseudoAnosov or to the

C

identity dieomorphism For example every pseudoAnosov element is pure as

is also any pro duct of Dehn twists ab out several disjoint circles In the latter

case the induced dieomorphism of S is isotopic to the identity in all other

C

cases F induces a dieomorphism isotopic to a pseudoAnosov dieomorphism

on at least one comp onent of S

C

Theorem E If m then m consists of pure elements

S

This theorem strengthens Corollary B and is a natural sharp ening of

it in the framework of Thurstons theory For a pro of see I Corollary

This theorem often allows the immediate elimination of all diculties related to

elements of nite order and to p ossible p ermutations of comp onents of the cut

surface by reducible dieomorphisms For example if one considers a subgroup

G of Mo d it is often p ossible to replace it by G m m

S S

Another imp ortant to ol for dealing with reducible elements is the notion of

essential reduction classes introduced in a slightly dierent form by J Birman

A Lub otzky and J McCarthy BLM For a pure element f Mo d we say that

S

a vertex of C S is an essential reduction class of f if f and f

for all such that cannot b e represented by disjoint circles For a general

element f Mo d we say that a vertex of C S is an essential reduction class

S

of f if it is an essential reduction class of some nontrivial and than as one may

n

prove of any nontrivial p ower f n which is pure Note that f has pure

p owers in view of Theorem E and the fact that m has nite index in

S

Mo d It turns out that the set of all essential reduction classes of f is a simplex

S

of C S which we call the canonical reduction system of f and denote by f

If f is pure it xes all vertices of f In general f leaves f invariant The

crucial fact is that f is non empty if f is a reducible element of innite order

f is empty for any element of nite order f reducible or not In particular

cutting along some realization of f provides a canonical way to simplify a

reducible element f If f is pure then this leads to a dieomorphism of the cut

surface which leaves all comp onents invariant and is isotopic on any comp onent

either to the identity or to a pseudoAnosov dieomorphism

Measuring against circles and its applications

In this section we discuss one of the basic technical to ols of Thurstons theory

measuring various geometric ob jects on a surface S against the isotopy classes of

nontrivial circles ie the vertices of C S We will denote by V S the set of all

vertices of C S ie the set of all isotopy classes of nontrivial circles One of the

fundamental ideas of Thurston was to consider measurements with resp ect to all

vertices simultaneously The natural place to record all such measurements is

the set RS of all functions V S R endowed with the pro duct top ology

The mapping class group Mo d acts on V S and this action induces an action

S

of Mo d on RS The next imp ortant idea is to pass from the space RS

S

to a pro jectivization of it namely to the quotient PRS of RS n fg by the

natural action of the multiplicative group R on it Clearly the

action of Mo d on RS induces an action of Mo d on PRS

S S

Let us illustrate this approach by the example of the Teichmuller space T

S

A natural but not the only p ossible way to measure the p oints of T ie

S

the classes of hyperb olic metrics on S with resp ect to the vertices V S

is to use the length functions l T R from The collection of all

S

length functions denes an embedding T RS Clearly this embedding

S

is Mo d equivariant Its image is contained obviously in RS n fg The

S

comp osition T PRS of this embedding with the canonical pro jection

S

RS n fg PRS is still an embedding which is also Mo d equivariant It

S

turns out that the embedding T PRS is a homeomorphism onto its image

S

this image has a compact closure in PRS and moreover this closure is home

omorphic to a closed disc of the dimension equal to the dimension of T under

S

a homeomorphism taking the image of T into the interior of this disc In this

S

way we achieve a natural compactication of T by a sphere called Thurstons

S

boundary of Teichmuller space The compactication itself is called Thurstons

compactication of Teichmuller space An advantage of this compactication

is that the action of Mo d extends to it by continuity as is clear from the

S

construction S Kerckho Ke proved that this desirable prop erty do es not

hold for the classical compactications of Teichmuller space by a sphere the so

called Teichmuller compactications A Teichmuller compactication dep ends

on a choice of a base p oint in T and Kerckho s result holds for any choice of

S

the base p oint In fact Kerckho proved that compactications corresp onding

to dierent base p oints are often dierent and this is closely related to the

imp ossibility to extend the action of Mo d to any of them by continuity

S

Another example of the idea of measuring geometric ob jects against the

isotopy classes of circles is provided by measuring the isotopy classes of circles

