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Encounter with a Geometer, Part I, Volume 47, Number 2

Encounter with a Geometer, Part I, Volume 47, Number 2

fea-berger.qxp 12/6/99 3:46 PM Page 183

Encounter with a Geometer, Part I Marcel Berger

Editor’s Note. Mikhael Gromov is one of the leading mathematicians of our time. He is a professor of at the Institut des Hautes Études Scientifiques (IHÉS), and, as an article elsewhere in this issue of the Notices reports, he recently won the prestigious Balzan Prize. The present article, with Part I in this issue and Part II in the next, discusses Gromov’s mathematics and its impact from the of view of the author, Marcel Berger. It is partially based on three interviews with Gromov by Berger, and it first appeared in French in the Gazette des Mathématiciens in 1998, issues 76 and 77. It was translated into English by Ilan Vardi and adapted by the author. The resulting article is reproduced here with the permission of the Gazette and the author.

he aim of this article is to communicate nical tools of the proofs, but almost al- the work of Mikhael Gromov (MG) and ways never study them deeply enough its influence in almost all branches of in order to understand the underlying contemporary mathematics and, with a thought. leap of faith, of future mathematics. Be- T Adrien Douady said of his own reading of Homo- cause of its length, the article will appear in two logical Algebra by Cartan and Eilenberg that he read parts. It is not meant to be a technical report, and, it “until the pages fell off.” The works of MG should in order to make it accessible to a wide audience, be read in that same way. I have made some difficult choices by highlight- The exposition is classical: algebra, analysis, ing only a few of the many subjects studied by MG. . One could object to the use of the word In this way I can be more leisurely in my exposi- “geometer” in the title, since we will be discussing tion and give full definitions, results, and even oc- algebra and analysis, but it will become clear to the casional hints of proofs. reader that the method MG uses to attack problems I wrote this article because I believe that MG’s is to turn them into ones formulated in the lan- work is greatly underrated. I will not analyze the guage of geometry. I owe to Dennis Sullivan the re- reasons for this phenomenon, although a general mark, “It is incredible what MG can do just with idea of why this is so will become clear from a read- the triangle inequality.” Yet MG’s results outside ing of the text. Apart from , of Riemannian geometry are really in algebra and notable exceptions to this phenomenon occur in analysis, even though they have not received much two fields: hyperbolic groups and symplectic geom- recognition, especially in analysis. etry via pseudoholomorphic curves; these areas will In order to shorten the text, I have omitted es- be discussed in the first two sections below. In any sential intermediate results of varying importance case, MG expressed his own view to me: and have therefore neglected to include numerous names and references. Although this practice might The readers of my papers look only at lead to some controversy, I hope to be forgiven for corollaries, sometimes also at the tech- the choices.

Marcel Berger is emeritus director of research at the Cen- Algebra: Geometry of Groups of Finite Type tre National de la Recherche Scientifique (CNRS) and was We consider here only “discrete groups of finite director of the Institut des Hautes Études Scientifiques type”, i.e., finitely generated discrete groups. These (IHÉS) from 1985 to 1994. His e-mail address is groups are of course interesting by themselves, but [email protected]. they appear also as transformation groups in var- The author expresses his immense debt to Ilan Vardi and ious situations: notably in number theory as the Anthony Knapp—to Vardi for translating the article into modular SL(2, Z) acting on the upper half English and for improving the clarity of the mathemati- cal exposition, and to Knapp for editing the article into its plane or, more generally, as discrete subgroups current form. acting on homogeneous spaces G/H, yielding

