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Automorphisms of Complexes of Curves and of Teichm Uller Spaces

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Automorphisms of Complexes of Curves and of Teichm Uller Spaces Automorphisms of complexes of curves and of Teichmuller spaces Nikolai V Ivanov To every compact orientable surface one can asso ciate following Harvey Ha Ha a combinatorial ob ject the socalled complex of curves which is analogous to Tits buildings asso ciated to semisimple Lie groups The basic result of the present pap er is an analogue of a fundamental theorem of Tits for these complexes It asserts that every automorphism of the complex of curves of a surface is induced by some elementoftheTeichmuller mo dular group of this surface or what is the same by some dieomorphism of the surface in question This theorem allows us to give a completely new pro of of a famous theorem of Royden R ab out isometries of the Teichmuller space In contrast with Roydens pro of which is lo cal and analytic this new pro of is a global and geometric one and reveals a deep analogy between Roydens theorem and the Mostows rigidity theorem Mo Mo Another application of our basic theorem is a complete description of isomorphisms b etween subgroups of nite index of a Teichmuller mo dular group This result in its turn has some further applications to mo dular groups Acknowledgments The rst version of this work cf I was nished during my stay at IHES BuressurYvette France in Summer of I would like to thank the Director of IHES Professor M Berger and Profes sor M Gromov for their hospitality Also I am grateful to Professors F Bonahon M Gromov D Kazhdan G Mess F Paulin L Sieb enmann D Sullivan M Troyanov and J Velling for stimulating discussions of this work at BuressurYvette Orsay and MarseillesLuminy Later on I had enjoyed numerous discussions of this work with other mathematicians to o many to name them all here I am esp ecially grateful to J S Birman R Hain Supp orted in part by the NSF Grant DMS W Harvey W Goldman H Masur J D McCarthy R Penner and M Seppala FinallyI thank the referee for several suggestions how to improve the exp osition Statement of the main results Let S be a compact orientable surface p ossibly with nonempty b oundary The complex of curves C S ofS is a simplicial complex in the sense given to this term in S Chapter for example Thus it consists of a set of vertices and a set of simplexeswhich are nonemptysets of vertices The vertices of C S are isotopy classes hC i of simple closed curves also called circles C on S which are nontrivialie are not contractible in S to a pointorto S A set of vertices is declared to b e a simplex if and only if these vertices can b e represented by pairwise disjoint circles Every dieomorphism S S takes nontrivial circles to nontrivial circles and obviously preserves the disjointness of circles Thus it denes an automorphism C S C S Clearly this automorphism dep ends only on the isotopy class of the dieomorphism S S Hence we get an action of the group of isotopy classes of dieomorphisms of S on C S This group is known as the mapping class group of S or as the Teichmul ler modular group of S We denote this group by Mo d Note S that we include the isotopy classes of orientationreversing dieomorphisms in Mo d Often this version of the mapping class group is called the extended S mapping class group Theorem If the genus of S is at least then al l automorphisms of C S are given by elements of Mo d That is Aut C S Mo d S S If S is either a sphere with four holes or a torus or a torus with one hole then C S is an innite set of vertices without any edges ie dim C S and the conclusion of this theorem is obviously false If S is a sphere with at most tree holes then C S is empty and the conclusion of the theorem is vacuous In the remaining cases of genus or surfaces the question ab out validityof the conclusion of the theorem was op en till recently Cf Section for further details The role of the complexes of curves in the theory of Teichmuller spaces is similar to the role of Tits buildings in the theory of symmetric spaces of noncompact typ e Originally only cohomological asp ects of this analogy were discovered cf for example Ha H or I Theorem together with other results of this pap er exhibits new sides of this analogy It is similar to a wellknown theorem of Tits T asserting that all automorphisms of Tits buildings stem from automorphisms of corresp onding algebraic groups In its turn this theorem of Tits extends the basic theorem of pro jective geom etry according to which all maps of a pro jective space to itself preserving lines planes etc are pro jectively linear Theorem Let be subgroups of nite index of Mo d If the genus S of S is at least and S is not closed surfaceofgenus then al l isomorphisms have the