«1 .. ° ;§._u. .. é WWW....?x!..~, "Hi. I a. .1 4%: . s ‘ if. I :u. 43‘» , .3 1...... 3W. _ *v.....fl.u.|nfl. , .Jm binary...i 3} we 3.31, :3 g 31...? e .. as ‘.:.:;3 r . . ..;.i.;. .s. .1 «9..., . ,Vguptqll vw ,,— .... 1a.»: 51.} .3 ‘ , . i 1.1:. 1a.}. 211:. 2 \ LIB’RARY 52009 ' Michigan State University This is to certify that the dissertation entitled Abelian Subgroups and Automorphisms of the Torelli Group presented by William R. Vautaw has been accepted towards fulfillment of the requirements for Ph . D . degree in Mathgmggicg & Major professor Date _Augus_t_1_._ZQ_0_2_ MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 PLACE IN REI‘URN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 6/01 c:/ClRC/DateDue.p65-p.15 #4 - --————— ABELIAN SUBGROUPS AND AUTOMORPHISMS OF THE TORELLI GROUP By William R. Vautaw A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 2002 ABSTRACT ABELIAN SUBGROUPS AND AUTOMORPHISMS OF THE TORELLI GROUP By William R. Vautaw Let S be a closed, connected, oriented surface of genus g 2 3. The mapping class group M of S is defined to be the group of isotopy classes of self-homeomorphisms of S, while the Torelli group T of S is the subgroup of M consisting of the isotopy classes of those self- homeomorphisms of S that induce the identity permutation of the first homology group of S. This work considers two aspects of the Torelli group. The first is the Abelian subgroups of T. This portion of the work, where graph theory is the principal tool, contains two primary theorems. One gives a complete description of the multitwist subgroups of T, and the other states that any Abelian subgroup of T has rank at most 29 — 3. The second subject of investigation is automorphisms of the Torelli group; specifically, we ask whether any automorphism \II : T —> T is induced by a homeomorphism of S. In several formal and informal announcements made between October 2001 and March 2002, Benson Farb stated that he was able to prove that this is indeed the case for g 2 4. In this work, we lay the foundation for proving that it is also true for g = 3. This involves three basic steps. The first is to characterize algebraically certain elements of the Torelli group, namely powers of Dehn twists about separating curves and powers of bounding pair maps. The characterization given by Farb is valid for g _>_ 4, while our’s is valid for g 2 3. The second step is to show that \I/ induces an automorphism ‘11... of C, the complex of curves of S. This difficult step remains incomplete at the time of this writing. In the last step we use a theorem of Ivanov which states that \II,.. is induced by a homeomorphism h of the surface S, and conclude, under the assumption that it is possible to complete the second step, that the automorphism of the Torelli group induced by the homeomorphism h agrees with our automorphism \II. I will sing unto the Lord, because he hath dealt bountifully with me. iv ACKNOWLEDGMENTS Professor John McCarthy provided immeasurable assistance in this project. He spent countless hours over the course of many months working to help me prove of Conjecture 2, and when I finally find a proof, it will surely be with his guidance. After many years of being my teacher, he has become my good friend, and I will always be proud to say that he was my thesis adviser. Professor Ron F intushel had no direct influence on this work, but he certainly made my years at Michigan State University much happier. I am thankful for all that he has shared with me: topology, humor, advice and encouragement, lots of music (i), but most of all his friendship. For their sound advice, I would like to thank Professors Bill Brown and Nikolai Ivanov. I would also like to thank the other members of my thesis committee, Professors Jon Wolfson, Ulrich Meierfrankenfeld, and especially John Hall for stepping in at the last minute. Finally, I would like to thank Charles Morgan for all of his 'IEX-nical assistance. TABLE OF CONTENTS List of Figures vii Introduction 1 1 Reduction Systems and Reduction System Graphs 9 1.1 Graph Terminology ....................... 9 1.2 Reduction Systems and Reduction System Graphs ...... 11 1.3 Results on Graphs ....................... 15 2 Abelian Subgroups of the Torelli Group 20 2.1 Multitwists in the Torelli Group ................ 20 2.2 The Rank of Abelian Subgoups of the Torelli Group ..... 25 3 Automorphisms of the Torelli Group 30 3.1 The Algebraic Characterization of Elementary T Classes . . 31 3.2 The Complex of Curves .................... 49 3.3 Proof that Conjecture 2 implies Conjecture 1 ........ 51 References 55 vi 1.1 1.2 2.1 3.1 3.2 3.3 3.4 3.5 LIST OF FIGURES On left: A surface with a reduction system. On right: The reduction system graph. .................. On left: Curves in and in. On right: The cycle H “pro- duced” by [)1 and b2. .................... The rank of the Torelli multitwist group on this reduction system is 29 — 3. ...................... A symplectic basis for H1(S’) completed to a symplectic basis for H1(S). ....................... A basis for H1(S') containing b completed to a symplectic basis for H1(S). ....................... The Torelli multitwist group on a reduction system with this graph has rank 29 — 3. ................... The Torelli multitwist group on a reduction system with this graph has rank 29 — 3. ................... g pairwise disjoint separating curves of genus 1, each disjoint from a ............................. vii 15 32 33 38 38 47 INTRODUCTION The subject of this work is surface mapping class groups, generally classified as within the realm of geometric topology along with other areas of mathematics such as knot theory, cobordism theory, the theory of retracts, and so on. This in spite of the fact that, although surface topology plays an essential role, our true object of interest is an alge- braic one - a group. For the key technique is to consider configurations of curves on surfaces. This strategy will lead us to graph theory in chapters 1 and 2, and in chapter 3 to a well-known abstract simpli- cial complex first introduced by William Harvey [8], the “complex of curves.” In this introductory chapter we give the notation, definitions, and essential foundational theorems that we will rely on in the later chap- ters. We also present the specific problem we will be addressing and some historical background. Throughout this work, S represents a closed, connected, oriented surface of genus g 2 3. We use the symbols a, b, c, etc, to denote (the isotopy classes of ) simple closed curves on S. In general, we confuse a curve and its isotopy class, and thus to say that two curves are distinct means that they are not isotopic. We use the symbol 6 to denote the set of all nonoriented (isotopy classes of ) homotopically nontrivial simple closed curves on S. The geometric intersection number of a and b is given by 2'(a, b) = min{card{a’fl b'} : a' is isotopic to a and b' is isotopic to b}. If a E 6 is separating, then S\a has two components Sl and 82, and we define the genus of a to be the smaller of the two numbers genus(Sl) and genus(Sg). A reduction system Q? on S is a set of distinct, mutually disjoint, homotopically nontrivial simple closed curves on S, and we use Se; to denote the natural compactification of S\ U a; that is, “S cut 066 along Q3.” If QE 2 {a}, then we write 8,, instead of S{a}' Note that a reduction system consists of at most 3g — 3 elements, and a reduction system containing 39 — 3 elements gives a pants decomposition of S. The mapping class group of S, M(S), is the group of isotopy classes, or mapping classes, of orientation-preserving self-homeomorphisms of S. In general, our notation will not distinguish between a homeomor- phism f : S —> S and its mapping class. Given a E G and f E M(S), we let f (a) denote the isotopy class of the image of any curve repre- senting the class of a under any map representing the class of f. The symbol Da will denote the right Dehn twist about a E (‘3. Two standard results are that Dehn twists Da and Db commute if and only if z'(a,b) = 0, and that if f E M(S) then fDaf‘1 = Df(a). Given a reduction system Q? on S, we define a multitwist on 6 to be a compo- sition of left and right Dehn twists about the curves in Q3, and denote the group of all multitwists on Q3 by De. Clearly De is a free Abelian group with basis {Du : a 6 QB}, and so rank(D¢) = card(€). The mapping class group acts in an obvious way on the first homology 2 group of S: given an oriented simple closed curve b on S, f sends the homology class of b to the homology class of f (b) In the case of a right Dehn twist we have a specific formula for the image class. If a and b are oriented simple closed curves on S, then in H1(S) we have Da(b) = b + (a, b)a, where (a, b) denotes the algebraic intersection number of a with b. The