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Z must be precisely defined, and this can be achieved by considering everything in nothing less than the set of all separa- R ble complete metric spaces. On this, MG defines a metric, denoted by dGH and Figure 1. Seen from infinity—practically speaking, from farther and farther now called the Gromov-Hausdorff met- away—the two discrete pictures above look more and more like the ric. We postpone its definition to the be- continous real . ginning of Part II. Very delicate techni- cal computation is required in order to geometric quotients \G/H. They appear also as show that there is a convergent subsequence in the fundamental groups of compact differentiable ( ,1d), that the limit is a Lie group, and that G manifolds, in particular those of negative curva- operates suitably on it. Finally, one succeeds in as- ture. Here I highlight two results. sembling in a single entity all the structure at in- The first is an affirmative answer in [3], “Groups finity, and this entity is continuous. An important of polynomial growth and expanding maps”, to Mil- remark is in order: in the above proof MG uses the nor’s 1968 conjecture: Every finite-type group of solution of Hilbert’s Fifth Problem to obtain a Lie polynomial growth contains a subgroup of finite group. This problem has a fascinating history; it index that is itself a subgroup of a nilpotent Lie was solved in 1955 by Gleason, Montgomery, and group. The function Growth with domain the set1 Zippin, but there had been practically no applica- of finitely generated groups is defined on a group tions until this result. The problem was to show L as the number Growth(L) of elements of the that a group without arbitrarily small nontrivial group that can be expressed as words of length L subgroups is necessarily of Lie type. MG used the in these generators and their inverses. A change following corollary: The isometry group of a locally of generators does not affect growth type: poly- compact, connected, locally connected metric space nomial, exponential, etc. In many cases the result of finite is a Lie group with only a fi- was known. For example, suppose that G is linear. nite number of connected components. To navigate By a result of Tits, either G contains a free non- through the technical details of the proof, the in- abelian subgroup and is then of exponential growth terested reader may look at the informative ex- or it is solvable up to finite index. In the solvable pository text of Tits in the Séminaire Bourbaki for case a 1968 theorem of J. A. Wolf shows that ei- 1980–81. ther a linear solvable group contains a nilpotent This might be the right time to make the fol- subgroup of finite index and is of polynomial lowing remark, communicated to me by Hermann growth or it does not and is of exponential growth. Karcher. Almost all of MG’s big results have three For a general abstract group the problem is to features. At the start is a very simple, even naive, find a Lie group, i.e., a continuous object, into idea, so simple that it does not seem possible to which the discrete group can be embedded. MG’s do anything serious with it! The second feature is solution is to look at the group from infinity (Her- that the development to the conclusion is very el- mann Weyl said that “mathematics is the science egant and technical, sometimes using subtle com- of infinity”). For example, the group Zd seen from putations but always introducing new tools. The infinity will appear as the Euclidean space Rd, be- third feature is that MG in the course of his proof cause this lattice from far away has a very small introduces one or more new invariants, i.e., new mesh that ultimately looks continuous. Even under concepts. These are often as naive as the starting a change of generators, this structure from infin- idea, but they play a basic role in the proof and ity will remain the same. For example, take Z, first most often have the paradoxical property that with the trivial generators {1}, and second with they are impossible to compute explicitly, even the generators {2 ∪3}. The corresponding Cay- for the simplest spaces. In many cases it is also im- ley graphs are shown in Figure 1. possible to decide whether they vanish or not; we It is clear that the second graph, seen from in- will meet many of these below. finity, is squeezed down to a line. For any abstract We return to the Gromov-Hausdorff distance be- group , one naively generalizes this as follows: first tween metric spaces, which has become a basic tool make into a homogeneous metric space ( ,d) by of Riemannian geometry. It was introduced in full letting the distance d(g,h) between two elements detail in Gromov’s address at the 1978 Interna- be the length of h1g when expanded in terms of tional Congress of Mathematicians2 and then used generators; the Cayley graph of then consists in heavily in his 1981 preliminary French version of joining two points by a line if and only if their dis- [13]. Today it is of primary importance when look- tance equals one. Looking at from infinity ing at a Riemannian manifold (M,g) from infinity, amounts to looking at the limit of the sequence of i.e., looking at the sequence (M, 1g) with going metric spaces ( ,1d) as goes to zero. This limit to zero.

1If isomorphic groups are identified, then the set of finitely 2The address was entitled “ of Rie- generated groups is a set. mannian manifolds”.

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The second algebraic topic is the concept of y z hyperbolic group. It was invented in [9], “Hy- perbolic groups”, which gives many of the properties of these groups as well as the spec- x tacular probabilistic statement: almost all y groups are hyperbolic. This long paper trig- = 0 gered an immense wave of research and results. Apart from work in Riemannian geometry, this x and the paper [7], z “Pseudoholomorphic curves in symplectic man- ifolds”, are, in my opinion, the only Gromov works to achieve full recognition. For the paper Figure 2. A triangle is -thin when any point of any of its sides is [7] Gromov received the AMS Steele Prize in universally within of the union of the two other sides. In hyperbolic 1997. geometry, there is a so that all triangles are -thin, but in Euclidean The first naive notion to come into play for geometry this is not the case. Any tree is trivially thin for =0. hyperbolic groups is that of space, sometimes also called length space; this is a • the universal cover of a compact Riemannian metric space where any pair of points of distance manifold of negative curvature (a continuous a can be joined by at least one segment of length object) and its (a discrete a, i.e., an isometric map of the interval [0,a] into object). the space with ends at the two given points. The MG calls a hyperbolic metric space a geodesic basic example is a complete Riemannian manifold space for which there exists a constant such (the Hopf-Rinow theorem). At the other end of the that all triangles of the space are -thin; i.e., any spectrum is the Cayley graph of a group in which point on an edge of the triangle has a maximum one lets every edge be isometric to [0, 1].3 distance to the union of the two other edges that The second notion is to isolate for our group a is . In other words, for every triangle concept defined in terms of the metric that is in- [x, y] ∪ [y,z] ∪ [z,x] with vertices x, y, z and every variant under change of generators and that there- u ∈ [y,z], one should have fore will respect the various growth types. Such a concept is that of quasi-isometry. This concept d(u, [x, y] ∪ [z,x]) = inft∈[x,y]∪[z,x]d(t,u) . was introduced explicitly by Margulis in 1969 for It is not hard to check that this notion is stable discrete groups and by Mostow in 1973 for the gen- under quasi-isometry and that Euclidean spaces are eral case. The general definition is quite involved: Two metric spaces X and Y are quasi-isometric if not hyperbolic. The classical hyperbolic spaces of there exist two maps f : X → Y and g : Y → X and in the ordinary sense, e.g., the two constants >0 and C 0 such that simply connected complete spaces of constant negative curvature, are indeed hyperbolic. Another basic example is the fundamental group of a com- dY (f (x),f(y)) dX (x, y)+C pact Riemannian manifold of negative curvature (or the manifold itself) and, more generally, any for every x, y ∈ X, polyhedron of negative curvature. Trees are an in- 0 0 0 0 dX (g(x ),g(y )) dY (x ,y )+C teresting case as they are hyperbolic for the con- 0 0 for every x ,y ∈ Y, stant =0. A group is said to be hyperbolic if its Cayley graph is hyperbolic. Evidently the Cayley d (g(f (x)),x) C X graph depends on the chosen system of generators, for every x ∈ X, but the property of being hyperbolic does not, 0 0 dY (f (g(x )),x ) C since the differences are only by quasi-isometries. 0 We avoid any detailed exposition of [9], “Hy- for every x ∈ Y. perbolic groups”, and instead will give a brief In other words, f and g are Lipschitz at large dis- overview. What is the structure of a hyperbolic tances, and f and g are almost inverses. One can group? MG shows that such a group is always fi- check that for Cayley graphs a change of genera- nitely related; i.e., it can be defined by giving only tors yields quasi-isometric metric spaces. Here are a finite number of relations among its generators. three more typical examples of quasi-isometry: This is a corollary of the existence, for any hyper- • a group and any subgroup of finite index, bolic group, of a polyhedron of finite dimension • R and Z, that is contractible and on which the group acts simplicially with compact quotient. The 3We shall use this metric on a Cayley graph throughout contractibility of this polyhedron, which is indis- this article. pensable, is due to Rips.