themselves For V S let us dene the geometric intersection number

i as the minimal number of p oints in the intersections A B where A B

run over all circle representatives of the isotopy classes resp ectively For

any V S the function I i b elongs to RS Moreover it is

easy to see that for any V S there is some V S such that i

Hence I RS n fg for every V S By assigning I to we get a map

V S RS n fg It is easy to see that this map is injective Moreover its

comp osition with the pro jection RS n fg PRS is still injective this is a

little more dicult The remarkable fact is that the image of V S under this

comp osition V S PRS is contained in Thurstons b oundary of T and

S

is dense in this b oundary In some sense this image plays the role of rational

p oints of Thurstons b oundary

It turns out that other p oints of Thurstons b oundary also admit a geometric

interpretation The relevant geometric ob ject is a measured with sin

gularities or more precisely the Whitehead equivalence class of such a foliation

on S Usually one sp eaks simply ab out measured on S The set of

all Whitehead equivalence classes of measured foliations on S is denoted by

MF S We omit the denitions referring to FLP and Th and restrict

ourselves to the following remarks A measured foliation on S is a co dimension

foliation on S so the leaves have dimension Since the Euler characteristic

of S is almost never zero S usually do es not admit honest foliations and we

have to allow some singularities in them The term measured means that our

foliation is equipp ed with a socalled transverse measure which allows us to

measure lo cally say within a ow b ox of the foliation the distance b etween

the leaves The transverse measure allows us to dene the transverse length of

curves and in particular of circles if a is contained in a leaf its transverse

length is

The notion of the transverse length of a circle allows us to deal with the

measured foliations in a way similar to the way we dealt with hyperb olic met

rics or rather p oints of T and circles or rather vertices of C S ab ove

S

Namely given a vertex V S and a measured foliation we dene l

to b e the inmum of transverse lengths with resp ect to of all circles in S rep

resenting the isotopy class In contrast with the hyperb olic metrics compare

this inmum is not always achieved but it is achieved on a not necessarily

embedded curve homotopic to circles representing For any measured folia

tion the function L l b elongs to RS It turns out that for every

measured foliation there exists some V S such that l Hence

L RS n fg for every MF S By assigning L to we get a map

MF S RS n fg It is not very hard to see that this map is injective In

particular we can equip MF S with the top ology induced from RS by this

map The comp osition of this map with the pro jection RS n fg PRS

turns out to b e not injective The reason is very simple the multiplication of

transverse measures by p ositive constants denes a natural action of the multi

plicative group R on MF S and our map MF S RS n fg is obviously

R equivariant So it is only natural to consider the quotient P MF S of

MF S by this action of R Clearly our map MF S RS n fg leads

to a map P MF S PRS which turns out to b e injective The image of

this map is equal to Thurstons b oundary of T The interpretation of p oints of

S

Thurstons b oundary in terms of measured foliations plays a crucial role in the

pro of of the fact that Thurstons b oundary is a sphere forming a ball together

with the Teichmuller space

Note that any isotopy class of a nontrivial circle naturally leads to an equiv

alence class of measured foliations and the images in RS of an isotopy class

of a circle and of the corresp onding measured foliation are the same So a

measured foliation turns out to b e in some sense a generalized circle

It is worth p ointing out that Thurstons b oundary can b e constructed and

its main prop erties can b e established using only the notion of a measured

foliation without any reference to Teichmuller spaces and for many applications

it is sucient to use only Thurstons b oundary itself together with the action

of Mo d on it

S

Action of the mapping classes on Thurstons compact

ication of Teichmuller space

Let us denote by T Thurstons compactication of Teichmuller space ie the

S

union of T with its Thurstons b oundary In this section we describ e the main

S

features of the action of various classes of elements of Mo d introduced in

S

T on

S

First of all the elements of nite order can b e characterized by the fact

that they have a xed p oint in the Teichmuller space T itself The set of xed

S

p oints in T can b e identied with some other Teichmuller space A similar

S

T result probably holds for the set of xed p oints in

S

A pseudoAnosov element f has exactly two xed p oints in T and b oth

S

of them b elong to Thurstons b oundary there are no xed p oints in T itself

S

One of the xed p oints is in a natural sense attracting and the other one is

repelling In the following theorem a denotes the attracting p oint and r denotes

the rep elling p oint

Theorem A Let f be a pseudoAnosov element There exist two points a

T r in Thurstons boundary of T such that if U is any neighborhood of a in

S S

n

T n fr g then f K U for al l suciently large and K is any compact set in

S

n

So all p oints in T n fr g converge under the action of the iterations of f to a

S

uniformly on compact sets This justies the terms attracting and rep elling

Since b oth T and its b oundary are obviously invariant a similar result holds

S

T One can nd a somewhat weaker result for Thurstons b oundary instead of

S

not asserting the uniformity of the convergence in FLP cf FLP Exp ose

Corollaire I I Theorem A itself easily follows for example from Theorem

B b elow

The action of reducible elements is more complicated If a reducible pure

element f is not a comp osition of Dehn twists ab out several disjoint circles

so if we cut S along a realization of f then the induced dieomorphism

of the cut surface is isotopic to a pseudoAnosov dieomorphism on at least

one comp onent then we have a picture similar to the ab ove description of the

action of a pseudoAnosov element We state the result only for the action on

Thurstons b oundary Its set of xed p oints consists of two comp onents one of

which is in a natural sense attracting while the other one is repelling In the

following theorem A denotes the attracting set and R denotes the rep elling

set

Theorem B Let f be a pure element of Mo d which is not a composition

S

of Dehn twists about several disjoint circles There exist two disjoint subsets

A R of Thurstons boundary of T such that if U is any neighborhood of A in

S

Thurstons boundary and K is any compact set in the complement of R then

n

f K U for al l suciently large n

This result is proved in I App endix where also the comp onents of the

xed p oint set are completely describ ed Theorem A is a sp ecial case of

Theorem B and this description Note that a weaker result in which the

attracting and the rep elling sets are not disjoint in general cf I Theorem

is sucient for most applications Another version of this weaker result

was proved by J McCarthy Mc cf Mc Uniform Convergence Lemma

Some applications

We start with a famous result of Thurston ab out manifolds which in fact

served as the motivation for Thurstons theory of surface dieomorphisms Re

call that the mapping torus M of map f X X is the quotient space

f

X x f x If X is a manifold and f is dieomorphism then

M is also a manifold

f

Theorem A Let f S S be a dieomorphism of a closed surface S The

mapping torus M admits a riemannian metric of constant negative curvature

f

if and only if the isotopy class of f is pseudoAnosov

A slightly more complicated result holds for surfaces with b oundary as well

This theorem is the most dicult sp ecial case of Thurstons famous geometriza

tion theorem for Haken manifolds Of course the pro of involves much more than

Thurstons theory of surface dieomorphisms The rst account of this theorem

is due to D Sullivan Su the exp osition in Su is very demanding Recently

new exp ositions of this theorem were published by C McMullen McM see

McM Chapter and JP Otal O C McMullens approach is close to

the original ideas of Thurston and refers to some still unpublished preprints

of Thurston The approach of JP Otal follows the one of Thurston in the

general outline but provides a completely new pro of of one of the key steps

the socalled double limit theorem his exp osition is essentially selfcontained

Hop efully Thurstons own pro of Th will also app ear so on See also notes by

M Kap ovich Kap

The next theorem due to S Kerckho Ke Ke provides a solution of

the socalled Nielsen realization problem

Theorem B Let G be a nite subgroup of Mo d Then there is a nite

S

subgroup G of Di S which maps isomorphically onto G under the natural

map Di S Mo d Moreover one can choose G to be a subgroup of the

S

group of isometries of some metric of constant curvature on S with geodesic

boundary

One says that G realizes G Note that this theorem generalizes Theorem

A from cyclic to arbitrary subgroups of Mo d The key ingredient of the

S

pro of is provided by the socalled earthquake ows on the Teichmuller space

T These ows are related to measured foliations in the same manner as the

S

FenchelNielsen ows from are related to circles In fact earthquake ows

are the limits of suitably chosen and parameterized FenchelNielsen ows

The main prop erties of the earthquake ows used in the pro of of Theorem

B are the convexity of the length functions from along the ow lines of

earthquake ows and the fact that every two p oints can b e connected by the

ow line of an earthquake ow We refer to Ke for the details see also Ke

Some further developments resulting from this solution of the Nielsen realization