Torelli group of S, which we denote by T(S) or simply T, is defined to be the kernel of this action of M(S) on H1(S). The Torelli group is torsion-free, and is trivial in the case of the sphere or torus. Also, the center of the Torelli group of any closed surface is trivial. We define the mapping class group and Torelli group of a surface with boundary analogously; we simply require that homeomorphisms and admissible isotopies fix each component of the boundary setwise. Note that the Torelli group of a surface with boundary (with the exceptions of a disc and an annulus) is still torsion-free, and that the Torelli groups of a pair of pants and a one-holed torus are trivial. Given a reduction system Q3 on S, we use the symbol MAS) to denoted the stabilizer of QB in M(S), and define the reduction homo- morphism A : MAS) —+ M(Se) as follows: For f E M¢(S), there exist a representative f of f and a set 53 of disjoint representatives of Qi such that T(é) = 53. We let A( f) be the unique extension of 13'5““; to Se. The kernel of A is De. We say that f E M(S) is reducible if f E M¢(S) for some nonempty reduction system Q. In this case we call Q3 a reduction system for f and 3 each a E 6 a reduction class for f. Nikolai Ivanov has proved a theorem ([9], Theorem 1.2) which implies the following useful result concerning the Torelli group, which we refer to as Ivanov’s Theorem: Theorem 0.1 (Ivanov’s Theorem) Let f E T and suppose GE is a re- duction system for f. Then f leaves each curve in Q3 invariant and A( f ) leaves each component and each boundary component of SQ invariant. Hence if f E T reduces along 6, then A( f) restricts to each compo- nent of Se. We call each component of Se; as well as each restriction of A( f ) a component of f ( determined by Q3). In particular, if S’ is a component of Se, and the restriction of A( f) to S' is a pseudo—Anosov element of M(S’), then we say that S’ is a pseudo-Anosov component of f. The notion of an adequate reduction system, which we describe presently, was first introduced by Birman, Lubotzky, and McCarthy in [2]. However, their paper deals with a more general situation (the mapping class group of a disconnected surface) than we will need to consider (the Torelli group of a connected surface), and so the defini- tion we give is specialized to suit our needs. We say that a reduction system Q3 for an element f E T(S) is an adequate reduction system for f if each component of f is either trivial or is pseudo-Anosov, and in this case, we say that A( f), the reduction of f along 6, is adequately re- duced. Using this concept and fact that the Torelli group is torsion-free, Thurston’s classification of mapping classes (cf. [7]) implies that every element f of the Torelli group is either reducible or pseudo-Anosov, and 4 if f is reducible, then it has an adequate reduction system. We say that a reduction class a for f E M(S) is an essential reduction class for f if and only if for each h E (‘3 such that i(a, b) 75 0, the isotopy classes fm(b) and b are distinct for all m 75 0. Given f E M(S), we define the essential reduction system for f by 6f 2 {a E 6 : a is an essential reduction class for f}. The following facts are proved in [2]: Theorem 0.2 (Birman, Lubotzky, and McCarthy) i) Let a and b be reduction classes for f E M(S). If a is essential, then i(a, b) = 0. ii) Let 6 be an adequate reduction system for f and let a E G. Then a is essential if and only if €\{a} is not an adequate reduction system for fm, for any m 7é 0. iii) For all g E M(S), g(€f) = Qigfg-l. iv) Qifm : 6f for all m 75 0. v) 6f is an adequate reduction system for f. vi) Q3 f Q Q? for each adequate reduction system 6 for f. Note that ( ii) above implies that the essential reduction system of an infinite order reducible mapping class is nonempty, and (iii) implies that if f and g commute, then the essential reduction system for f is a reduction system for g. A standard result is that if f 2 D211 DE; - - - D2: is a multitwist about the reduction system Q3 2 {(11, a2, . . . , ak}, where each exponent is nonzero, then 6 f 2 QB. In particular, QED“ = {a}. John McCarthy has proved the following, which we refer to as Mc- Carthy’s Theorem: Theorem 0.3 (McCarthy’s Theorem)([11], Theorem 1 and Corol- lary 3) Let f be a pseudo-Anosov element in the mapping class group M of a compact, connected, orientable surface with possibly nonempty boundary. Then the centralizer CM( f) of the cyclic subgroup of M generated by f is a finite extension of an infinite cyclic group, and consequently every torsion-free subgroup of C M( f) is infinite cyclic. This theorem implies the following useful corollary concerning the Torelli group: Corollary 0.4 Let A be an Abelian subgroup of the Torelli group of a compact, connected, orientable surface with possibly nonempty bound- ary. If A contains a pseudo-Anosov mapping class, then A is infinite cyclic, and any generator is pseudo-Anosov. The specific issue addressed in this work concerns automorphisms of the Torelli group of S. Namely, are all automorphisms of T topological (i.e., induced by homeomorphisms of the surface)? In the 1980’s Nikolai Ivanov answered affirmatively the analogous question for the mapping class group (genus 2 3). One of the inspira- tions for Ivanov’s work was the results concerning Abelian subgroups of the mapping class group published in 1983 by Birman, Lubotzky, and McCarthy [2]. The idea of the essential reduction system, introduced in their paper, played a central role in Ivanov’s arguments as he char- acterizes certain elements of the mapping class group in terms of their centralizers and centers of centralizers. He then proceeded by using algebraic methods involving relations in the mapping class group. Using Harvey’s complex of curves, Ivanov later proved a stronger theorem [9], which includes as a special case the result about automor- phisms of the mapping class group. Specifically, he first showed that an automorphism \II of the mapping class group induces an automor- phism of the complex of curves, and then that any automorphism of the complex of curves is induced by a homeomorphism h of the surface. The theorem is proved when he shows that the automorphism of the mapping class group induced by h is the same as the automorphism ‘11. We follow the same basic steps in the case of the Torelli group. First we investigate the Abelian subgroups, proving two primary theorems. The first gives a complete description of “Torelli multitwists,” and the second states that any Abelian subgroup of T has rank at most 29 — 3. In this portion of the work, in chapters 1 and 2, graph theory is the principal tool. We then deal with the automorphism question directly. In several formal and informal announcements made between October 2001 and March 2002, Benson Farb announced that he was able to prove that every automorphism of the Torelli group is induced by a homeomor- phism of S, for genus g 2 4, [4], [6]. He outlined his proof in [5]. In chapter 3 we lay the foundation for proving that this is also true for g r- 3. Following Ivanov, we use centralizers and centers of cen- tralizers to give algebraic charactertizations of certain elements of the Torelli group, namely powers of Dehn twists about separating curves and powers of bounding pair maps. We note that Farb’s characteriza- tion is valid only for g 2 4, whereas the characterization we give is valid for g 2 3. The second step, showing that any automorphism ‘I! of the Torelli group induces an automorphism ‘1'... of the complex of curves, remains incomplete at the time of this writing. In the last step we use Ivanov’s theorem stating that ‘11... is induced by a homeomorphism h of the surface S to conclude, under the assumption that it is possible to complete the second step, that the automorphism of the Torelli group induced by the homeomorphism h agrees with our automorphism \II. The apparent difficulty in proving this result purely algebraically (i.e., without using the complex of curves) suggests that the automorphism theorem for the Torelli group is a deeper result than for the mapping class group. CHAPTER 1 Reduction Systems and Reduction System Graphs 1.1 Graph Terminology We use graph-theoretic terminology consistent with its use in [3]. We re- mind the reader of the less familiar terms, and give the graph-theoretic definitions of those terms that may be used in different ways in ordinary topology. We let G denote a connected, finite linear graph. We include the possibility that G may contain loops or parallel edges. E = E(G) will denote the edge set of G, and we use the symbols a, b, c, etc. to denote edges of G. For E’ C E(G), G -— E’ denotes the subgraph obtained from G by deleting the edges in E’, while G + E” is the graph obtained from G by adding a set of edges E”. If E’ = {e}, then we