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We now give the reader a sense of the great loop (of length L()), one always has A() CL() power of the hyperbolicity notion by discussing for some constant C. many conditions that are related to it. First is the The second way to use isoperimetric inequali- viewpoint of “complexity and algorithms” for dis- ties is to prove that amenable groups5 can be de- crete groups: If a group has a finite presentation fined as those enjoying an isoperimetric inequal- and is of “small cancellation” C0(1/6), then it is hy- ity in maximal dimension with an exponent equal perbolic. Small cancellation means that propor- to zero. One can also define this concept on Cay- tionately little cancellation is possible in the prod- ley graphs: volume will be the number of points uct of two defining relations. More precisely, in a contained in the set, while the boundary of a set group defined by generators and relations, the re- X will be the set of points that are not in X but at lations being certain words in the generators, let distance 1 from X. Thus amenable groups are not R be the set of all cyclic permutations of these “too big”. MG proves that a nonelementary hyper- relations. The hypothesis C0(1/6) is that the set R bolic group, if not amenable, possesses a normal can contain two distinct words of the form uv and subgroup that is free and nonabelian. For those uv0of the same length only if the length of u is groups MG goes on by defining on them a type of < 1/6 of the common length of uv and uv0. An im- “geodesic flow”, a dynamical object but only topo- portant consequence of MG’s results is that the de- logical at first, because there is not initially an in- cision problem for discrete groups is only a local variant measure. These flows have the same prop- one.4 erties of ergodicity, mixing, etc., as the geodesic The second concept is that of isoperimetric con- flows on manifolds with negative curvature. In stant. For Euclidean spaces Rd the isoperimetric in- this and a number of other unexpected situations equality says that there exists a constant k(d) such in mathematics, dynamics is occasionally invading that for all bounded domains D the boundary ∂D mathematical objects in which time is not present always satisfies in the definition and yet the object can be en- dowed with a dynamical system structure, dis- (d1)/d Vol(∂D) k(d) (Vol(D)) . crete or continuous. This can be expressed by saying that Rd satisfies Hyperbolic groups, like manifolds of negative an isoperimetric inequality for maximal dimen- curvature, admit natural compactifications, so sion domains with exponent (d 1)/d. One gets have a boundary “at infinity” (sometimes a sphere, this type of inequality also for the volume of com- but not always). This compactification and the pact closed submanifolds N of a given dimension “sphere” at infinity are to be distinguished from s and the minimum volume of the (s +1)-sub- the limit of ( , 1d) as above. I make only the manifolds whose boundary is N. For Euclidean rather vague assertion that, seen from infinity, hy- spaces, the exponent is equal to s/(s +1). For clas- perbolic groups “look like trees”. sical hyperbolic spaces there is again an isoperi- metric inequality in maximum dimension, but the exponent now equals 1 because the volume of In [9], “Hyperbolic groups”, MG wrote the heuris- balls and of spheres grows exponentially and with tic statement, “A group chosen at random is hy- the same factor as a function of the radius. For dis- perbolic.” This statement was completely proved crete objects, the volume of a domain is understood by Ol’shanskii in 1992. It should be compared with to be the number of points it contains, and the MG’s heuristic vision that “almost all polyhedral same for its boundary. have negative curvature.” For geome- What MG shows concerning isoperimetric be- tries this certainly is true in dimension two, but for havior is twofold. First, if a group with finite pre- higher dimension more study is required, and the sentation enjoys a linear isoperimetric inequality assertion remains an open problem. in dimension 2, then it is hyperbolic. To define this In [11], “Asymptotic invariants of infinite notion precisely, one considers any finite polyhe- groups”, MG considerably refined the result about dron X whose fundamental group is isomorphic randomness. The first thing is to define precisely to and calls its universal cover X . Consider in how a group is chosen at random, and the next is X a simplicial loop contained in the 1-skeleton to define, for any group under study, a suitable no- of X ; simple connectivity implies that there exist tion of the ratio d of the number of relations to simplicial disks whose boundary is . Call the area the number of generators. The result is this: For a of a disk the number of its 2-simplices, and let A() random group, if d<1/2, then the group is hy- be the minimal area of disks with boundary . The perbolic, but if d>1/2, then the group is trivial. isoperimetric inequality one wants is: for every MG calls such a result an example of “phase transition”. It fits very well with his philosophy con- 4For the various algorithmic notions of simplification, the cerning the three states: solid, liquid, and gas. He reader may consult Strebel’s appendix in the 1990 book Sur les Groupes Hyperboliques d’après Mikhael Gromov 5A group G is amenable if whenever G acts continuously by Ghys and de la Harpe, which is an excellent detailed on a compact metrizable space X, X has a G-invariant presentation of many of MG’s results. probability measure.