problem are presented in Ke After Kerckho s work S Wolpert Wo and

A Tromba Tr see Tr Section Tr found other pro ofs of Theorem

B indep endent of Thurstons theory but still based on convexity prop erties

of some functions in fact the same length functions in Wolperts pro of but

along dierent paths in T

S

The rst applications of Thurstons theory to the algebraic prop erties of

mapping class groups are due to J McCarthy Mc who noticed that Thurstons

theory provides a go o d description of normalizers and centralizers of pseudo

Anosov elements and to J Birman A Lub otzky and J McCarthy BLM who

proved in particular the following theorem

Theorem C If a subgroup of Mo d contains a solvable subgroup of nite

S

index then it contains an abelian subgroup of nite index

So on after BLM the author proved I the following more strong theorem

Theorem D Let G be a subgroup of Mo d and let m Then either G

S

contains a free nonabelian group with two generators or G m is abelian

S

Since m is of nite index in Mo d and a solvable group cannot contain

S S

a free group with two generators Theorem D implies Theorem C The

following corollary of Theorem D which is only slightly weaker than Theo

rem D itself was simultaneously and indep endently proved by J McCarthy

Mc

Theorem E Let G be a subgroup of Mo d Then either G contains a free

S

group with two generators or G contains an abelian subgroup of nite index

A similar prop erty with ab elian replaced by solvable holds for arbitrary

linear groups ie subgroups of GL k for a eld k according to a famous

n

theorem of J Tits Tits In fact the search for an analogue of Tits theorem

for subgroups of the mapping class groups led to Theorems D and E

Later on the author I I discovered that some other fundamental theorems

ab out the linear groups have analogues for subgroups of mapping class groups

In order to state them recall that a subgroup H of a group G is called maximal

if for every subgroup K of G such that H K either H K or K G Recall

also that the Frattini subgroup F G of a group G is dened as the intersection

of all maximal subgroups of G An intuitive characterization of F G is provided

by the following elementary result F G is the set of all nongenerating elements

of G ie such elements g G such that for every generating set X of G the

set X n fg g is also generating

Theorem F Let G be a nitely generated subgroup of Mo d Then ei

S

ther G contains a maximal subgroup of innite index or G contains an abelian

subgroup of nite index

It is easy to see that if a group G contains an ab elian subgroup of nite in

dex then all maximal subgroups have nite index While usually it is very easy

to nd a maximal subgroup of nite index at least when there are nontrivial

subgroups of nite index take a subgroup of minimal index it is hard to

prove the existence of maximal subgroups of innite index For linear groups

a theorem similar to Theorem F with solvable instead of ab elian was

proved by G A Margulis and G A Soifer MaSo their result motivated The

orem F

Theorem G Let G be a nitely generated subgroup of Mo d Then the

S

Frattini subgroup F G is nilpotent

This is an analogue of a theorem of V P Platonov Pl to the eect that

Frattini subgroups of nitely generated linear groups are nilp otent It was

proved by the author I after some partial results of D Long Lo

The pro ofs of Theorems CG are based on the prop erties of the action

of the elements of Mo d on Thurstons b oundary of the Teichmuller space T

S S

and in particular on the results discussed in A crucial role is also played

by the following result in which we call a subgroup G of Mo d irreducible if

S

there is no nonempty simplex of C S such that g for all g G

Theorem H If G is an innite irreducible subgroup of Mo d then G con

S

tains a pseudoAnosov in particular an irreducible element

Note that a similar statement for nite subgroups is false according to J

Gilman Gi there are nite irreducible subgroups of Mo d consisting entirely

S

of reducible elements In view of Theorems B and H it is natural to

classify the subgroups of Mo d in a manner similar to Thurstons classication

S

of elements of Mo d Namely a subgroup of Mo d can b e reducible ie

S S

not irreducible nite or innite irreducible Note that these classes are not

completely mutually exclusive a subgroup can b e nite and reducible The

following theorem provides a useful complement to Theorem H and this

classication

Theorem I If a subgroup G of Mo d contains a pseudoAnosov element

S

then either G contains an innite cyclic subgroup of nite index generated by

a pseudoAnosov element or G contains a free group freely generated by two

pseudoAnosov elements

Theorems CG are proved using the same general outline First we

replace our group G by G m for some m If the resulting subgroup

S

still denoted by G is reducible then one may cut S along some realization

of a nonempty simplex of C S such that g for all g G This

reduces the problem to a similar problem on the cut surface which has simpler

comp onents If a subgroup G is irreducible and innite then Theorems H

and I provide us with pseudoAnosov elements in G which play the key role

in the rest of the pro of Note that nite groups present no problems in these

theorems For the details see I

Commuting elements and Dehn twists

Thurstons theory allows us to formulate a very useful criterion for two elements

of Mo d to commute We state it only for the pure elements this case usually

S

b eing sucient for the applications

Theorem A Let f g be two pure elements of Mo d These elements com

S

mute ie f g g f if and only if there is simplex of C S its realization

C S and representatives F G S S of f g respectively such that

i F and G preserve setwise every component of C and every component

of S n C

ii for every component R of the result S of cutting S along C the dieo

C

morphisms F G R R induced on R by F G respectively are isotopic to

R R

some powers of the same pseudoAnosov dieomorphism of R or to the identity

The pro of is based on the results of see I Section As an

application we give a purely algebraic characterization of p owers of Dehn twists