write G — e and G+e instead of G — {e} and G+ {e}. A bond E’ in G is a minimal subset of E(G) such that G — E’ is disconnected. Note that G — E’ consists of precisely two components. We say that the edge e is a cut edge if G — e is disconnected. We use the symbols u, v, :c, y to denote vertices of G. The degree of a vertex v is the number of edges incident with v, each loop counting as two edges. A (v0,vn)—walk W of length n is a finite nonempty alternating se- quence, W = voelvlegvg . ..e,,v.,,, of vertices and edges such that the ends of the edge e,- are the vertices v,-_1 and v,- for 1 g i g n. If the edges of W are distinct, W is called a trail. A cycle in G is a closed trail of positive length whose origin and internal vertices are distinct. Thus a cycle is an embedded circle in G. For our purposes, to denote a trail or cycle, it will be enough to give its sequence of edges, and we do not distinguish between a closed trail W and another closed trail whose sequence of edges is a cyclic permutation of W’s. A spanning tree T is a subgraph of G with the same vertex set as G such that T contains no cycles. The number of edges in any spanning tree is equal to one less than the number of vertices of G. Note that if T is a spanning tree, and e is an edge of G not in T, then T + e contains a unique cycle C, and e is an edge of C, so the rank of 7r1(G) is equal to the number of edges of G outside any spanning tree. Every connected graph contains a spanning tree. Given a subgraph H of G, we let G 0H denote the graph obtained by deleting every edge e of H and identifying the ends of e. Equivalently, thinking of G as a CW—complex and H as a subcomplex, G o H is the complex obtained from G by crushing each component of H to a point. Thus, we have a quotient (“contraction”) map p : G —> G o H. Next, by a cut vertex of G, we mean a vertex v of G such that when v, and only v, is removed from the topological space G, the resulting space is disconnected. (This is not the definition used by graph theorists, but 10 is an equivalent topological one.) A block is a connected graph without cut vertices, and a block of a graph is a subgraph that is a block and is maximal with repsect to that property. Any graph is the union of its blocks. 1.2 Reduction Systems and Reduction System Graphs Let QB be a reduction system on S, as defined in the introduction. We partition the set QE 2 {e1,e2, . . . ,en} according to the equivalence relation ~ generated by the rule e,- = e,- e, ~ e,- if or {e,-, ej} is a minimal separating set in QB Here, “{e,-, 3,} is a minimal separating set” means that Show} is dis- connected, but both 8,, and Se]. are connected. There are three types of ~-—equivalence classes: i) Singleton classes {a1}, {a2}, . . . , {up} consisting of the separating curves a1,a2,...,ap in Q3. Such a curve wil be called an a—type CUT’UG. ii) Classes {b11, . . . , blql}, {(121, . . . , {32.1,}, . . . , {hm . . . , brqr} ofcardinal- ity at least 2. Each such class {En-1, . . . , him} is characterized by the following three properties: a) No curve by is separating. b) [1,,- is homologous to hijr for every pair [1,3, but. 11 c) Maximal with respect to (a) and (b). A curve in such a class will be called a b—type curve. iii) Singleton classes {c1}, {c2}, . . . , {c3} where each c,- is non-separating and is homologous to no other curve in 6. Such a curve will be called a c—type curve. According to (i), (ii), and (iii) above, we write Q3: {a1,...,ap,b11,...,b1q,,b21,...,b2q,,...,brl,...,b,.qr,c1,...,c3}. We use QB to define a graph G6, which we call the reduction system graph of 6, as follows: Vertices of G9; correspond to the components of Se. Edges of Ge; correspond to the curves in the reduction system Q3, with (Links) Two distinct vertices are connected by the edge e,- if and only if the curve e,- in 6 is a common boundary curve of the two components of Se which correspond to the vertices in question. (Loops) A vertex has a loop e,- if and only if the curve 2,- in Q? represents two boundary curves of the component of Se which corresponds to the vertex in question. Note that G9; is connected, and that any connected graph G is Ge for some surface S and some reduction system QB on S. However, the genus of S is not determined by G, any two possible S’s differing by the genera of their complementary components. But, unless G is the 12 graph consisting of a single vertex and either no edges or a single loop, then genus(S) Z rank(7r1(G)) + (number of vertices of degree S 2). Since S and 6 will be fixed, we will denote Ge simply by G. The equivalence relation ~ on the curves in 6 induces a equivalence relation ~ on the edge set E(G) = {e1,e2, ...,en} of G. It is generated by e,- = e,- 6,- ~ e, if or {e,-, 6,} is a bond (Again, it should be noted that this equivalence relation may be defined for any graph G.) The three types of equivalence classes described above become, for G, i) Singleton classes {a1}, . . . , {ap} consisting of the cut edges a1, . . . ,ap of G. Such an edge will be called an a-type edge. ii) Classes {b11, . . . ,blql}, {b21, . . . ,bng}, . . . , {b.1, . . . ,brq.} of cardinal- ity at least 2. Each such class is characterized by the following three properties: a) No edge bi]- is a cut edge. b) {bij, bijr} is a bond for every pair bij, bijl. c) Maximal with respect to (a) and (b). An edge in such a class will be called a b-type edge. iii) Singleton classes {c1}, . . . , {c3} where each c,- is not a cut edge, and 13 forms a 2—edge bond with no other edge of G. Such an edge will be called a c-type edge. According to (i), (ii), and (iii) above, we write E(G) = {a1,...,ap,b11,...,b1q1,b21,...,b2q2,...,b,.1,...,brqr,c1,...,cs} Figure 1.1 shows a typical example of a reduction system and its graph. Figure 1.1: On left: A surface with a reduction system. On right: The reduction system graph. Now let i) be a simple closed curve on S that intersects each element of Q transversely at most once. Starting at any point on i) and travelling in either direction gives a cyclic ordering of the reduction curves which i) intersects, thus defining a closed trail H in G. Note that H is a cycle in G if and only if l) D S,- is either empty or is a single (that is, connected) arc, for every component S,- of Se. Likewise, given a closed trail H in G, there is such a curve b on S defining H. The fact that the isotopy class of l) is never unique is not important for our purposes. Figure 1.2 shows a typical example. Note that In and [)2 are noniso— topic curves which both define the cycle H = b11c2b12. 14 Figure 1.2: On left: Curves in and in. On right: The cycle H “produced” by in and in. 1.3 Results on Graphs This section presents some purely graph-theoretic results, concluding with Theorem 1.4, which is used to prove the main theorem of section 2.1. Here, G denotes an arbitrary connected, finite linear graph. Lemma 1.1 If G has no cut edges, then any two vertices of G are connected by two edge-disjoint paths. Proof: We prove by induction on d(u, v) that any two vertices u and v in G are connected by two edge-disjoint paths. d(u, v) = 1 : In this case, uv is an edge, and so by hypothesis uv is not a cut edge. By [3], page 27, uv is contained within a cycle G. Then uv and G — uv are two edge-disjoint (u, v)—paths. N ow suppose the theorem is true for any two vertices at distance less than k, and let d(u, v) = k 2 2. Consider a (u, v)—path of length k, and let w be the vertex preceding v on this path. Then d(u, w) = k — 1, so by the induction hypothesis, there are two edge-dijoint (u, w)-paths P and Q in G. Since G has no cut edges, G —wv is connected, so there is a 15 (u, v)-path P’ in G — wv. Lew a: be the last vertex of P’ lying in P U Q, and suppose that :1: E P. Then G has two edge-disjoint (u,v)-paths: one composed of the portion of P from u to :13, the other is Q + wv. Cl Lemma 1.2 Let b1 and b2 be edges ofG such that {b1, b2} is a bond. If C is a cycle in G, and b1 is an edge of G, then so is b2. Proof: Let b1 have ends ul and v1, and let b2 have ends ug and v2. Since {b1, b2} is a bond, G — b1 and G - b2 are connected, but G — {b1,b2} is not, having two components which, without loss of generality, separate ug and v2. Let G = 016162 - --e,,, and suppose b2 is not an edge of G. Then e1e2 - - - en is a path in G 2 {b1, b2} connecting ul and v1, so ul and v1 lie in the same component of G — {b1, b2}. Since there is no (uz, v2)-path in G — {b1, b2}, but there is a (U2, v2)-path P in G — b2, P must contain b1. Let P1 be the portion of P from 7.1.2 to, say, ul, and let P2 be the portion of P from v1 to v2. Then P1 + 8182 . - . en + P2 is a (U2, v2)-path in G — {b1,b2}. This is a contradiction. Hence b2 must also be an edge of C'. E] Lemma 1.3 Let c be a c—type edge in G that is not a loop. Then c is contained within two cycles of G, the