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compares it to that for conics: ellipses, parabolas, ity”, “sheaves”—language that is not familiar to and hyperbolas. For groups, the elliptic ones are every analyst. In addition, the writing is extremely the finite groups, the hyperbolic groups are the hy- dense and hard to master. This explains why quite perbolic groups in the above sense, and the para- regularly in analysis journals results appear that bolic groups are yet to be defined. The trichotomy are a small part of some statement of [8]. To de- arises also with partial differential equations (PDEs): scribe the essence of the book, that of the h-prin- elliptic, parabolic, and hyperbolic. We shall discuss ciple, it is best to quote from the preface: MG’s contribution in this area in the next section. Number theorists will appreciate the following The classical theory of partial differ- property of hyperbolic groups. Fix a system of ential equations is rooted in physics, generators for such a group, and for any integer where equations (are assumed to) de- n let (n) denote the number of elements of the scribe the laws of nature. Law abiding functions, which satisfy such an equa- group Pof length equal to n. Then the formal series (t)= (n)tn is a rational function of the vari- tion, are very rare in the space of all ad- able t. missible functions (regardless of a par- It is now time to talk about [11], “Asymptotic ticular in a function space). invariants of infinite groups”, which is a new de- velopment growing out of [9], “Hyperbolic groups”. Moreover, some additional (like initial It studies asymptotic properties of infinite groups. or boundary) conditions often insure Briefly, asymptotic properties of groups refers to the uniqueness of solutions. The exis- properties that are stable under passage to sub- tence of these is usually established groups of finite index. The same kind of thought with some a priori estimates which lo- applies to Riemannian manifolds considered only cate a possible solution in a given func- up to finite coverings. This is the first text by the tion space. author of a very new kind of mathematical writ- ing. A later paper of this type is [12], “Positive cur- We deal in this book with a completely vature, macroscopic dimension, spectral gaps and different class of partial differential higher signatures”. These works amount to more equations (and more general relations) of a research program than a standard paper, al- which arise in though they still contain new notions and results. rather than in physics. Our equations The article [11] contains in particular two re- are, for the most part, undetermined markable geometric notions, both due to MG: that (or, at least, behave like those) and their of filling and that of simplicial volume. We shall re- solutions are rather dense in the space turn to these a little later in the present article. My of functions. belief is that these works will influence many gen- erations of mathematicians. We solve and classify solutions of these The above classification of groups and the prob- equations by means of direct (and not abilistic result do not address two essential ex- so direct) geometric constructions. amples of groups. These are discrete uniform sub- I add to this a portion of my interview with MG: groups of Lie groups associated with symmetric spaces of rank at least two and the fundamental This book is the cornerstone for build- groups of compact manifolds of nonpositive cur- ing a geometric theory of PDEs (the vature. Right notions, such as “parabolic group” or space of solutions, etc.). The book is “semi-hyperbolic group”, are still awaiting their cre- practically ignored because it is too ator. conceptual. However, it is so universal! For more details (and solid classical mathe- I maintain that most underdetermined matical writing) the 1990 book of Ghys and de la PDEs are soft, while the rigid ones are Harpe is invaluable, even if it contains only a part exceptional. In my book one can get al- of MG’s work on the subject. most anything, even fractals. Therefore there is a certain robustness; one knows Analysis: The h-Principle and Applications that robustness is essential in physics, Having roots in his dissertation and articles in the because “one sees only what is robust”. 1970s, the h-principle (“h” stands for homotopy) is the hardest part of MG’s work to popularize. It The h-principle: it is indeed hard to be- was developed fully in his long book [8], Partial Dif- lieve; analysis experts do not believe in ferential Relations. But it was only in 1998 in a book it, and as a result prove from time to by Spring that parts of [8] were first treated in de- time parts of what is already in the tail in book form. Gromov’s work [8] is visionary book. Hard to believe, because it but very dense, and it tackles many subjects. It is contradicts mathematical intuition and written with the language of “jets”, “transversal- physical intuition at the same time. For