inside subgroups of pure elements Recall that the centralizer C g of an

G

element g of a group G is dened as the subgroup ff G f g g f g and that

the center C G of a group G is dened as the subgroup ff G f g g f for

all g Gg

Theorem B Let G be a subgroup of Mo d consisting entirely of pure el

S

ements for example let G m for some m An element g G is

S

a nontrivial power of a Dehn twist if and only if the center C C g of the

G

centralizer C g of g is isomorphic to Z and is not equal to the centralizer

G

C g of g

G

In fact C C g is isomorphic to Z only for nontrivial p owers of Dehn

G

twists and for pseudoAnosov elements The second condition serves to exclude

pseudoAnosov elements This theorem is a version of a characterization of

Dehn twists in I

Since p owers of Dehn twists obviously are pure elements one can apply

Theorem A to them and get the following result admittedly known b efore

Thurstons theory

Theorem C Let t t be two Dehn twists and let m n be two nonzero

n m n m

t if and only if and can be represented by t integers Then t t

disjoint circles

Theorem A also allows us to easily nd centers of the mapping class

groups

Theorem D If S is neither a sphere with holes nor a torus with

holes nor a closed surface of genus then the center C Mo d is trivial ie

S

C Mo d If S is a closed surface of genus then the center C Mo d is

S S

isomorphic to ZZ and is generated by the hyperelliptic involution from

The corresp onding result for the cases excluded in this theorem is only

slightly more complicated Theorem D is well known but apparently there

is no go o d reference for it

Corollary E If S is neither a sphere with holes nor a torus with

holes nor a closed surface of genus then the canonical action of Mo d on

S

C S is eective If S is a closed surface of genus the kernel of the canonical

action of Mo d on C S is equal to the subgroup of order generated by the

S

hyperelliptic involution from

This easily follows from Theorem D if one uses the fact that Dehn twists

generate PMo d by Theorem C

S

Automorphisms of complexes of curves

The theorem ab out automorphisms

The action of Mo d on C S induces a homomorphism Mo d Aut C S

S S

This homomorphism is injective if S is neither a sphere with holes nor a

torus with holes nor a closed surface of genus cf Corollary E It

turns out that this homomorphism is also surjective in these cases and also for

closed surfaces of genus

Theorem A Suppose that S is not a sphere with holes and not a torus

with holes Then al l automorphisms of C S are induced by the elements

of Mo d In particular if in addition S is not a closed surface of genus

S

then Aut C S is canonically isomorphic to Mo d If S is a closed surface of

S

genus then Aut C S is canonically isomorphic to the quotient of the group

Mo d by the subgroup of order generated by the hyperelliptic involution

S

This theorem is due to the author I for surfaces S of genus and to

M Korkmaz Kor in the remaining cases See also I for an expanded and

up dated version of I Most of the pro of namely the ideas discussed in

for the genus case works also in the genus and cases The main diculty

in treating these low genus cases overcome by M Korkmaz consisted in nding

and proving an analogue of Lemma A b elow

If S is either a sphere with holes or a torus with holes then C S

is an innite set of vertices without any edges ie dim C S and hence

Aut C S is an innite Obviously dieomorphisms of S

cannot p ermute vertices of C S arbitrarily even in these cases and thus the

conclusion of the theorem is not true for such S If S is a sphere with at

most holes then C S is empty but Mo d is not trivial it includes at least

S

one orientationreversing mapping class Hence the conclusion of the theorem

is not true by trivial reasons for such S also F Luo Luo observed that

Aut C S is not equal to Mo d if S is a torus with holes The reason is very

S

simple If S is a torus with holes and S is a sphere with holes then

in is not isomorphic to Mo d C S is isomorphic to C S but Mo d

S S

05 12

acts transitively on the set of vertices of C S and the action fact Mo d

S

05

on the set of vertices of C S has two orbits corresp onding to of Mo d

S

12

separating and nonseparating circles cf for a similar phenomenon F Luo

Luo also suggested another approach to Theorem A still based on the

ideas of I outlined b elow and also on a multiplicative structure on the set

of vertices of C S introduced in Luo

Theorem A is similar to and was motivated by a well known theorem

of J Tits Tits to the eect that all automorphisms of a Tits building with

some exceptions as in our case stem from automorphisms of the corresp onding

algebraic group see Tits Introduction Problem B The theorem of Tits

in turn has its ro ots in the Fundamental Theorem of Projective Geometry

asserting that maps of a pro jective space to itself which map lines to lines

planes to planes etc are in fact pro jectively linear

In we will outline the main ideas of the pro of of Theorem A

After this we will discuss applications of this result to mapping class groups

and Teichmuller spaces

Intersection number prop erty

We start by introducing some terminology Two isotopy classes of non

trivial circles on S ie two vertices of C S are said to have geometric inter

section number resp ectively if they can represented by two circles inter

secting transversely in exactly one p oint resp ectively by two disjoint circles

If this is the case we write i resp ectively i These de

nitions form a sp ecial case of a general notion of geometric intersection number

from Clearly two isotopy classes have geometric intersection number if

and only if they are connected by an edge in C S Obviously these intersection

numbers are preserved by the action of Mo d The starting p oint of the pro of of

S

Theorem A at least for surfaces of genus is the fact that the prop erty

of having geometric intersection number is preserved by all automorphisms of

C S which a priori preserve only the prop erty of having geometric intersection

number The key part of the pro of of this fact is contained in the following

lemma at least when the genus is

Lemma A Suppose that the genus of S is at least Let be isotopy

classes of two nontrivial circles on S Then the geometric intersection number

i if and only if there exist isotopy classes of nontrivial

circles having the fol lowing two properties

i i if and only if the ith and j th circles in Fig are

i j

disjoint

ii if is the isotopy class of a circle C then C divides S into two parts

one of which is a torus with one hole containing some representatives of the

isotopy classes

Pro of Fig provides a pro of of the only if part In view of the prop erty

ii the pro of of the if part can b e reduced to an examination of a torus with

one hole which is not dicult We omit the details 2

Corollary B Suppose that the genus of S is at least Every automor

phism of C S maps pairs of vertices having geometric intersection number

into pairs of vertices also having this property

Pro of It is sucient to show that the prop erties i and ii of Lemma A

are preserved by every automorphism of C S As we noted b efore the lemma

two vertices have geometric intersection number if and only if they are con

nected by an edge in C S It follows that the prop erty i is preserved by all

automorphisms In order to prove that the prop erty ii is also preserved we

will show how to express it in terms of the structure of C S only

Let us consider a vertex hC i of C S Let L b e the of in

C S ie the set of all vertices of C S connected by an edge with with the

induced structure of a simplicial complex Consider the graph L having the

same vertices as the link L and having as edges exactly those pairs of vertices

which are not edges of L It is easy to see that the connected comp onents of

L corresp ond exactly to the connected comp onents of the result S of cutting

C

S along C at least if none of them is a sphere with holes After detecting the

comp onents of S with the help of L when one of the comp onents is a sphere

C

with holes we can detect only the other but this turns out to b e sucient

we can return to the link L itself and try use Corollary B or the table

preceding it to recognize the top ological type of these comp onents

Let us apply these remarks to The part of S not containing represen

tatives of should have genus one less than the genus of S and one more

Fig

Fig

Fig Fig

b oundary comp onent If the genus of S is these prop erties can b e recog

nized with the help of Corollary B The case of genus is only a little more

dicult one should use not only this Corollary but also the table preceding

it It follows that the prop erty ii can b e expressed in terms of the structure

of C S alone and hence is preserved by all automorphisms of C S This

completes the pro of 2

Now we turn to the genus case The following analogue of Lemma A

was proved by M Korkmaz Kor

Lemma C Suppose that S is a torus with at least two holes Let be

isotopy classes of two nontrivial circles on S Then the geometric intersection

number i if and only if there exist isotopy classes having

the fol lowing two properties

i i if and only if the ith and j th circles in Fig are

i j

disjoint

ii if is the isotopy class of a circle C then the circles C C and C

i i

are non separating and both C and C divide S into a torus with one hole and

a sphere with b holes where b is the number of boundary components of S

Pro of Similarly to the pro of of Lemma the pro of of theonly if part is

provided by the Fig The pro of of the if part is a little more dicult

than the pro of of the if part of Lemma but we again omit the details 2

Corollary D Suppose that S is a torus with at least two holes Every auto

morphism of C S maps pairs of vertices having geometric intersection number

into pairs of vertices also having this property

The pro of is more dicult than that of Corollary B partly b ecause

the table preceding Corollary B is not sucient to distinguish b etween the

surfaces of genus and We omit fairly complicated details

Surprisingly the vertices of b oth Lemmas A and C with

the edges connecting them form a p entagon presented in Fig and a similar

p entagon plays a role in the case of genus

The case of a sphere with holes

At rst sight none of the ab ove ideas can b e applied when S is a sphere with

holes all circles on a sphere with holes are separating and hence two isotopy

classes never have geometric intersection number It turns out that instead

of non separating circles one can use circles b ounding a disc with two holes

in S Then the role of pairs of isotopy classes having geometric intersection

number is taken over by the pairs of isotopy classes of such circles having the

simplest p ossible nontrivial intersection namely intersecting as in Fig

where the shadowed discs represent holes Such pairs of isotopy classes can b e

characterized in a manner similar to Lemmas A and C A p entagon as

in Fig app ears again but its ve vertices are not sucient now We refer

to Kor for the details

Note that the circles b ounding discs with two holes and pairs illustrated in

Fig are exactly what is needed to make the machinery of the next section

work cf the discussion of co dings in

Complexes of curves and ideal triangulations

Now we describ e the ideas involved in the remaining part of the pro of of Theorem