intersection of whose edge sets is precisely c. 16 Proof: Assume that G is a block. If G has exactly two vertices, then each edge of G is a link, and G must have at least three edges, since c is a c—type edge. The result is clear in this case. Otherwise, G has at least three vertices and no cut edges. Consider the graph G — c. If G — c has a cut edge e, then G — {c, e} is not connected, so {c, e} is a bond of G. This contradicts the fact that c is a c—type edge. So G — c has no cut edges. By Lemma 1.1, there are two edge-disjoint paths P and P’ in G —- c connecting the ends of c. Then the cycles G = P + c and C" = P’ + c have exactly the edge 0 in common. In the case that G is not a block, we let B be the block of G containing c. It is easy to see that c is a c—type edge of B, so we apply the first case to B and find two such cycles within B. Cl Theorem 1.4 Let G have edge set E(G)={a1,...,ap,b11,...,b1q1,...,b,.1,...,b,qr,c1,...,cs}, notated according to a— , b-, and c—type equivalence classes. Let w : E(G) ——> Z be a weighting of G. Then w(H) = 0 for every cycle H in G if and only if i) w(ci) = 0, 13 i g s, and ii) w(b,1)+ w(bjz) + - . - + w(quj) = 0, 1 g j g r. Proof: 2?) Assume that w(H) = 0 for every cycle H in G. 17 i) Let c be a c—type edge with ends u and v. If c is a loop, then w(c) = 0, by hypothesis. Otherwise, there are two edge-disjoint (u, v)—paths, P and P7, in G — c. We have three cycles: P + c, P’ + c, and P + P’. Thus, w(P) + w(c) = w(P + c) = 0 w(P’) + w(c) = w(P’ + c) = 0 => w(c) = 0 w(P) + w(P’) = w(P + P’) = 0 ii) Let B be the equivalence class of the b—type edge b, and B = B\{b}. Let p : G —> G o B be the contraction map. Suppose that b is a cut edge of G 0B, separating it into two components G1 and G2. Then the restriction of p to G -— b maps onto the disconnected space G1 U G2, and so G - b is disconnected. This is a contradiction to the hypothesis that b is a b—type edge of G. We obtain a similar contradiction if we suppose {b, e} is a bond in G o B. Thus b is a c—type edge in G o B. If b is a loop in G o B, then p’1(b) = B, which therefore forms a cycle in G. So equation (ii) holds for the equivalence class of b. If b is not a loop in G o B, then by Lemma 1.3 there are two cycles H and H7 in G a B, the intersection of whose edge sets is {b}. Lemma 1.2 implies that p‘1(H) and p‘1(H7) are cycles H and H’, respectively, the intersection of whose edge sets is precisely B. Thus we have 0 = w(H) = w(B)+w(H—B) 0 = w(H’) = w(B)+w(H’—B) => 0 = 2w(B)+w(HAH’) = 2w(B) And so, w(B) = 0. Here we have used the fact that the symmetric difference H AH ’ of the cycles H and H’ is a disjoint union of cycles 18 (regarded as sets of edges). <2) Assume that i) w(ci) = 0, 1 _<_ i S s, and ii) 10(bj1) + 10(bj2) + ' ' ° + 1U(quj) I O, 1 S j S 7‘. Let H be a cycle in G. H contains no a—type edges, since they are cut edges, and by Lemma 1.2, if H contains one edge of a b—type class, then it contains the whole class. So the assumptions imply that w(H) = 0. El 19 CHAPTER 2 Abelian Subgroups of the Torelli Group 2.1 Multitwists in the Torelli Group We at first consider a specific type of Abelian subgroup of the Torelli group T(S), namely one consisting of multitwists — that is, composi- tions of left and right Dehn twists about a fixed reduction system Q? on S. Theorem 2.1 Let S be a closed, connected, oriented surface, and let QB:{a1,...,ap,b11,...,b1q,,b21,...,b2q,,...,brl,...,b,.qr,c1,...,c,} be a reduction system on S, notated by a—, b—, and c—type ~—equivalence classes as in section 1.2. Let Do; be the multitwist group on Q3, and let f = D311 - - - ngDé’: ...DBIQIDBZI ”1352202 .. . DEWrDcvll _ , , DZ: b191 b21 (:2 ”11' be an element of D93. Then f is an element of De (7 T E Te; which we call the Torelli multitwist group of QB, if and only if i)7,-=0,1_§i_<_s, and z"U5j1+5j2+--'+5jq,-=0, ISjST- 20 Consequently, Te is a free Abelian group of rank p+(91"1)+(Q2—1)+'”+(Qr“1)ZP+CI1+Q2+'°°+Qr‘—To Proof: =>) Assume that f 6 T6. Let G be the reduction system graph of Q3 with edge set E (G) We weight each edge of G according to the exponent in f of the twist about its corresponding curve in Q3, giving w : E(G) —~> Z. Let H = e1e2,...,en be a cycle in G. Then, as in section 1.2, H is defined by any simple closed curve f) on S that intersects each of the corresponding curves e1, 22, . . . , an of Q3 exactly once, and does not intersect any of the other curves of Q3. Orient I). Then orient the curves 21,22, . . .,e,, so that (l),e,-) =1. So we have 0 = ([1,1]) = (b,f(b)> = = €1+€2+- - -+€n, where e,- = w(ei). Hence the weight of every cycle in G is zero. The conclusion follows from Theorem 1.4. <=) Assume that 07,20,13135, and ii);3j1+5j2+-"+5jq,=0,lfijST- Since H1(S) has a basis consisting of simple closed curves, in order to prove that f E T, it suffices to show that in H1(S), we have f (b) = f) for any simple closed curve l) on S. Note that for any such l), we have ((1,, b) = 0, 1 S i S p, and after orienting l) and then each [1,,- so that 21 (525,0) = (bah), W6 have by = bib 2 S j S (Ii, 1 S i S T- La 6,- : (bu, h). Then in H1(S) we have b) 2 Diff "'DE’KDIX "'05:,1053"'05::"'D€:::D31""Dc’.’(b) = 0+ 511(511, IJ>b11 + ' ° ' + qu1b1q1 + ' " +5r1br1 + ' ° ' + firq.(brq., b>brq. 2 [7+ 51(511 + ° ' ' +51q1)511 + ° ° ' + 514574 + ° ' ' + firqr)brl =6 Theorem 2.2 Let S be a closed connected oriented surface, and let Q3 2 {21, e2, . . . , en} be a reduction system on S. Let f = 020:: - - - DE: be a multitwist on 6. Let G be the reduction system graph of Q3, and define a weighting w : E(G) —> Z ofG by w(ei) = 6,. Then f is in the Torelli multitwist group 7}; if and only if the weight of every cycle in G is zero. Proof: Partition G? into ~—equivalence classes and write 6 = {a1,...,ap,b11,...,b1q,,b21,...,b2q,,...,b,1,...,b,.qr,c1,...,c3}. Theorems 1.4 and 2.1 show the conditions to be equivalent. E] Given a pair, el and 82, of disjoint, non-separating, but homologous simple closed curves on S, we call Dengl a bounding-pair map. Powell 22 [12] has shown that the Torelli group T is generated by bounding pair maps and Dehn twists about separating simple closed curves. Corollary 2.3 Let S, 6, D6, and 753 be as in Theorem 2.1. Let D’ be the subgroup of M(S) generated by i) bounding pair maps about bounding pairs in Q3, and ii ) Dehn twists about separating curves in 6. Then D’ =D¢flT=T€. Proof: By the definition of De, it is clear that every generator of D’ is in De. By Powell’s result noted above, every generator of D’ is in T. Thus D’ Q De 0 T. We must show that De 0 T E D’. Let f E De 0 T. By Theorem 3.1, we know that f : D311 . . . 03:05:: . . . 0510119521 . . . DB2” . . .135: . . . 05"", blql b2] b2q2 TQr where 511 +Bi2+' ° '+/8iq.- = 0, 1 _<_ i _<_ r. Since each D3: is a product of tYPG—(ii) generators of D’, we will be done if we write D5::1’D522 - - - DB‘“ biqu‘ as a product of bounding pair maps. We do this: i 1 Bi 1' — i — i ’ i ' 05305.: - - - D " = (Db..D.,3)52(Db..D.,:)"3 - . - (Db 0.3%. biqi ’ql where W8 note that —,B,'2 — ,82'3 — ' ° ' — :8ng = ,8“. [:1 Corollary 2.4 Let S be a closed, connected, oriented surface, and let 6:{a1,...,ap,b11,o..,blql,b2l,...,b2q2,o..,br1,...,brqr,C1,oo.,CS} 23 be a reduction system on S, notated by a-, b—, and c—type ~-equivalence classes as in section 2. Let De be the multitwist group on Q3, and let f = 13311 . . . 035051111 . . . D61.“ 0521 . . . D522” . . . D5”. D2: . . . D3: blql b21 <12 Tar be an element of De. Let m 2 2 be an integer. Then f E Fs(m) E {g E M(S) :9 acts trivially on H1(S;Zm)} if and only if i)’y,-EO(modm),1§iSs, and iiffij1+fij2+~-+quj50(mod m),1$j_<_r. Let S be the surface of genus g 2 2 and Q3 the reduction system on S shown in Figure 2.1. Since Q3 consists of 29 — 3 a—type curves, rank(7&) 2 2g— 3. This example, along with Theorem 2.7 below, shows that the maximal rank of an Abelian subgroup of the Torelli group is attained by a multitwist group. Figure 2.1: The rank of the Torelli multitwist group on this reduction system is 29 - 3. 