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example, for a C1 equation, the solu- Part II, positivity has many implications, but one tions are dense in C0. still does not know all the topological consequences of this condition. As for the negativity of Ricci But there is still much to do: find a curvature, Lohkamp’s result is: “negative Ricci cur- measure for the space of solutions vature amounts to nothing”. More precisely, any (something like Markov fields), trans- manifold admits such a metric, but also on such form the h-principle into measure the- a manifold the space of all metrics with negative ory. In fact, to be able to say something Ricci curvature is contractible as a topological statistical about solutions, one must space. There are more surprises in store: on any find a measure. manifold each Riemannian metric can be approx- imated by negative Ricci curvature metrics. Of The h-principle is too long to formulate in detail, course, this can hold only in the C0-topology, since and moreover it has to be defined and proved ex- we can start with positive curvature, which is a C2- plicitly for each specific problem. But what it says, object, whereas the metric is a C0 object. Again, roughly, is that for the equations under consider- using suitably adapted h-principles, Lokhamp very ation there are no obstructions to constructing recently proved that on any compact manifold the space of solutions other than those of topo- there exists a dense set of Riemannian metrics logical and transversal nature found in a suitable whose geodesic flow is Bernoulli, hence is mixing, 6 set of jets. Here, as mentioned above and as will ergodic, etc. Previously, for the sphere S2 , indi- be mentioned again later, the initial idea is simple vidual ergodic metrics had been constructed very and geometric, and beyond that is a long techni- painfully. Lohkamp’s results illustrate the power cal and subtle path. of the h-principle. Without further definitions, here are some ap- plications of the philosophy of the h-principle taken either from the book itself or from subse- Practically at the same time that he was pub- quent texts. lishing the book [8], Partial Differential Relations, First, MG made almost complete progress in • MG completely revolutionized symplectic geome- the quest for the optimal dimension N(d) of the try in his article [7], “Pseudoholomorphic curves isometric embedding problem of abstract Rie- in symplectic manifolds”. This tidal wave, which mannian manifolds Md of a given dimension into is not yet finished, is the subject of the 1994 book some RN(d) . Nash’s famous theorem yielded such Pseudoholomorphic Curves in Symplectic Geometry an isometric embedding, but for of by Audin and Lafontaine. We shall borrow here magnitude like d3, which is too large. from the introduction, and we refer the reader to • Another contribution is the extension of Oka’s the book for a detailed exposition. principle for Stein manifolds.7 Oka’s principle says Symplectic geometry (sometimes called sym- that every continuous section of a holomorphic vec- plectic topology by those who use the word geom- tor bundle is homotopic to a holomorphic one. In etry exclusively in situations where there is a met- MG’s way of looking at it, this is nothing but the ric) is the study of manifolds M2n endowed with h-principle for the Cauchy-Riemann equations. a symplectic form, namely, a differential 2-form ω Like every application of the h-principle, this has that is closed (dω =0) and nondegenerate (one to be stated precisely in the context with which one also says of maximal rank), i.e., is such that the is working. V maximal exterior product form n ω is every- • In Riemannian geometry Lohkamp, in his se- where nonzero. This product is a volume form ries of papers starting in 1992, was inspired by the and hence defines a canonical measure on the h-principle and proved it in the cases he worked manifold. We will speak of symplectic manifolds on. His results illustrate in a spectacular way the deep nature of the h-principle, as outlined by MG and symplectic structures, since there is an obvi- above. The problem is to decide whether the hy- ous notion of symplectic diffeomorphism (= sym- pothesis of negative Ricci curvature for a Rie- plectomorphism). Symplectic structures are found mannian manifold has strong topological conse- in the mathematical world in a number of differ- quences. For sectional curvature, negativity has a ent settings, but two are especially important. The drastic consequence: the manifold’s universal cover first is that of Kähler manifolds, in which the sym- gotten by the exponential map is diffeomorphic to plectic form is nothing but the Kähler form. The Rd. For Ricci curvature, which will be discussed in second is that of , where the form under study is the “Liouville form” on the 6A jet is a system of values of Taylor coefficients through cotangent bundle (= “”) of the mani- a certain order. fold where mechanics is studied. As a consequence of a theorem of Darboux, one 7MG waited a long time to publish his result in 1992 in a joint paper with Eliashberg, “Embeddings of Stein mani- knows that, locally, every symplectic form ω is folds of dimension n into the affine space of dimension symplectomorphic to the canonicalP symplectic R2n n ∧ 3n/2+1”. structure on given by i=1 dxi dxi+n . It

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follows that there is no local invariant CP2 for symplectic geometry. In dimension B U B1 2 two a symplectic form is just a smooth (a) (b) orientable measure, so it is legitimate to ask whether in dimension four or more b there are more symplectic global in- a variants besides ones that come only

from the measure. What freedom do we have? MG’s contributions to this ques- C -1 ø C 2 C C 1 tion are various. 1 P A Lagrangian submanifold is one where the form ω vanishes when re- -1(C) stricted to it. In his 1986 book Partial Dif- ferential Relations, MG used the h-prin- ciple and addressed the question of freedom by constructing an immense Figure 3. In the complex projective space, for any reasonable symplectic number of Lagrangian immersed sub- structure, pseudoholomorphic curves mimic exactly the projective manifolds. On the other hand, he ex- geometry axioms for the projective lines. hibited a certain rigidity in symplectic geometry; this is the essence of the 1985 paper [7], “Pseudoholomorphic curves in sym- J that are compatible, i.e., have ω(x, Jx) > 0 for plectic manifolds”, where MG proved a conjecture every nonzero tangent vector. In this situation a of Arnold: in R2n there exists no compact La- is a submanifold of di- grangian embedded submanifold. Now, among the mension two whose tangent bundle is stable under many results and new tools introduced in [7], I J. In the holomorphic case this is nothing but a choose a typical one and treat it in detail so that complex curve, but the interest is in the noncom- the reader might appreciate the full depth of MG’s plex case. The condition for a two-dimensional ideas, both creative and technical. submanifold to be a pseudoholomorphic curve is We will consider the problem of packing balls a PDE. Motivated by the h-principle, MG showed via symplectic diffeomorphisms and try to see that there are numerous global pseudoholomorphic whether there are obstructions beyond their mea- curves. sure. MG has many results about this problem, In particular, MG studied the case of the com- and I choose the simplest one, both to state and plex projective space CP n endowed with its stan- 2n to prove. Let U be an open set in R (endowed with dard symplectic structure. MG considered any the standard symplectic structure) that contains compatible almost complex structure J, and in two Euclidean disjoint balls B1(a, r1) and B2(b, r2) particular he was interested in the case when J is of respective radii r1 and r2 and respective centers not the standard complex structure on CP n. For a and b. The result is this: if there is a symplectic CP 2 the h-principle reflects the fact in projective R2n embedding of U into a ball of radius R of geometry that through any pair of points there (still endowed with the standard symplectic struc- passes a unique projective line. Indeed, MG shows 2 2 2 ture), then one necessarily has r1 + r2 R . For that through any pair of points there passes a n 4, this condition is stronger than the measure pseudoholomorphic curve (for any J) and there is condition; in dimension 2, it is exactly the measure uniqueness. For CP n,n 3, MG shows existence, condition. We cannot resist mentioning the less but uniqueness no longer holds. According to Gro- symmetric but more striking result: with their mov this is the best example illustrating that the induced standard symplectic structures in R2n, a h-principle can be, despite its general softness, ball and a half ball (a ball cut in two parts through sometimes soft and sometimes rigid. More re- its center) having the same volume are never sym- cently, in 1996, Donaldson used the h-principle to plectomorphic as soon as n>1. study the similarity between pseudoholomorphicity We now sketch the proof of the two-ball pack- and genuine holomorphicity. ing problem, which uses a good deal of the tech- Returning to our open set U in R2n and the two niques introduced in the 1985 paper. We need some definitions. An almost complex structure on balls B1 and B2 packed symplectically into the ball R2n a manifold (necessarily of even dimension) is an B of radius R in , we now explain the proof of R4 automorphism J with square minus the identity the ball-packing result in ; things are the same (J2 = Id) on its tangent bundle. As soon as the in higher dimensions. We easily construct a sym- dimension is larger than two, such a structure is plectic embedding of B into the complement C 2\C 1 C 2 typically not integrable; i.e., there is no holomor- P P , where P has its canonical symplectic phic complex structure whose J will be multipli- structure ω but the structure is normalized in √ 2 4 cation by i = 1. Every symplectic structure ad- such a way that the total volume equals R /2, R4 mits some (in fact many) almost complex structures the volume of the ball B of radius R in . Let