A These ideas work almost equally well for surfaces of any genus we

restrict ourselves to the genus case

We need a minor mo dication AS of the ideal triangulations of the Te

ichmuller spaces from The vertices of the complex AS are the isotopy

classes hI i of properly embedded nontrivial cf arcs I in S The isotopies

are not required to b e xed at the ends but are required to run in the class

of prop erly embedded arcs A set of vertices f g forms a simplex of

n

AS if and only if the isotopy classes can b e represented by pairwise

n

disjoint arcs I I The extended mapping class group Mo d acts on AS

n

S

in an obvious way

We can collapse each b oundary comp onent of S into a p oint dierent for

dierent b oundary comp onents and then consider these p oints as punctures

Let R b e the resulting surface with punctures There is a natural map S R

establishing a dieomorphism b etween the complements of the b oundary and

of the punctures Clearly this map induces an isomorphism b etween AS and

the complex AR from Using this isomorphism and Corollary B we

immediately get the following result

Lemma A Every automorphism of AS is determined by its action on

any single top dimensional simplex In particular if an automorphism of AS

agrees with the action of some element of Mo d on some simplex of codimen

S

sion then this automorphism agrees with this element of Mo d on the whole

S

complex AS 2

In view of the last Lemma the automorphisms of AS seem to b e much more

accessible than the automorphisms of C S The main idea of the remaining

part of the pro of of Theorem A is to reduce the study of automorphisms of

C S to a study of automorphisms of AS The main to ol is a co ding of the

vertices of AS in terms of the vertices of C S and some additional data

Let us describ e this co ding The vertices of AS are naturally divided into

three types and a vertex is co ded by its type and one or two vertices of C S

dep ending on its type The types of vertices are the following a the isotopy

classes of arcs I connecting two dierent comp onents D D of S b the

isotopy classes of arcs I connecting a comp onent D of S with itself such that

the image of this arc in R this image is a circle passing through a puncture do es

not b ound a disc with one puncture c the isotopy classes of arcs I connecting

a comp onent D of S with itself such that the image of this arc in R do es b ound

a disc with one puncture In addition to the type a vertex of type a is co ded

by hC i where C is the b oundary of some regular neighborho o d of D I D

in S a vertex of type b is co ded by the pair hC i hC i where C C are

two comp onents of a regular neighborho o d of D I and a vertex of type c is

co ded by hC i where C is the nontrivial comp onent of a regular neighborho o d

of D I one of the comp onents is trivial in this case

Now we need to prove that every automorphism of C S maps vertices

co ding vertices of AS of type a into similar vertices b ehaves similarly with

resp ect to types b and c and also maps co dings of pairs of vertices connected

by an edge in AS into similar co dings Fig may serve as an illustration of

co dings of a pair of vertices connected by an edge This is the most technically

dicult part of the pro of which splits into many cases according to the types

of the vertices involved After this is done we can assign an automorphism

of AS to any automorphism of C S Moreover one can prove that on an

explicitly given co dimension simplex of AS every automorphism of AS

induced by an automorphism of C S agrees with an element of Mo d at least

S

if the number of b oundary comp onents is In this case Lemma A implies

that the induced automorphism of AS is equal to some element of Mo d And

S

if the induced automorphism of AS is equal to some element of Mo d then

S

the original automorphism of C S is equal to some element of Mo d as it is

S

easy to see

This proves Theorem A for surfaces with b oundary comp onents

The case of surfaces with b oundary comp onents can b e reduced to the

case of surfaces with b oundary comp onents as follows Note rst that

every automorphism of C S takes the isotopy classes of non separating circles

into isotopy classes of non separating circles for example b ecause C is non

separating if and only if there exist a vertex such that i where

hC i Since all non separating circles are in the same orbit of the group of

dieomorphisms of S we can assume that our automorphism of C S xes some

vertex hC i where C is non separating Such an automorphism induces an

automorphism of the link L of and hence of the complex C S where S

C C

is the result of cutting S along C compare with the pro of of Corollary B

Since S has b oundary comp onents this automorphism of C S is equal

C C

Considering dierent non separating circles C in to some element of Mo d

S

C

fact all of them one can deduce that the original automorphism of C S is

equal to some element of Mo d If the genus of S is an additional diculty

S

is caused by the fact that the genus of S will b e But our automorphisms

C

of C S are induced by automorphisms of C S and hence retain all the nice

C

prop erties of the latter such as the preservation of the intersection number

This completes our outline of the pro of of Theorem A

An application to subgroups of nite index

As the rst application of Theorem A we give a complete description of

isomorphisms b etween subgroups of nite index in Mo d This result will nd

S

its own applications in cf Corollary B

Theorem A Suppose that S is not a sphere with holes and not a torus

with holes Let and be two subgroups of nite index of Mo d If S

S

in addition is not a closed surface of genus then al l isomorphisms

have the form x g xg for some g Mo d If S is a closed surface of

S

genus then al l isomorphisms have the form x g xg x for

some g Mo d and some homomorphism from to the center of Mo d

S S

ie the subgroup of Mo d of order generated by the hyperelliptic involution

S

see Theorem D can be nontrivial only if contains the hyperelliptic

involution

Pro of First note that for some natural number N the N th p owers

N

of all Dehn twists t are contained in b ecause is of nite index t

Next Theorem B easily implies that any isomorphism maps

suciently high p owers of Dehn twists into p owers of Dehn twists Taking into

account the fact that two nontrivial p owers of Dehn twists commute if and

only if the corresp onding isotopy classes of circles have geometric intersection