24 2.2 The Rank of Abelian Subgoups of the Torelli Group We prove that for any closed oriented surface of genus g 2 2, the general Abelian subgoup of its Torelli group has rank 3 2g — 3. We first give two lemmas. Lemma 2.5 Let S be a closed, connected, oriented surface, and Q3 a reduction system on S with reduction system graph G. Let Te be the Torelli multitwist group on QB, as in Theorem 2.1. Then rank(T¢) g 1/ — 1, where 1/ is the number of vertices of G, or, equivalently, the number of components of Se. Proof: Let G have edge set E(G) = {a1,...,ap,b11,...,b1q1,b21,...,b2q2,...,b,.1,...,b,.qr,c1,...,cs} notated according to a- b-, and c-type equivalence classes. Let E’ : {b11, . . . ,b1(ql_1),b21, . . . ,b2(q2_1), . . .,br1, . . . ,b,(qr_1)} (_I E(G) and let G’ = G [E’] T hen G’ contains no cycles, since any cycle con- taining one edge of a b—type class contains the whole class. Therefore, G’ is contained in a spanning tree T. T contains each cut edge a,, 1 g i g p, so T contains the set of edges E’ U {a1,a2, . . . ,ap}, and by Theorem 2.1, the cardinality of this set equals the rank of T93. Hence I/ — 1 = card(E(T)) 2 p +(q1— 1) + ° ° ° + (Gr — 1) = rank(TQ;) 25 Lemma 2.6 Let S be a closed, connected, oriented surface of genus g 2 2, and let Q be a reduction system on S. Let 9 denote the number of components of So; not homeomorphic to a pair of pants or a one-holed torus. Let Tc be the Torelli multitwist group on QB. Then rank(T¢)+Q 5 2g — 3. Proof: Let G be the reduction system graph of Q. We use the following nota- tion: 0 F is the maximum genus of any component of Sg. o A is the maximum degree of any vertex of G, or, equivalently, the maximum number of boundary curves of any component of Se. 0 Vb is the number of vertices of G of degree b, or, equivalently, the number of components of Sc»; with b boundary curves. 0 12,] (V?) is the number of components of Se of genus 7 (2 7) having b boundary curves, or, equivalently, the number of vertices of G of degree b corresponding to a component of Se of genus 'y (2 7). So we have P A Vb = E V5 and V: E Vb b=1 7:0 But the assumption that each element of GE is homotopically nontrivial means V? = 0, and the assumption that the elements of Qi are pairwise nonisotopic means V3 2 0. So, in fact, 1/ : V121+V221+V3+V4+° - °+VA. Now, V11 is the number of one-holed tori, and V3 is the number of pairs 26 of pants, so by the definition of 9, we have 92111-2 +1/2-1 +V3—1 +V4+---+I/A. Hence 2g—2 = —X(S) components V of Se I‘ F A I‘ I 2(27 — 1)1/1’+ :9ng + Z 2(27 + b - 2%- By Lemma 3.1, rank(7'é) S 1/ — 1, so we have rank(T93) +QSV+Q—1 (2.1) (v?1+v?’+- - -+vA)+(v?’+v§’+z/§’+V4+- - -+VA)—1 =[(u,1+2u,?2)+2u 221+(u§+2u 321)+2u4+2u5+ +21/A]- 1 F S[Z(27)—13+Z(27)1/27+ZZ(27+b— 2)1/b]—1 :1 721 b: 3 7:0 = —X(S) - 1 229—3 Theorem 2.7 Let S be a closed, connected, oriented surface of genus g 2 2, and let A be an Abelian subgroup of T, the Torelli group of S. Then rank(A) S 2g — 3. Proof: Let f E A, f 75 0. As mentioned in the introduction, by Thurston’s classification, f is either reducible or pseudo-Anosov. 27 Case 1: f is pseudo-Anosov. By the corollary to McCarthy’s Theorem given in the introduction, A is infinite cyclic. Case 2: f is reducible. Let Q? = U Gh. Then 6 is an adequate reduction system for each heA h E A ([2], Lemma 31(1)), and f reducible implies Q? 3i (0, so every element of A is reducible. Let A : M¢(S) —> M(S@) be the reduction homomorphism. Then ker(A) 2 D6, the multitwist group on 6, and thus ker(AlA) = ker(A) nszanzvenTnAzflnA. We now have a short exact sequence 0——>7IgflA—+Ai’l4A(A) -—>O of free Abelian groups, which shows that rank(A) = rank(7}; H A) + rank(A(A)) S rank(TG) + rank(A(A)) By applying Lemma 2.6, we will be done if we show that rank(A(A)) _<_ Q, the number of components of Se; not homeomorphic to a pair of pants or a one-holed torus. Ivanov’s theorem, given in the introduction, implies that A( f) re- stricts to each component S1,Sg,...,S,, of Se, giving “projections” p,- : A(A) ——> M(S,-) induced by restricting representatives. Set A,- = p,(A(A)) Q M(S,-). Then A(A) Q @A, so rank(A(A)) S E rank(A,-). We make the following observations: i) If S, is a pair of pants, then M(S,—) is finite, so rank(Ai) = 0. 28 ii) If S, is a one-holed torus, then the homomorphism H1(S,-) ——> H1(S) induced by inclusion is injective. Any homeomorphism f representing an element of A maps a circle c in S,- to a circle c’ in Si, so A: lies within the Torelli group of S,, which is trivial in this case. iii) If S,- is neither a pair of pants nor a one-holed torus, then A,- is either trivial or is an adequately reduced torsion-free Abelian subgroup of M(S,~). So again by McCarthy’s theorem, rank(A) S 1. These observations tell us that rank(A(A)) g Zrank(A,—) g n. (2.2) i=1 29 CHAPTER 3 Automorphisms of the Torelli Group Our reason for investigating Abelian subgroups of the Torelli group is to provide the necessary algebraic information about the Torelli group so that we may prove the following, the principal problem of this work. Conjecture 1 Let S be a closed, connected, oriented surface of genus g 2 3, and let \I’ : T —> T be an automorphism of.the Torelli group T of S. Then ‘11 is induced by an homeomorphism h : S —> S. That is, for any f E T, we have ‘I’(f) = hfh‘l. As stated in the introduction, our strategy for proving this conjecture is comprised of three basic steps. The first, in section 3.1, is to give algebraic characterizations of power of Dehn twists about separating curves and powers of bounding pair maps. From this it will follow that \II must permute the set of left and right Dehn twists about separating curves and that \II must permute the set of bounding pair maps. The second step, in section 3.2, is to show that \I/ therefore induces an automorphism of C (S), the complex of curves of S. At the time of this writing, this step is incomplete. Finally, a theorem of Ivanov states that any automorphism of C(S) (in particular, \II) is induced by a 30 homeomorphism h of S. In section 3.3 we apply this theorem and show, assuming that it is possible to complete step two, that the two automorphisms of T, \I' and the one induced by the homeomorphism h, are the same. This method of proof was first employed by Ivanov in [9] to prove the analogous theorem about the mapping class group. In several formal and informal announcements made between Octo- ber 2001 and March 2002, Benson Farb stated that he is able to prove this conjecture in the case g 2 4, [4], [5], [6]. His strategy is the same, but we note that his characterization of powers of Dehn twists about separating curves and powers of bounding pair maps is only valid for genus at least 4, whereas our characterization is valid for genus 3 and above. Also, his proof that an automorphism of the Torelli group in- duces an automorphism of the complex of curves appears to rely on the ability to produce three curves on the surface, each pair of which is a bounding pair. This is possible only for genus 4 and above. 3.1 The Algebraic Characterization of Elementary T Classes In this section assume that the genus of S is at least 3. Recall that if f E T(S) is reducible along 6, and A : M¢(S) ——> M(S¢) is the reduction homomorphism, then, by Ivanov’s Theorem given on page 4, A( f) leaves each component of Se invariant, and so A( f) restricts to the mapping class group of each component. Also, f(a) = a for each a 6 Q3. Lemma 3.1 i) Let f E T(S) be reducible along the separating simple 31 closed curve a, and A the reduction along a. Let S’ be a component of 8,, and f’ the restriction of A(f) to M(S’). Then f’ E T(S’). ii) Let f E T(S) be reducible along the bounding pair {a, b}, and A the reduction along {(1, b}. Let S’ be a component of Sfab} and f’ the restriction of A(f) to M(S’). Then f’ E T(S’). Proof: i) Since a is separating, we may consider 8’ to be a subsurface of S. Then a symplectic basis for H1(S’) may be extended to a symplectic basis for H1(S) (see figure 3.1). Thus H1(S’) is a direct summand of H1(S), and since f acts trivially on H1(S), its restriction to S’, which is f’, acts trivially on H1(S’). That is, f’ E T(S’). Figure 3.1: A symplectic basis for H1(S’) completed to a symplectic basis for H1(S). ii) Since the two boundary components of S’ correspond to the distinct curves a and b in S, we may again consider 8’ to be a subsurface of S. Again, a basis for H1(S’) may be extended to a basis for H1(S) (see figure 3.2) so H1(S’) is a direct summand of H1(S). As in (i), f’ e T (S’). 32 Figure 3.2: A basis for H1(S’) containing b completed to a symplectic basis for H1(S). Lemma 3.2 Let 6 consist of either a single separating curve or a single bounding pair, and let A : M¢(S) —> M(S¢) denote the reduction along Q3. Let S’ and S” be the components of Se. If f E T(S) is reducible along Q3, and if f’ is the restriction of A( f) to M(S’), then there exists h E T(S), reducible along Q3, such that the restriction of A(h) to M(S’) is equal to f’ and the restriction of A(h) to M(S”) is trivial. Note: By Lemma 3.1, all mapping classes mentioned in this lemma actually lie in the Torelli group of their respective surfaces. Proof: As in the preceding lemma, we may consider 8’ and S” to be subsur- faces of S. Now, take a homeomorphism F’ : S’ —> S’ representing f’ and extend F’ trivially to a map F” : S —> S. In the case that E is a separating curve, we may let h be the mapping class of F”. In the case that 6 is a bounding pair, we may compose F” with a multitwist about 33 6 so that the resulting mapping class h lies in the Torelli group of S. In either case then, h E T(S) is reducible along 6, and the restriction of A(h) to T(S’) is equal to f’, while S” is a trivial component of h. C] We use the symbol Cg(:z:) to denote the centralizer of :1: in the group g, and Z (g ) to denote the center of 9. Lemma 3.3 Let a be a separating simple closed curve on S and 0 35 n E Z. Then CT(D;’) consists of all elements of T that are reducible 010719 a, and Z(CT(DZ)) = Z(Cr(Da)) = (Du)- Proof: Let f E T be reducible along a. Then ng’f‘l 2 D5“) = D: :> sz,’ = Dgf, so f E CflDQ). Now let f E C7-(D3). That is, f E T and fo,’ 2 Diff. Then f(a) = f(QEDQ) = Emu—1 = 60?