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= : U → CP 2 be the symplectic embedding damental group and puts on them an arbitrary obtained by the composition of and , where Riemannian metric. As was true in the section is the given symplectic embedding. We now above on algebra, it is important to realize that, for 8 push the canonical almost complex structure J0 the current purpose, the metric that is used is 4 of R into a partial almost complex structure (J0) quite irrelevant. This is because two metrics on a 2 on CP . It is easy to extend (J0) into a global and compact manifold are always quasi-isometric. The compatible almost complex structure J on CP 2. The dilatation of f is defined to be the supremum of object yielding the result is the pseudoholomor- the ratios 2 1 0 phic curve C in CP that is homologous to CP and dY (f (x),f(x )) 0 passes through the two points (a) andR (b). The dX (x, x ) 2 total area of C equals R since it is C ω. This 0 0 integral is over all the pairs (x, x ) with x =6 x . The main re- Z Z sult of the article is this: for Y simply connected, > ω + ω. the number of homotopically distinct maps grows (B1) (B2) polynomially with respect to dilatation. 1 In fact, [2] also has a completely new approach Consider the two surfaces C1 = ((B1) ∩ C) 1 4 to . Briefly, Morse theory considers dif- and C2 = ((B2) ∩ C) in R . Then Z ferentiable functions f : M → R on a given compact manifold M and expresses the existence of criti- Area(C1)= (ω) C1 cal points (those where the derivative of f vanishes) and as a function of the topological complexity of M. Z The precise statement needs the notion of index Area(C2)= (ω). of a critical point, which is nothing but the num- C2 ber of negative eigenvalues in the second differ- ential of f. Morse proved that the number of crit- Moreover, both the surfaces C1 and C2 are pseudo- ical points of index equal to i is always the i-th holomorphic curves for the standard J0 structure of R4 , which comes from a complex structure on Betti number bi(M). But in many cases, typically 4 in the study of periodic , Morse’s result R . So C1 and C2 are in fact holomorphic, hence 4 leaves untouched the values f (x) at these critical minimal surfaces in R . By construction, C1 con- points. Are they very close to each other, very tains the center a of B1. Thus the monotonicity principle for minimal surfaces implies that the sparse, or what? Moreover, Morse theory imposes 2 a technical restriction: all the critical points have area of C1 is larger than or equal to r1 . The same 2 2 2 to be nondegenerate; i.e., the quadratic form given applies to C2, and hence r + r R . 1 2 by the second differential of the function has to Introducing Quantitative Methods in be everywhere nondegenerate. Algebraic Topology Some of the results of [2] have been treated in A fundamental but immense program is to study more detail in the book [13] Metric Structures for quantitative (as opposed to qualitative) algebraic Riemannian and Non-Riemannian Manifolds, which topology and . An example is is an enlarged translation of the French 1981 ver- constructing a quantitative theory of homotopy. A sion, Structures Métriques pour les Variétés Rie- typical example of the need, arising from a ques- manniennes, known to all Riemannian geometers tion of Thom, is: if catastrophe theory is not pre- as the “little green book”. The idea is, as always with dictive, it is precisely because this theory does not MG, naive at first: get metric control over alge- take into account a metric or a measure. Another braic topology operations. But complete proofs aspect of the need is the common definition of al- are hard. MG introduced new intermediate invari- gebraic topology as the study of properties in- ants, which are surprisingly subtle. We discuss a variant under “any” deformation. It has been MG’s typical example. constant concern to tackle such questions. We de- scribe some of MG’s major contributions to quan- tization of algebraic topology. In the pioneering work [6] “Volume and bounded cohomology”, MG introduced the simplicial vol- ume of a compact manifold M: consider all possi- In [2], “Homotopical effects of dilatation”, two ble ways to write the fundamental class [M] in the subjects are tackled. The first is the complexity of top degree of theP homology of M as a linear com- the space of all maps f : X → Y from one manifold bination [M]= i aii of singular simplexes i into another. To study this setting, one works with with real coefficients ai . By definition, the simpli- k k manifolds that are compact and have finite fun- Pcial volume M of M is the infimum of the sum | | 2 i ai . For a first example, take the sphere S , 8The canonical almost complex structure refers to the where a single simplex will suffice. Map one side one coming from regarding R4 as C2 . of the standard 2-simplex to a great from pole