number by Theorem C we see that every such isomorphism induces an

automorphism C S C S By Theorem A this automorphism of C S

is given by some element g Mo d This means that for some suciently large

S

N we have

M

N

t t

g

for some M for all vertices of C S The p otential dep endence of M

on is irrelevant in what follows and we write simply M for M

Now let f Then for any vertex

M N N

t f t f t

g f f

On the other hand

N N M M

f t f f t f f t f t

g f g

Comparing the results of these two computations we conclude note that two

n m

nontrivial p owers t t of right Dehn twists are equal if and only if and

m n that f g g f for all and after putting g

that f g f g for all vertices of C S If S is not a closed surface

of genus this implies that f g f g ie has the required form see

Corollary E If S is a closed surface of genus then f can dier from

g f g by the hyperelliptic involution It is easy to see that this dierence can

b e describ ed by some homomorphism as stated 2

Corollary B Suppose that S is not a sphere with holes and not a torus

with holes Let be a subgroup of nite index of Mo d Then the outer

S

automorphism group Out is nite 2

Theorem A was proved rst by the author I for the case

Mo d or Mo d and for closed surfaces only Another but closely related pro of

S

S

in this case was provided by J McCarthy Mc who also noticed additional

automorphisms resulting from the hyperelliptic involution in the genus case

which were overlooked in I An alternative approach was suggested by R T

Tchangang T Shortly after I the author extended these results to surfaces

with b oundary and to some natural subgroups of Mo d like PMo d

S

S

see I A crucial element of these early approaches namely the fact that any

isomorphism takes p owers of Dehn twists to p owers of Dehn twists also plays

a crucial role in the ab ove pro of of Theorem A

Recently J McCarthy and the author extended the results of I Mc to

some injective homomorphisms b etween mapping class groups Let us measure

the size of a surface S by the number g b where g is the genus and b is

the number of b oundary comp onents of S this number is equal to the maximal

number of pairwise nonisotopic disjoint circles on S The main result of IMc

IMc asserts that if the sizes of two surfaces S S dier by at most and they

0

are in fact have the genus then injective homomorphisms Mo d Mo d

S S

isomorphisms and S is dieomorphic to S with p ossibly a few exceptions

An application to Teichmuller spaces

In this section we consider the Teichmuller spaces of punctured surfaces cf

The Teichmuller space T of a punctured surface R carries a natural struc

R

ture of a metric space given by its Teichmuller metric we note in passing that

Teichmuller spaces have a couple of other natural metrics This structure of a

metric space is not derived from a riemannian metric although it can b e derived

from a nslerian metric but nevertheless has some nice geometric prop erties

In particular T is a complete metric space every two p oints can b e connected

R

by a geo desic ie a lo cally shortest path and moreover geo desics have a

nice description in terms of the conformal geometry of surfaces For an intro

duction to this theory see W Abiko s b o ok A The fact that the metric is

naturally dened implies that Mo d acts on T by isometries A remarkable

R

R

result proved by H Royden R for closed surfaces and isometries preserving

the natural complex structure of T and then extended by C Earl and I Kra

S

EK to the surfaces with b oundary and general isometries asserts that there

are no other isometries More precisely we have the following theorem

Theorem A Suppose that R is not a sphere with punctures and not a

torus with punctures Then any isometry of T is induced by some element

R

of Mo d

R

Note that the conclusion of this theorem is not true for a sphere with

punctures and a torus with or punctures b ecause for these surfaces the

Teichmuller space is isometric to the hyperb olic plane which has a continuous

group of isometries For a sphere with punctures the conclusion is vacuous

b ecause the Teichmuller space consists of just one p oint For a torus with

punctures the conclusion of the theorem is not true b ecause if S is a torus

is isometric with punctures and S is a sphere with punctures then T

S

12

cf See EK is not isomorphic to Mo d but Mo d to T

S S S

05 12 05

H Roydens pro of and its extension by C Earl and I Kra is analytic and

is based on a detailed investigation of the nonsmo othness prop erties of the

nslerian metric underlying the Teichmuller metric It is essentially lo cal in

nature A detailed exp osition of this pro of is presented in F Gardiners b o ok

Ga cf Ga Chapter Our Theorem A can b e used to prove this theorem

in a completely dierent manner in all cases except that of a torus with two

punctures This new pro of outlined in I reveals a deep analogy b etween

Roydens theorem and G D Mostows rigidity theorems Most Most In

fact Theorem A plays a role in this pro of similar to the role played by

the theorem of Tits mentioned in in Mostows pro of Most In addition

to Theorem A this pro of of Theorem A naturally relies heavily on the

theory of the Teichmuller spaces and in particular on the results of S Kerckho

Ke and H Masur MasMas Unfortunately this theory falls outside the

scop e of the present pap er

Mapping class groups and arithmetic groups

Arithmetic groups

The analogy b etween arithmetic groups and mapping class groups was probably

suggested rst by Harvey Harv and had guided much of research ab out the

mapping class groups since then Let us give a downtoearth version of the

denition of arithmetic groups

The basic example of an arithmetic group is SL Z the group of all integral

n

n n matrices with the determinant note that the inverse of an integral

with the determinant is automatically an integral matrix All other

arithmetic groups can b e in some sense constructed from this one Supp ose

that G is a semisimple Qalgebraic subgroup of SL R This means that

n

G is simultaneously a semisimple Lie subgroup of SL R and a subset of

n

the space of all real n n matrices which can b e describ ed by p olynomial

equations for the matrix entries with rational co ecients For any such G

the intersection G of G with the set of integral matrices is by denition an

Z

arithmetic group Next let G b e the connected comp onent of the identity of

G and G H b e a homomorphism from G onto a connected semisimple

Lie group H with compact kernel to b e honest we should also assume that

H has trivial center and no compact factors but the reader may ignore this

remark Then G G is by denition an arithmetic group as is also

Z

any subgroup of H commensurable with it ie any subgroup such that

the intersection has nite index b oth in and More precisely we say

that any such is an arithmetic subgroup of H We also call arithmetic any

group isomorphic to any such

So any arithmetic group can b e obtained from SL Z in three steps taking

n

an intersection with a semisimple Qalgebraic real Lie group then taking the

image under a surjective homomorphism with compact kernel this step is the

most dicult to control and nally taking any subgroup commensurable with

this image In particular any subgroup of nite index of an arithmetic group is

also an arithmetic group

In addition to SL Z a typical example of an arithmetic group is the sym

n

plectic group S p Z which shows up in the top ology of surfaces as the group

g

of all automorphisms of the rst homology group H S of a closed orientable

g

surface S of genus g preserving the intersection pairing In particular we

g

S p Z which is well known to b e have a natural homomorphism Mo d

g S

g

surjective

In Harv W Harvey asked if the mapping class groups are arithmetic

They turned out to not b e as was proved by the author I I On the other

hand the years after Harv profoundly demonstrated that there is indeed a

deep analogy b etween mapping class groups and arithmetic groups esp ecially in

their cohomology prop erties the main results of Section are in fact motivated

by analogous results ab out arithmetic groups Although the rst pro of of the

nonarithmeticity of the mapping class groups was quite short cf it did

not exhaust the problem and the question of why the mapping class groups

are not arithmetic led to further results In the next section cf we will

describ e what seems to b e the b est answer to this question

Abstract commensurators

The notion of the abstract commensurator of a group has its ro ots in the the

ory of arithmetic groups The author learned it from NeR see also BasK

App endix B where it is attributed to JP Serre and W D Neumann It al

lows us to express one of the most striking dierences b etween arithmetic and

nonarithmetic groups cf Theorem A b elow and the remarks after it

Let b e a group Let us consider all p ossible isomorphisms

b etween subgroups of nite index of Let us identify two such iso

morphisms dened on resp ectively if they agree on a subgroup of

nite index in the intersection We can comp ose them in an obvious

manner the comp osition of with is dened

on Under this comp osition op eration the classes of such isomor

phisms form a group which is called the abstract commensurator of and is

denoted by Comm There is a natural map i Comm sending an

element of to the class of the inner automorphism g g This map

is injective if the centralizers of subgroups of nite index in are trivial as is

the case for arithmetic groups and for mapping class groups see Theorem D

for the latter