“) 2 Q3193 = a, so f is reducible along a. This proves the first assertion. Now we show that Z(C7-(Da)) : (Du). First, Da E Z(C(Da)), so (Du) Q Z(C(Da)). Now, 8,, consists of two components, 81 and Sg, at most one of which is a one—holed torus. If 8,, i E {1, 2}, is not a one-holed torus, then it is a surface of genus at least 2 with one boundary component. In each such component of So, perform the following construction: Choose a separating curve c, and a family {21, 62, . . . , em} of curves which together with c fill 3,. Let D = DemD ---D¢,(c), where e,- D D D --D,,(c) 75 0. Then 0 is cm—1 Cj—l €j-2 ° a separating curve in S,- and c and D fill Si, since the complementary components of c U D are the same as the complementary components of 34 cUe1 U82 U---U em. Let f,- = D.D,,‘1 and g,- = DEDgl. Then f, and g,- do not commute, and both f,- and g,- are pseudo-Anosov elements of the Torelli group of S,- (cf. [7]). It follows from the two preceding lemmas that there exist elements f and g of T(S) that are reducible along a such that the reduction of f to each component S,- which is not a one-holed torus is f,- and the reduction of g to S,- is g,-. (The reduction to any one-holed torus component of 8,, must be trivial.) Let h be a nontrivial element of Z (CT(Da)). Then h E T, so h has infinite order, and h E CT(Da), so h is reducible along a. As remarked in the introduction, then, 93;, 75 0. We claim that 93;, = {a}. Suppose a 75 b E 93),. Since 5 is essential and a is a reduction class for h, a and b are disjoint (Theorem 0.2 (i)), so 5 lies in either 31 or 82. If S,- is a one-holed torus, then h is trivial on Si, and in this case, b C S,- implies that b is not essential, since the removal of b from any adequate reduction system for h leaves an adequate reduction system contradicting Theorem 0.2 (ii). So 5 lies within S,, where S,- is not a one-holed torus component of So. Since each of f and g constucted above are reducible along a, we know that f and g are elements of CT(Da). Thus, both f and g commute with h. In particular, the facts that f and h commute and b is an essential reduction class for h imply that b is a reduction class for f. But f has no reduction class within 8,, since by construction the restriction of f to S,- is pseudo-Anosov. We conclude that Qih = {a}. 35 So Z(C7~(Da)) is a finitely generated, torsion free Abelian subgroup of T, and U 6,, = {a}. h€Z(Cr(Da)) Let A : Ma(S) -—> M(Sa) be the reduction homomorphism, and denote Z(C7—(Da)) by simply Z. Now, ker(A) = (Du), so ker(Alz) :2 (Du) fl Z = (Du). We have the short exact sequence of Abelian groups: 1—>(Da)——>Zfl§+A(Z)—+1 Since h commutes with both f and g, A(h) commutes with both A( f) and A(g). Let S, be a component of S, that is not a one-holed torus, and let h, be the restriction of A(h) to S,. Then h, commutes with both f, and g,. By McCarthy’s theorem given in the introduction C7-(s,)(f,), the centralizer of f, in T(S,), is infinite cyclic since T(S,), and thus also CT(S,)(f,), is torsion-free. Similarly, CT(S,)(g,-) is infinite cyclic. Since f, and 9, do not commute, the cyclic subgroups CT(S,)(f,) and CT(S,)(g,-) intersect trivially. But h, is in this intersection, and hence we see that each component of h is trivial. Therefore A(Z) = {1}, so that (Du) —> Z (C7-(Da)) is an isomorphism. Finally, it is clear that Z (67(03)) 2 Z (C7-(Da)). Cl Lemma 3.4 Let {(1, b} be a bounding pair on S, and let 0 74 n E Z. Then CT((Dan_1)”) consists of all elements of T that are reducible along both a and b, and Z(Cr((DaD§1)")) = Z(Cr(DaD§1)) = (0an— 1)- Proof: 36 Let f E T be reducible along both a and b. Then fD,,Db_1f‘1 = fDaf lfD; 1f 1—— _) Assume f is an elementary T class. Case 1: f is a power of a Dehn twist about a separating curve a, say, f 2 D3, where n 75 0. In this case, a can be completed to a pants decomposition 6 on S whose reduction system graph is shown in Figure 3.3. 37 Figure 3.3: The Torelli multitwist group on a reduction system with this graph has rank 29 — 3. The Torelli multitwist group on 6, T93, is an Abelian subgroup of T with rank 2g — 3, and f E Tc, so f E 7}, C C7-(f). Since 9 2 3, it follows that 2g — 3 Z 3, so CT( f) is not infinite cyclic. So both (i) and (iii) hold. Finally, by Lemma 3.3, Z (07( f )) is infinite cyclic. Case 2: f is a power of a bounding pair map, say, f = (Dan’ I)”, where n 75 0. In this case, {(1, b} can be completed to a pants decom- position QE whose reduction system graph is shown in Figure 3.4. mo Figure 3.4: The Torelli multitwist group on a reduction system with this graph has rank 2g — 3. The Torelli multitwist group on Q3, 72,, is an Abelian subgroup of T with rank 2g — 3, and f E Te, so f E Te C C7~(f). Since g _>_ 3, it follows that 29 — 3 Z 3, so that C7-( f ) is not infinite cyclic. So both ( i ) and (iii) hold. Finally, by lemma 3.3, Z (C7-( f )) is infinite cyclic. <2) Assume (i), (ii), and (iii) hold. Since C7—(1)= T and Z(T) = {1}, we have by (ii) that f 75 1. Then f E T implies that f is either pseudo-Anosov or infinite-order reducible. But C7-( f ) is not infinte cyclic by ( i ), so f is not pseudo-Anosov. Hence 38 f is infinite-order reducible, and therefore 6f, the essential reduction system for f, is nonempty. Assume that f is not an elementary T class. We obtain a contradic- tion by considering 6f. First, suppose Qif contains a separating curve a. Let g E CT(f). Then g(€f) = Ggfg—l 2 Eye, so a is a reduction class for g. Ivanov’s theorem implies that 9(a) = a, and so 9Da9_1 = Dgu) = Du => gDa = Dag. Since f E CT( f), the paragraph above shows that Du E CT( f), as well as that D, E Z (CT( f )) Consider the short exact sequence of Abelian groups 0 —+ (Du) -> Z(Cr(f)) —+ Z(Cr(f))/(Da> —> 0 Claim 1: Z (07( f )) / (DC) is not a torsion group. Proof: Now, f E Z (CT( f )) and f E (Du) since f is not an elementary T class, but suppose that fm = D: for some integers m and n. Then {a} = QED, = QED: 2 6f": 2 6f, so the reduction of f along a is adequately reduced. But f E (Du) implies that the reduction of f along a is not trivial, so that f has a pseudo-Anosov component. But then fm = D: is impossible unless m = n = 0. Hence f generates an infinite cyclic subgroup in Z (CT( f )) / (Du), proving Claim 1. Therefore, the rank of Z (67( f )) / (D0,) is at least one, so the rank of Z (CT( f )) is at least 2, contradicting condition (ii). We conclude that Q? f can contain no separating curves. 39 Second, suppose that E f contains a bounding pair {a, b}. Let g E Z(CT(f)). Then g(€f) = Qigfg—i 2 6f, so both a and b are reduction classes for g. Ivanov’s theorem implies that g(a) = a and 9(5) = b, so gD.,D,,-lg-1 = Dg(a,Dg-({,, = Dan’1 :> gD,D,—1 = Dan—lg. Since f E CT( f), the paragraph above shows that Dan’1 E 67( f), as well as that Dan—1 E Z (C7—( f )) Consider the short exact sequence of Abelian groups 0 —> (Dan—ll -—> Z(Cr(f)) —> Z(Cr(f))/ —-> 0 Claim 2: Z (C7—( f )) / (DaDg 1) is not a torsion group. Proof: Now, f E Z(C7-(f)) and f E (Dan'l), since f is not an elementary T class, but suppose that f m = (DaD; 1)" for some integers m and n. Then {a,b} = 60:0: 2 €(Dan—1)n : Efm 2 6f, so the reduction of f along {a, b} is adequately reduced. If the reduction of f along {a, b} were trivial, it would follow that f E (De: Db) n T 2 7{a,b} : (Dan—ll which is a contradiction. So we conclude that f has a pseudo-Anosov component. But then fm = (Dan‘ 1)” is impossible unless m = n = 0. Hence f generates an infinite cyclic supgroup in Z (CT( f)) / (Dan‘ 1), proving Claim 2. Therefore, the rank of Z (C7-( f )) / (Dan— 1) is at least one, so the rank of Z (C7-( f )) is at least two, contradicting condition ( ii ) We conclude that 6 f contains no bounding pairs. 40 Therefore, 6; consists of a collection of c-type curves. Let A be an Abelian subgroup of T containing f, and let QE 2 U Q39. gEA Then 6f C C. By equations 2.1 and 2.2, we know that rank(A) S rank(Tg) +9 S u+Q — 1 S 29 — 3. By considering equation 2.1, we see that the last inequality above will be strict unless each of the following are true: 0 Each component of Se; having one or two boundary components has genus 1. 