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to pole, and map the other two sides to another and, in dimension 2,

such great circle. The angle between these great cir- 0 0 cles at a pole can be increased through any mul- kM#M k = kMk + kM k. tiple of 2 that we please, and in particular this Here # denotes the connected sum operation. In singular simplex can represent n[M] with n as particular, products of manifolds of negative cur- large as we please. The a =1/n, and we see that 1 vature have positive simplicial volume. Such man- the simplicial volume of S2 therefore vanishes. In ifolds have curvature that is only nonpositive. The fact, this happens any time a manifold admits other natural type of nonpositive-curvature man- maps into itself of degree larger than 1, since for 0 ifold is a space form of rank at least 2, i.e., a com- any map f : M → M of (topological) degree d, one 0 pact quotient of a noncompact Riemannian glob- has kMkdkM k. More generally, kMk =0 as ally symmetric space of rank at least 2, but it is still soon as the fundamental group (M) is “not too 1 unknown whether these manifolds have positive large”—more precisely, as soon as (M) is 1 simplicial volume. It remains almost a total mys- amenable.9 This is one of the results of [6], and it tery whether manifolds with nonvanishing sim- shows that the simplicial volume is an invariant of plicial volume are numerous or scarce. The tie the geometry at infinity of a manifold (actually of with groups is subtle: if (M) is hyperbolic, then its universal cover). 1 it does not necessarily follow that kMk > 0. Another good example is that of a compact In [6] Gromov introduced a second topological manifold of negative curvature. The geometric vi- invariant for compact differentiable manifolds. sion here is exactly the opposite of the spherical The minimal volume of M, written MinVol(M), is case. Namely, when one stretches a simplex, its vol- the lower bound of the volume over all Riemann- ume does not grow, since one easily proves that ian metrics one can put on M that have their sec- the volume is bounded above. Thus kMk > 0. A tional curvature in [1, 1]. One can think of it as classical principle of good mathematics is that measuring the extent to which a metric on M is after introducing a new object, one should use it bumpy. It could in principle be used to find the least to prove something deep. In [6] MG uses the sim- bumpy metric on a given M, but this would not be plicial volume to re-prove a part of the Mostow robust, since sectional curvature is not robust. rigidity theorem, one of the most spectacular re- The minimal volume is less subtle than the sim- sults in geometry. This part of the Mostow theo- plicial volume. It has recently become clear be- rem says that two discrete compact quotients of cause of results of Cheeger and Rong in 1996 that a hyperbolic space in dimension > 2 are isomet- most compact manifolds have zero minimal vol- ric if they have isomorphic fundamental groups. ume. Except for surfaces, where the Gauss-Bonnet Gromov’s result is in fact a variable-curvature ver- theorem gives the answer, and for space forms of sion of this part of Mostow’s theorem. rank 1, where the Besson-Courtois-Gallot result The simplicial volume is also interesting for applies, not a single explicit minimal volume is discrete groups, as was shown in [11], “Asymptotic known, even for the simplest geometric spaces: not invariants of infinite groups”. More generally, it ap- for the standard projective spaces over R or C, or plies to various asymptotic properties of mani- the quaternions, or the Cayley numbers, and not folds. It strongly inspired the 1995 results of even for the even-dimensional spheres starting Besson, Courtois, and Gallot as follows: Define a S4 space form of rank one to be any compact quotient with . of a noncompact Riemannian globally symmetric There is a basic link between these two volume space of rank 1 (i.e., a “hyperbolic space” over R, invariants, the main inequality of [6]: there is a con- C, the quaternions, or the Cayley numbers) by the stant c(d) such that, for any Riemannian manifold d 11 action of a torsion-free discrete subgroup.10 Their (M ,g) whose Ricci curvature satisfies Ricci results are that the standard Riemannian structure > (d 1), the simplicial volume satisfies minimizes all the usual invariants: entropy, growth Vol(g) >c(d)kMk. of lengths of periodic geodesics, asymptotic vol- ume of balls, etc. A sufficient condition for Ricci > (d 1) is Nevertheless, positivity of simplicial volume is that the sectional curvature satisfy K> 1. Con- 12 k k incredibly difficult to establish. Simplicial volume sequently MinVol(M) >c(d) M . Thus simpli- has the following functorial properties: there are cial volume fits quite well with Ricci curvature; it universal constants c and c0 depending only on the is an important open problem to see whether it also dimensions such that fits well with scalar curvature. The proof of the main inequality above is quite difficult, and to my 0 0 0 0 ckMkkM kkM M kc kMkkM k knowledge it has never been written up in detail. Moreover, it uses a new technique called “diffusion 9See footnote 5. 11This notion is defined in Part II. 10We take space forms to have sectional curvature nor- 12 malized to be in the interval [4, 1] . This corollary is much weaker than the initial statement.