n

As a nice example let us mention that Comm Z SL Q

n

Theorem A If is an arithmetic group then i is of innite index in

Comm

Pro of We only outline the main idea Let us consider SL Z and show that

n

any element of SL Q naturally denes an element of Comm SL Z

n n

Let g SL Q and let m b e the least common multiple of all denominators

n

of the entries of the matrices g g Let f SL Z j I mo d m g

n

where I is the identity matrix Clearly is of nite index in SL Z If

n

then I m h where h is an integral matrix and hence g g

I m g hg is also an integral matrix Thus g g SL Z Let us

n

show that g g is a subgroup of nite index in SL Z Consider the

n

subgroup f SL Z j I mo d m g If then I m h

n

where h is an integral matrix and hence g g I m m g hg

b ecause m g hg is an integral matrix It follows that g g and hence

g g Since obviously has nite index in SL Z we conclude that

n

g g is of nite index

Now it is clear that the inner automorphism x g xg induces an isomor

phism b etween two subgroups of nite index of SL Z and hence

n

an element of Comm SL Z This leads to a homomorphism SL Q

n n

Comm SL Z Clearly its image contains iSL Z as a subgroup of in

n n

nite index This proves the theorem for SL Z

n

The same argument works for the groups G where G is a semisimple Q

Z

algebraic group simply b ecause G like SL Z has a corresp onding group

Z n

of rational matrices asso ciated to it After this it is not hard to extend the

result to the groups of the form G G cf and then to any group

Z

commensurable with such a group We refer to R Zimmers b o ok Zim for the

details cf Zim Section 2

A converse to this theorem is also true if is a in a semisimple Lie

group G ie if G has nite invariant and i is of innite index

in Comm then is arithmetic This result is an immediate corollary of

an arithmeticity theorem of G A Margulis cf Zim Chapter and G D

Mostows rigidity theorem Most Most cf NeR Section This converse

is much more deep and dicult than Theorem A and we do not need it for

our applications

Corollary B Suppose that S is not a sphere with holes and not a torus

and al l subgroups with holes Then the mapping class groups Mo d Mo d

S

S

of nite index in them are not arithmetic

Pro of It follows from Theorem A that Comm Mo d Comm Mo d

S

S

iMo d and hence Mo d Mo d cannot b e arithmetic by Theorem A

S

S S

If is a subgroup of nite index in Mo d then again Comm iMo d

S S

by Theorem A Note that the additional automorphisms for the closed

surfaces of genus disapp ear when we pass to a subgroup of nite index not

containing the hyperelliptic involution 2

If S is a sphere with holes or a torus with holes then Mo d is

S

indeed arithmetic if S is a torus with holes then Mo d is isomorphic

S

to SL Z if S is a sphere with holes then Mo d is commensurable with

S

PSL Z SL ZfI g cf I Section for a detailed discussion of this

case if S is a sphere with holes then Mo d is nite If S is a torus

S

with two holes then Mo d is not arithmetic The ab ove argument do es not

S

currently apply to this case b ecause Theorem A do es not include it But

older approaches discussed in the next two sections cf and do apply

to this case also

It is worth stressing that this pro of of the nonarithmeticity of the mapping

class groups uses only very basic prop erties of arithmetic groups going not

much further than the denition and a deep result Theorem A from the

top ology of surfaces This contrasts strongly with the original approach to the

nonarithmeticity which relied on the deep est prop erties of arithmetic groups

cf

The original approach to the nonarithmeticity

Arithmetic groups like representations can b e divided into reducible and ir

reducible ones An arithmetic group is called reducible if it is close to a

pro duct of two innite groups in particular if is reducible then it contains

two subgroups of innite index commuting with each other and gen

erating together a subgroup of nite index in Using the last remark and

Theorem A it is easy to see that no subgroup of nite index in Mo d can

S

b e isomorphic to a reducible arithmetic group If is not reducible it is called

irreducible

By the denition any arithmetic group arises as a subgroup of a real

semisimple Lie group H without compact factors The rank of the latter ie

the dimension of a Cartan subalgebra of its is usually called the

rank of In this section we will not address the question of whether this rank

dep ends only on or also on the realization of as an arithmetic subgroup of

some semisimple Lie group cf and will consider our use of this notion as

a slight abuse of language The main p oint is that the prop erties of dep end

strongly on whether the rank of is or

In particular if is an irreducible arithmetic group of rank then any

normal subgroup of is either a central nite subgroup or a subgroup of nite

index This is the famous Margulis niteness theorem see Zim Chapter for

the details Now for a closed surface S of genus g the kernel of the natural

g

S p Z from is an innite and noncentral homomorphism Mo d

g S

g

subgroup of innite index Indeed this kernel usually called the Torelli subgroup

contains for example all Dehn twists ab out separating circles and of Mo d

S

g

the ab ove homomorphism is surjective as we noted in It follows that Mo d

S

g

cannot b e an irreducible arithmetic group of rank Similar arguments apply

to surfaces with non empty b oundary for example one can use the kernels of

homomorphisms from instead of the Torelli subgroup or a version of the

latter

It remains to show that Mo d cannot b e an irreducible arithmetic group

S

of rank One way to show this is to use the well known fact that any such

arithmetic group contains a subgroup of nite index actually any torsionfree

subgroup of nite index will work which is isomorphic to the fundamental

group of a complete of nite volume with curvature pinched

b etween two negative constants together with the following theorem

Theorem A If S is not a sphere with holes and not a torus with

holes then no subgroup of nite index in Mo d is isomorphic to the fundamental

S

group of a complete riemannian manifold of nite volume with curvature pinched

between two negative constants

Pro of We only indicate the main idea It is sucient to prove that any sub

group of nite index in Mo d contains a subgroup isomorphic to Z Z Z

S

and that none of the considered fundamental groups do es Subgroups of Mo d

S

isomorphic to Z Z Z can b e constructed in abundance with p owers of

Dehn twists as generators The absence of such subgroups in the considered

fundamental groups follows easily from the theory of P Eb erlein and B ONeill

EbO see I Pro of of Theorem for the details 2

Irreducible arithmetic groups of rank can also b e excluded in a couple of

other ways See Har Section for a cohomological approach due to Harer

and still another approach due to W Goldman The latter is similar in spirit

to the use of the Eb erleinONeill theory ab ove

The ideas outlined in this section if combined with Mostows rigidity the

orem Most Most lead also to some nonarithmeticity results for normal

subgroups of Mo d even for socalled subnormal subgroups For example the

S

Torelli subgroup like the mapping class group itself is not arithmetic See I

Finishing the discussion of this original approach to the nonarithmeticity

we note that it uses almost no nontrivial information ab out mapping class

groups but relies heavily on the Margulis niteness theorem one of the deep est

results ab out arithmetic groups at all Cf the remarks at the end of

Rank of the mapping class groups

As we mentioned in prop erties of an arithmetic group dep end to a consider

able extent on whether its rank is or Given the analogy b etween mapping

class groups and arithmetic groups it is imp ortant to ascertain what arithmetic

groupsof rank or of rank would b e b etter analogues of mapping class

groups The b est answer to this question would b e a notion of rank applicable to

b oth arithmetic groups and mapping class groups together with a computation

of this rank for the mapping class groups

A suitable notion of rank was prop osed by W Ballmann and P Eb erlein

BalE who developed the ideas of G Prasad and M Ragunathan PrR Let

b e an abstract group Let denote the set of elements of whose centralizer

i

contains a free ab elian subgroup of rank i as a subgroup of nite index so

that Let r b e the least natural number i

i

such that can b e presented as a nite union

i m i

of left translates of the set the elements are arbitrary Let

i m

us dene the rank of as the maximum of the numbers r where runs

over all subgroups of nite index in We denote it by rank

G Prasad and M Raghunathan PrR showed that if is an arithmetic

group then r is equal to its rank in the sense of In particular this

implies that the rank of an arithmetic group in the sense of dep ends only

on the structure of as an abstract group and not on its particular realization

as an arithmetic subgroup of a semisimple Lie group The replacement of r

by rank was suggested by W Ballmann and P Eb erlein BalE in order to

get a notion of rank invariant under passing to subgroups of nite index They

proved that if is a discrete group of motions of a complete simply connected

riemannian manifold X whose curvature is nonp ositive and b ounded from b elow

such that X has nite volume then rank is equal to a geometrically dened

rank of X This result in fact includes the case of arithmetic groups and means

that this rank of an abstract group gives the desired result even for groups more