0 Each component of Se having three or four boundary components has genus 0. e No component of Se has more than four boundary components. Suppose each of these conditions hold. Consider the pseudo-Anosov components of f; each of them must be a pseudo-Anosov component of each 1 75 h E A. None can have exactly one or two boundary curves, since such boundary curves are elements of Q3, and we have already seen that G contains no separating curves or bounding pairs. None can be a pair of pants, since pants do not support pseudo-Anosov mapping classes. So every pseudo-Anosov component of f must be a four-holed sphere, and we note that in the reduction system graph of 6, such a component would correspond to a non-cut vertex of degree 4. But by the following lemma, then, rank(A) < 29 — 3. That is, f cannot lie in 41 an Abelian subgroup of T having rank 2g — 3, contradicting condition We conclude that f must be an elementary T class. El Lemma 3.6 Let A be a reducible Abelian subgroup of T, and let G = U Eg be the essential reduction system for A. If the reduction system gEA graph for Q3 has a non—cut vertex of degree 2 3, then rank(A) < 29 — 3. Proof: In this proof, we use the notation and results found in Chapters 1 and 2. Let v be a non-cut vertex of degree 2 3. Then no two of the at-least three edges incident with v are ~-equivalent b-type edges and none are cut edges (a—type edges). So there exists a spanning tree T in the graph containing none of the b—type edges incident with v. Hence T contains a c-type edge, incident with v, so V—l = card(E(T)) > 19+ ((11 - 1) + ((12 - 1) + - ° ° + (qr — 1) = rank(T¢) But then rank(A) S rank(qu) + Q < V — 1 + Q S 2g — 3. Cl Theorem 3.7 Let f be an elementary T class. Then f is a power of a Dehn twist about a separating curve of genus 1 if and only if f is contained in an Abelian subgroup of T of rank 2 which is not contained within any Abelian subgroup of T having higher rank. 42 Proof: =>) Assume the f is a power of a Dehn twist about a separating curve of genus 1, say f 2 D5, where 0 75 p E Z. Let S’ be the component of So having genus > 1, and let S” be the genus 1 component of 8,. Let h E T be reducible along a such that S’ is a pseudo-Anosov component of h. Since h reduces along a, f and h commute, so (f, h) is Abelian. We have a short exact sequence of Abelian groups 0 —> (f) —> (fill) —> (AM/(f) —>0 Suppose that hm = f" for some integers m and n. Since hm has a pseudo—Anosov component for all nonzero integers m, whereas f n has no pseudo-Anosov component for any integer n, we must have n = m = 0, so that the coset of h has infinite order in (f, h) / (f) Thus (f, h) / (f) is not a torsion group, and hence has rank 1. By our sequence, then, we see that that (f, h) has rank 2. We must show now that if A is an Abelian group and (f, h) E A C T, then the rank of A is at most 2. We have a short exact sequence of Abelian groups 0 ——+ (f,h) —+ A —> A/(f,h) ———> 0 so rank(A) = 2 + rank(A/(f, h)). We show that A/(f, h) is a torsion group, and hence has rank 0. Let 1 75 a E A. We prove that am E (f, h) for some nonzero integer m, which will establish that A/ (f, h) is a torsion group. Now, f, a E A 43 imply that a and f commute, so a is reducible along a, and a E T implies that a has infinite order. Therefore, the essential reduction system for a, Q30, is nonempty. Suppose there exists a 34 b E Qia. Then a and b must be disjoint, so either b lies within S’, or b lies within 8”. But in the second case, since a is trivial on this component, b would not be essential. So b C 8’. But the fact that a commutes with h implies that h is reducible along b. This is a contradiction, since S’ is a pseudo-Anosov component of h. Hence Q30 = {a}. Let A : Ma(S) ——> M(Sa) be the reduction homomorphism. So f E ker(A) : (Du). Now, for any d E A, the restriction of A(d) to S” is trivial. Thus A(A) is isomorphic to a torsion-free Abelian subgroup of T(S’) (Lemma 3.1) containing the pseudo-Anosov class A(h). By the corollary to McCarthy’s Theorem, A(A) is cyclic. Hence if a E ker(A), then there are nonzero integers m and n such that A(a)m = A(h)”. But then amh‘" E ker(A), so have am 2 th" for some integer k, and therefore am 2 Dfikhp" = fkhnp. That is, apm E (f, h). In the case that a E ker(A), so that a = Df, it follows that a1” = f" E (f, h). <=) Assume that f is not a power of a Dehn twist about a separating curve of genus 1. Then f is a power of a Dehn twist about a separating curve of genus at least 2, say f = k" where k = D“, or else f is a power of a bounding pair map, say f = k" where k = Dan’ 1. In either case, we want to show that it is not true that f is contained in Abelian subgroup of T of rank 2 which is not contained in an Abelian subgroup of T with higher 44 rank. In other words, if f E A C T, where A is Abelian and of rank 2, then there exists an Abelian subgroup B C T such that A C B and rank(B) > 2. We consider the two possibilites for k simultaneously. Let p be the smallest positive integer such that k1” E A. Claim: A/ (kp) is torsion-free. Proof: Suppose that h E A\(k1’), but hq E (W) for some 0 34 q E Z. That is, hq :2 km for some 0 aé r E Z. Since Q3}, 2 Qihq = kar = 6;, by Theorem 0.2 ( iv), we see that h reduces along 61,, which is either the separating curve a or the bounding pair {(1, b}, whichever the case may be. Let A denote the reduction along 03],. So A(h)q = A(hq) 2 A04”) 2 1. By Ivanov’s Theorem, A(h) leaves both of the two components of Se, invariant, and since h is in the Torelli group of S and we are reducing along either a separating curve of a bounding pair, by Lemma 3.1 each restriction of A(h) lies in the Torelli group of its component, which is torsion-free. These facts imply that A(h) = 1, so that h is a Torelli multitwist about 61,, which then implies that h is a power of 1:. But then it is easy to see that this will contradict either the minimality of p or the assumption that h E (kp), proving the claim. Since A has rank two, it follows that there exists t E A such that A 2 (hp) 69 (t), and since t and k commute, t reduces along 6k. Let $1 and 82 be the two components of Se, (neither of which is a one-holed torus), and A the reduction along 6],. Note that A(t) is not trivial; we may assume without loss of generality that 81 is a nontrivial component of t. By Lemma 3.2, there exists t1 E T(S) reducible along 6;, such 45 that the restrictions of A(t) and A(tl) to T(Sl) are the same, and that 82 is a trivial component of t1. If 82 is also a nontrivial component of t, then we have t2 E T(S) defined analogously to t1. For the case that S2 is a trivial component of t, note that 82 contains a separating curve c, so let t2 2 DC. In either case, methods as employed in the preceding lemmas show that B = (W, t1, t2) is an Abelian subgroup of the Torelli group with rank 3. Then t = tltgk" or t = tlk’ for some integer r, so tEB.ThustACBCT,andwearedone. Cl Theorem 3.8 Let f be an elementary T class. Then f is a power of a Dehn twist about a separating curve of genus Z 2 if and only if there exist 9 Dehn twists Dal, D0,, . . . Dag about separating curves of genus 1 such that (f, Dal, D02, . . . Dag) is an Abelian subgoup of T with rank 9 + 1. Proof: =>) Assume f is a power about a separating curve of genus 2 2, say f = Di- Then there are g separating curves a1, a2, . . . , a9 of genus 1 that are pairwise disjoint and each disjoint from a. See figure 3.5. It follows that (f, Dal, Daz, . . . Dag) is an Abelian subgroup of T with rank 9 + 1. <=) Assume that f is not a power of a Dehn twist about a separating curve of genus Z 2. First, suppose that f is a power of a Dehn twist about a separating curve a of genus 1. In this case, there do not exist 9 separating curves 46 Figure 3.5: g pairwise disjoint separating curves of genus 1, each disjoint from a. that are pairwise disjoint and disjoint from a. Second, if f is a power of a bounding pair map, say f = (Dan‘ 1)1”, then there do not exist g separating curves of genus 1 that are pairwise disjoint and each disjoint from both a and b. So in either case, no such group (f, Dal, Daz, . . . Dag) can exist. Cl Theorem 3.9 Let \I' : T —> T be an automorphism of the Torelli group of S. Then ‘11 permutes that set of left and right Dehn twists about separating curves, and \Il permutes the set of bounding pair maps. Proof: Let f E T. If 67( f) is not infinite cyclic, then CT(‘II( f )) is not infinite cyclic. If Z (CT( f )) is infinite cyclic, then Z (CT(\I’( f ))) is infinite cyclic, and if f is contained within an Abelian subgroup A C T with rank 29 — 3, then \I!( f) is contained within the Abelian subgroup \II(A), which has rank 29 — 3. Thus, \II sends elementary T classes to elementary T classes. Now suppose that \I!