form x gxg g Mo d If is a subgroup of nite S index in Mo d and if the genus of S is at least then the group of outer S automorphisms Out is nite The second assertion of this theorem obviously follows from the rst one except when S is a closed surface of genus In the case of a closed surface of genus some additional automorphisms can app ear exactly as in McC I This theorem extends the authors theorem I I cf also McC to the eect that all automorphisms of Mo d are inner except when S is S closed surface of genus The assertion ab out niteness of Out proves a conjecture stated in I in connection with this theorem It is analogous to the Mostows theorem ab out niteness of outer automorphisms groups of lattices in semisimple Lie groups Mo Theorem is a simple corollary of Theorem given some ideas and results of I Another application of Theorem is concerned with the Teichmuller space T of the surface S We dene the Teichmuller space T as the space S S of isotopy classes of conformal structures on S nS without ideal b oundary curves only with punctures and consider T together with its Teichmuller S metric The mo dular group Mo d naturally acts on T as a group of isome S S tries Theorem If the genus of S is at least then al l isometries of T belong S to the group Mo d S This theorem is due to Royden R for closed surfaces S and to Earle and Kra EK for surfaces with nonempty b oundary Theorem allows us to give a completely new pro of of this theorem This new pro of follows the same general outline as Mostows pro of Mo of the rigidity theorem for symmetric spaces of rank at least In particular Theorem plays a role similar to the role of the ab ove mentioned theorem of Tits ab out automorphisms of buildings in Mostows pro of The analogy between Roydens theorem and the Mostow rigidity theorem is quite unexp ected and was not anticipated b efore Some recent remarks by Kra cf Kr p fo otnote suggest that this new pro of maybe in some sense the right one Note that the conclusion of Theorem is also true for almost all surfaces of genus and and the pro of of Earle and Kra EK works uniformly well in all cases Since Theorem was recently extended to most of these surfaces cf Section our pro of applies to most of the surfaces of genus and also Further results along these lines are discussed in Section Sketch of the pro of of Theorem The starting p ointofthe pro of is Fig Figure Lemma Let be isotopy classes of two nontrivial circles on S The geometric intersection number in Thurstons sense i is equal to if and only if there exist isotopy classes of nontrivial circles having the fol lowing two properties i i if and only if ith and jth circles on Fig are disjoint i j ii if is the isotopy class of a circle C then C divides S into two parts and one of these parts is a torus with one hole containing some representa tives of the isotopy classes Figure Note that i if and only if the vertices are connected i j i j by an edge in the complex C S It follows that the prop erty i can be recognized in C S and hence is preserved by all automorphisms of C S It turns out that the prop erty ii also can be recognized in C S To see this start with a vertex hC i of C S Let L be the link of in C S Let us consider the graph L having the same vertices as L and having as edges exactly that pairs of vertices that are not connected by an edge in L or what is the same in C S It is clear that the connected comp onents of L corresp ond to the connected comp onents of the result S of cutting S C along C After recognizing the comp onents we can return to the structure of the complex of curves on corresp onding sets of vertices If it is known b eforehand that the b oundary of a surface R is nonempty one can recognize the top ological typ e of R using only the structure of a simplicial complex of C R in the following way it is sucient to use the fact that if R then dim C R g b and C R is homotopy equivalenttoawedge of spheres of dimension g b where g is the genus and b is the number of b oundary comp onents of R at least if g The latter result is due to Harer H cf I for an alternative pro of By applying this remark to R S we see that the prop erty ii also can be recognized in C S and C is preserved by automorphisms of C S Hence Lemma implies that the prop erty of two isotopy classes to have the geometric intersection number can b e recognized in C S and so is preserved by all automorphisms of C S Lemma immediately implies that some useful geometric congurations such as chains with i i for j i j j n i i i j are mapp ed by automorphisms of C S into similar congurations More precisely every automorphisms of C S agrees on the set f g n with some element of Mo d Esp ecially imp ortant are the conguration of S circles presented in Fig in the case of b oundary comp onents
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