general than arithmetic ones

Theorem A If S is not a sphere with holes then rank Mo d

S

Pro of We will give only an outline of the pro of Let b e a torsion free

subgroup of nite index in Mo d such subgroups exist in view of Corollary

S

B Since rank is invariant under passing to subgroups of nite index

it is sucient to show that r for any such It is easy to see that

r Let denote the set of pseudoAnosov elements of Theorem

pA

A easily implies that the centralizers of pseudoAnosov elements contain an

innite cyclic subgroup as a subgroup of nite index Thus Hence

pA

it suces to prove that for some nite number of elements g g we

m

have

g g

pA m pA

In other words it is sucient to nd a nite number of elements g g such

m

g g g is pseudoAnosov that for any g one of the elements g

m

Using the compactness of the Thurston b oundary of the Teichmuller space

cf it is p ossible to show that there exists a nite collection of isotopy

classes such that for any isotopy class one of the pairs lls S

n i

in the sense of Fix such a collection Let t i n b e a

n i

contained in remember that is of nontrivial p ower of the Dehn twist t

i

nite index in Mo d Using Fathis Theorem D one can show that for any

S

d

g one of the elements t g i n d is pseudoAnosov It follows

i

that r We refer to I for the details 2

Corollary B For any compact orientable surface S the group Mo d is not

S

isomorphic to any arithmetic group of rank

One can replace all references to the Margulis niteness theorem in by

this Corollary This leads to still another pro of of the nonarithmeticity of the

mapping class groups On the arithmetic groups side it uses the results of G

Prasad and M Raghunathan PrR which are not as dicult as the Margulis

niteness theorem It also uses some quite nontrivial results on the mapping

class groups side Fathis Theorem D cf the remarks at the end of

The next theorem as we will explain in a moment also supp orts the idea

that the mapping class groups are similar to the arithmetic groups of rank as

opp osed to rank

Theorem C Suppose that S is not a sphere with holes Let us x a

nite set of generators of Mo d and let us denote by j j the minimal word

S W

length with respect to these generators If f is an element of innite order in

Mo d then there is a constant c such that

S

n

j f j c j n j

W

for al l n

It is easy to see that one can always replace f by a nontrivial p ower of f

in the pro of In particular we may assume that f is a pure element of Mo d

S

see Then there are two dierent cases to consider If the dieomorphism

induced on the result of cutting S along some realization of f is isotopic to

a pseudoAnosov dieomorphism on at least one comp onent then the theorem

follows from the results of L Mosher Mos Otherwise f is a comp osition of

Dehn twists ab out several disjoint circles This case was recently dealt with by

Y Minsky Min in resp onse to a question p osed by the author

Recall that an arithmetic group arises as a subgroup of a real semisimple

Lie group H The group is called cocompact if H is compact The sim

ilarity of Mo d with co compact arithmetic groups can b e easily excluded by

S

cohomological reasons as for example in Har Section cf But for

noncompact arithmetic groups of rank there are always elements f

n

of innite order such that the word length j f j grows slower than linearly in

W

fact logarithmically according to a recent result of A Lub otzky S Mozes and

M S Raghunathan LubMR Therefore Theorem C supp orts the analogy

b etween Mo d and rank arithmetic groups

S

The MostowMargulis sup errigidity

The Margulis sup errigidity theorem Ma cf also Ma or Zim Chapter

extends the Mostow rigidity theorem Most Most It asserts in particular

that any homomorphism b etween arithmetic subgroups of

semisimple Lie groups H H resp ectively without compact factors extends

to a homomorphism H H after p erhaps passing to a subgroup of nite

index in if rank Note that the restriction rank in this theorem

is necessary Here we meet one of the situations where arithmetic groups of rank

and rank b ehave dierently

The analogy b etween mapping class groups and arithmetic groups suggests

that one may extend this theorem to include mapping class groups on the equal

fo oting with arithmetic groups At rst sight this is imp ossible b ecause there

are no Lie groups naturally or to the b est of our knowledge otherwise con

taining mapping class groups But this simply means that there should b e no

homomorphisms or b etter taking into account the p ossible passage to a nite

index subgroup in the Margulis theorem that any homomorphism should have

nite image In addition since mapping class groups have rank and the source

group in the Margulis theorem should have rank we are forced to consider

only homomorphisms from arithmetic groups of rank to mapping class

groups Then it is natural to exp ect that any such homomorphism has nite

image

The rst step in the direction of this conjecture was made in I and the

conjecture itself was stated explicitly in I Further steps were made in I

Recently V Kaimanovich and H Masur KaM almost completely proved it

Following KaM let us call a subgroup of Mo d nonelementary if it leaves no

S

nite subset of the Thurston b oundary of the corresp onding Teichmuller space

invariant

Theorem A Any nonelementary subgroup of a mapping class group is not

isomorphic to an arithmetic group of rank

See KaM Theorem The pro of of V Kaimanovich and H Masur is

based on a theory of random walks on Teichmuller spaces and mapping class

groups developed by H Masur Mas and V Kaimanovich and H Masur KaM

and the results of H Furstenberg Fu Fu ab out random walks and harmonic

functions on discrete subgroups of Lie groups Fu provides an excellent in

tro duction to this circle of ideas More precisely H Furstenberg introduced a

prop erty of harmonic functions which are closely related to the random walks

on discrete groups which holds for all arithmetic groups of rank and do es

not hold for at least some arithmetic groups of rank V Kaimanovich and

H Masur show that this prop erty do es not hold also for all nonelementary

subgroups of mapping class groups thus they cannot b e arithmetic groups of

rank Note that these results of H Furstenberg as also the results of G D

Mostow preceded and inuenced G A Margulis approach to his sup errigidity

theorem

If combined with the Margulis niteness theorem and the results of I

cf ab out subgroups of mapping class groups Theorem A leads to the

following Corollary which proves the ab ove conjecture

Corollary B Let Mo d be a homomorphism from an arithmetic

S

group to a mapping class group Mo d If rank then the image of is

S

nite

A pro of of this Corollary along similar lines was also found by B Farb

and H Masur FM It seems that it is p ossible to prove this Corollary more

directly avoiding the use of the Margulis niteness theorem of random walks

and hence of Furstenbergs results and even of Teichmuller spaces but one

still needs their Thurstons b oundaries Such an approach would follow the

Margulis pro of of his sup errigidity theorem I hop e to return to these ideas in

a future pap er

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