( f) is an elementary T class. Since \I/’1 : T —9 T is also an automorphism, and thus also sends elementary T classes to elementary T classes, we know that \IJ‘1(\IJ(f)) = f is an elementary 47 T class. Since \II is a bijection of T, it follows that \IJ permutes the set of elementary T classes. Similarly, we see that ‘I/ permutes the set of powers of Dehn twists about separating curves of genus 1, permutes the set of powers of Dehn twists about separating curves of genus Z 2, and permutes the set of powers of bounding pair maps. If f 2 D0 is a right Dehn twist about a separating curve, we know that \II( f) is a power of a Dehn twist about a separating curve, say \Il(f) 2 D5, 0 7i p E Z. Then \IJ‘1(Db) is also a power of a Dehn twist about a separating curve, say \I"1(Db) 2 D3, 0 # q E Z. Hence D3” = \II'1(Db)p = \P'1\Il(f) = f 2 Da. Since QED“ = {a} and 6ng = {c}, we see that a = c. Hence Da 2 D3”. But since D, has infinite order, we must have qp = 1, so that p : :l:1. So \II permutes the set of left and right Dehn twists about separating curves. If f = Dan‘1 is a bounding pair map, we know that ‘11( f) is a power of a bounding pair map, say ‘I’( f ) = (DCDO—IV, 0 79 p E Z. Then \I’"1(D,D;1) is also a power of a bounding pair map, say ‘II’1(DCD;1) = (D,D{1)q, 0 75 q E Z. Hence (D,D,—1)qp = \It‘1(D.D,—1)P = \II‘hI/(f) = f = 0,0,4. Since ED“): =2 {a,b} and 6(Dch-l)” = {2,f}, we see that {a, b} =2 {e,f}. Hence DaD51 = (DaD51)iqp. But since Dan—1 has infinite or- der, we must have qp = :i:1, so that p = :l:1. So \II also permutes the 48 set of bounding pair maps. D 3.2 The Complex of Curves First introduced by William Harvey [8], the complex of curves, C (S), of S is an abstract simplicial complex whose vertex set is G, the set of unoriented isotopy classes of homotopically nontrivial simple closed curves on S, and a collection of vertices forms a simplex if and only if the vertices have a set of mutually disjoint representative curves. In other words, a simplex is a reduction system as defined in the intro- duction. By an automorphism C (S) —> C (S) we mean a bijective map 6 —> G of the vertex set of C (S) such that the image of any simplex is a simplex (i.e., the map is simplicial.) Notation: Let a be a nonseparating simple closed curve on S. We write a = {b E 6 : {a, b} is a bounding pair}. We want to show that our automorphism \II : T —> T of the Torelli group induces an automorphism \I'... : C(S) —> C(S), where we make the following definition: An automorphism \II, : C (S) —+ C (S) is induced by ‘11 if and only if i) for all a separating, ‘11,..(a) = a’ 4:) \II(D,,) = (Dar)€(“), where 6(a) = i1, and 49 ii) for all a nonseparating, \II..(a) = a’ 42> there exist functions owl : 6,, —> Ga: and 6,, : 6,, ——> {i1} such that for all b E 6,, we have Our definition above makes it clear where ‘11... should send the vertex corresponding to a separating curve. Conjecture 2 below says that there exists a unique well-defined image vertex for each vertex corresponding to a nonseparating curve. The proof of uniqueness is straight-forward and is stated and proved as the following lemma. The proof of existence, on the other hand, seems quite difficult, and is incomplete at the time of this writing. We must also show that the bijection (‘3 ——> 6 we get is simplicial. Lemma 3.10 Let a be a nonseparating simple closed curve on S. Then there exists at most one nonseparating simple closed curve a’ on S with the property that there exist functions 0a,a’ : 6,, —+ Car and 6,, : 6,, —+ {:lzl} such that for all b E 6a, we have \I'(DaD51) = (DalBga’a’(b))”a(b). Proof: Suppose there also exists a” with 0a,," : 6,, ——> 6,," and 7a : 6,, -—> {:El} such that for all b E 6a, we have \I’(Dan‘ 1) = (DairD;:a,l(b))%(bl. Then for each b E 6,, we have (DG’D;,],,(b))6a(b) 2 (Da”D;,],u(b)l’°(b’ and hence {a’, swim} -_— {a”,oa,a~(b)}. ' (3.1) 50 Claim: fl {a’, (1,303)} = {a’} bESQ Proof: Suppose there exists a’ aé c E {a’,oa,a:(b)} for all b E 6,. That is, c = oa,az(b) for all b E 6a. This implies that card{(DaID;a:a,(b))5°(b) : b E Ga} S 2. But this is a contradiction since 6a, and thus also {Dan—1 : b E Ga}, is infinite, and {(DarD;1,(b))"°(b) : b E 6,} is the image of the set {Dan‘1 : b E 6,} under the injective map ‘11. This proves the claim. Likewise, n {a”, oa,air(b} = {0”}. But equation (3.1) implies that hes. fl{a’,oa,ar(b)} 2 fl{a”,oa, a”(b)}. That is, a’ = a”. Conjecture 2 Leta be a nonseparating simple closed curve on S. Then there exists a unique nonseparating simple closed curve a’ on S with the property that there exist functions 0a,a’ : 6,, ——> Go: and 6a : 6,, ——> {:izl} such that for all b E 6,, we have \II(D,,D;1) : (DaID’1 ))"°(b). 0a,a’(b 3.3 Proof that Conjecture 2 implies Conjecture 1 We now prove Conjecture 1, stated on page 30, under the assumption that Conjecture 2 is true. Conjecture 2 implies that our automorphism \I! : T ——> T induces an automorphism \II... : C (S) —> C (S), given by 51 o for a E G separating, ‘11,,(a) = a’ 4:) @(Da) = (Du/)4“), and o for a E G nonseparating, \II,(a) = a’ 4:) there exist unique functions 0,, : 6,, ——> 6“: and 6,126,, ———> {i1} such that for all b E 6,, we have \I/(Dan—1)=(DC,I‘D 1 ) W l. 0.. a(b) Now, if h : S —+ S is a homeomorphism, then h induces two functions: 0 an automorphism h... : C(S) —> C(S) defined for a E (‘5 by a i—> h(a) c an automorphism h, : T ——> T defined by f +——> hf h‘l. By Ivanov ([9], Theorem 1), \I', = h... for some homeomorphism h : S —> S. Our goal is to show that ‘11 2 hit- Here, ‘II... = h... means that if a E 6, then \II...(a) = h...(a). Specifically, if a is separating, then \IJ...(a) = h...(a) means that @(Da) = (Dh(a))€(°), while if a is nonseparating, \II,.. (a) = h...(a) means that for all b E Ga, mam-1) = (Du )Dgi. ,> < >. Let f E T, and let a E 6. We have two cases. Case 1: a is separating. In this case we have \I/’(fD.f >= ‘I’(Df< ))= 02332., On the other hand, we also have w(foarl) = @(f)‘P(Da)‘I’(f)" = \P(f)DZ[Z’)\I’(f) — —D;‘(°},(,(.,,. and so \Il(f)(h(a)) = hf(a). If we write a = h‘1(b), then this shows that \I/(f)(b) = hfh‘1(b) for all b E 6 separating. 52 Case 2: a is nonseparating. Let b E 6,. Then we have w(fDa Db 1f 1:) ’I’a(Df( )Df fb( 111:)l (th(a )D 1 f(b)))6’(a)(f(b))- 011a) ( On the other hand, we also have @(fDan’lf—l) ‘I’(f)‘1’ (f)(Dh(a))D;,1 (1,5)) “MIND—1 (f)D§’, railing)‘: ) b (Danl)\1’(f)‘1 ll (f D2: )w ) ’\P(f)D;j’(.(,’"1’(f)" ll U666 6( ) —6 (b) wow: (u >>Dwv>} = {hf(a)} Proof: Sfigapose there exists hf(a) 75 c E {hf(a),of(a)(f(b))} for all b E Ga. That is, c = of(a)(f(b)) for all b E 6,. This implies that 6 a , card{(th(a)Daf1(a) (f(b))) f‘ WW) . b E Ga} S 2. But the set 6,, is infinite, and since f is injective, so is {Df(a )DT(l1) b E 6.1} And Since {(th(a)Da,1(,,(f a») of the infinite set {Df(,,)Df(1b f(b)’ b E 6a} under the injective map \II, we df(a)(f(b)) ; b E 60} IS the image obtain a contradiction. This proves the claim. Likewise, fl {@(f)(h(a)),\11(f)(oa(b))}={‘I’(f)(h(a))}. So by (3.2) 66,, we have hf(ba) = \I/(f)(h(a)). If we write a =2 h‘1(D), then this shows that hfh‘1(D) = \IJ(f)(D) for all 0 E (‘3 nonseparating. 53 Cases 1 and 2 together show that \Il(f)(a) = hfh‘1(a) for all a E (‘3, and since the genus g of S is at least 3, this implies that \II( f) = h f h—1 for all f E T. That is, ‘I/ = h, : T—> T. 54 REFERENCES [1] Joan S. Birman: Braids, Links, and Mapping Class Groups, Prince- ton University Press and University of Tokyo Press, Princeton, New Jersey, 1974. [2] Joan S. Birman, Alex Lubotzky, and John McCarthy: Abelian and Solvable Subgroups of the Mapping Class Group, Duke Mathematical Journal, Volume 50, Number 4, pp. 1107 — 1120; December 1983. [3] J. A. Bondy and U. S. R. Murty: Graph Theory with Applications, North-Holland, New York; 1976. [4] Benson Farb: Private electronic mail correspondence to Nikolai V. Ivanov and John D. McCarthy, October 30, 2001. [5] Benson Farb: Private electronic mail correspondence to Nikolai V. Ivanov, November 11, 2001. [6] Benson Farb: Automorphisms of the Torelli Group, American Math- ematical Society abstract number 974—57-58, received December 28, 2001. [7] A. Fathi, F. Laudenbach, and V. Poenaru, et. al.: Travaux de Thurston sur les surfaces, Seminaire Orsay, 1975, Asterisque No. 66/67. [8] W. J. Harvey, Boundary Structure of the Modular Group, in Rie- mann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, edited by Irwin Kra and Bernard Maskit, Prince- ton University Press and University of Tokyo Press, Princeton, New Jersey, 1981. [9] Nikolai V. Ivanov: Automorphisms of Complexes of Curves and of Teichmiiller Spaces, International Mathematics Research Notes, 1997, No. 14. [10] Nikolai V. Ivanov: Subgroups of Teichmiiller Modular Groups, Translations of Mathematical Monographs, Volume 115, American Mathematical Society, 1992. 55 [11] John D. McCarthy: Normalizers and Centralizers of Pseudo- Anosov Mapping Classes, (preprint), March 5, 1982. [12] Jerome Powell: Two Theorems on the Mapping Class Group of a Surface, Proceedings of the American Mathematical Society, Volume 68, Number 3, pp. 347 — 350; March 1978. 56 1111111111111