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manifolds, these relations being valid for A A A any metric. For example, for surfaces of A genus , one would expect an inequality of the form A/L2 c() with a constant c() depending on the genus, and one would ex- pect the constant to grow with the genus. Before the Filling paper nobody could ob- L L tain any result of this form, apart from in- dependent work of Accola and Blatter in 1960 that for surfaces yields a constant Figure 4. It is very intuitive that, for surfaces with one or more handles going to zero tremendously fast with the (say the genus is positive) as in the picture, the ratio between the area genus. and the squared length of the shortest noncontractible curves is larger The Filling paper solves these two prob- and larger when the number of holes grows. But in fact this turns out to lems. The one for surfaces of genus yields be extremely difficult to prove. a constant growing like /log2 , and un- published examples of MG show that one of cycles”; this technique is used in the “Filling cannot do better. The second problem, the vol- paper”, which we consider next. ume as a function of the systole of curves for higher-dimensional manifolds, is solved for so- called “essential manifolds”. A universal inequal- The preceding results were mainly concerned ity Vol/(Systole)d >c>0 is hopeless for general with “topology and metric”. The next result con- manifolds; just take the product of a circle with any cerns “topology, metric, and measure”. It first ap- other compact manifold. What is needed is a con- peared in the huge work [5] “Filling Riemannian dition expressing the geometric fact that the non- manifolds”, which is called the “Filling paper”. contractible curves generate the fundamental class This text is a seminal work, and many of its results (which yields the volume). Essential manifolds con- have been used, but to my knowledge nobody has stitute a large class, containing all tori, all real gone completely through it. In particular, the con- projective spaces, and all compact manifolds whose tent of the last four of the nine sections has never universal covering is diffeomorphic to Rn (e.g., been elaborated in complete detail. We will now ex- space forms and other manifolds with nonpositive plain in detail the first topic of the Filling paper, curvature). MG’s result for surfaces uses the tech- since it is extremely simple to visualize and for- nique of “diffusion of cycles” mentioned above. The mulate. It starts with a result of Loewner con- situation for surfaces is exceptional, since the re- cerning the torus: We consider a two-dimensional sult for surfaces is optimal yet is obtained by “el- torus with an arbitrary Riemannian metric.13 We ementary” geometry. At the time, the uniformiza- define the systole of this torus to be the infimum tion theorem and other techniques of theory of one of the lengths of the noncontractible closed curves complex variable were yielding very poor results. on this torus. This minimum is classically achieved For higher dimensions one needs new ideas and by a periodic geodesic, but this is not the point. invariants. The first thing that MG does in the Fill- Let us call this systole L. Then Loewner proved that ing paper is to embed the Riemannian manifold the total area A of our torus always satisfies the under consideration in the infinite-dimensional √ 0 inequality A/L2 3/2. Moreover, equality is at- Banach space C (M) of all continuous functions on d tained if and only if the torus is flat and regular M by the map f given by distance functions to hexagonal (i.e., corresponds to the lowest point of points, f (x)=d(x, ). The basic property of this map is that it is an isometric embedding of Md into the classical modular domain). Loewner never pub- 0 lished this result, since he did not consider it deep C (M) with the supremum norm. The isometry enough. condition is stronger than the usual one for iso- The mathematical mind will immediately ask for metric embeddings into Euclidean spaces: the in- generalizations—first to surfaces of higher genus, duced metric is not the Riemannian one based on then to higher-dimensional manifolds, and then to length of curves, but the induced one in the ele- mentary sense: higher-dimensional submanifolds. As Thom pointed out to me in Strasbourg in 1960, Loewner’s d(x, y)=d(f (x),f(y)) = sup |d(x, z) d(y,z)|. result is the simplest case of the general problem z∈M of finding universal relations between norms of var- Such a strong isometric embedding is impossible ious homology classes when putting metrics on in finite dimensions, but it is precisely the one 13 R3 needed for the proof. This can be taken as the one induced by when we 0 have a “real” torus. On the other hand, if the torus is flat In C (M), an infinite-dimensional setting, MG (that is to say, locally Euclidean), the torus has to be proves an isoperimetric inequality for the volume d+1 d treated as an abstract manifold or else embedded in some of the manifolds N that fill the image f (M ); i.e., higher-dimensional Euclidean space. the boundary of Nd+1 is precisely f (Md). The

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infimum of these vol- umes is called the filling d d+1 f (M ) d+1 volume of Md. Another N N new invariant is needed: the filling radius of Md, the smallest number such that if an Nd+1 has d U (f (M )) Md as boundary, then it cannot be contracted to f (Md) f (Md) inside the -neigh- borhood of f (Md). Now MG proves (this is the rel- atively easy part) that for essential manifolds the Boundary of Nd+1 does Boundary of Nd+1 systole is 6. The proof not contract to contracts to f(Md) of the isoperimetric in- f(Md) in U(f (Md)) in U(f (Md)) equality in infinite di- mensions is hard, and MG considers it as belonging to “geometric surgery”. Figure 5. It is intuitive that the tubular neighborhood of radius R of a subvariety M, for Some comments are small R, can be retracted to M, but that there is a limit value (called the filling radius) for now in order. We have al- which that property stops. Then M will bound some submanifold of one dimension ready mentioned the fact higher. that MG’s invariants are often impossible to compute for standard mani- The above concerned situations in which the sys- folds. The filling volume and the filling radius il- tole was “one-dimensional”. We return to Thom’s lustrate this point. The filling radius has been com- program, stated above, for higher-dimensional sys- puted only for spheres, real projective spaces, and toles, i.e., with homology or homotopy classes in the complex projective plane (by M. Katz in 1983 dimension > 1. Natural candidates are complex and 1991). The filling volume is more baffling, be- projective spaces, products of manifolds, etc. Un- cause its exact value is still unknown for any man- happily, in a 1992 paper, “Systolic and intersystolic ifold, even for the simplest one, namely, the circle inequalitites”, MG laid the foundations of a pro- S1 (the expected value is 2, the area of a hemi- gram that kills every possible hope for these man- sphere). ifolds. Starting with dimension 2, no universal sys- Let us continue with the Filling paper, which is tolic inequality can hold. We have here systolic 150 pages long and is written in a very succinct softness, whereas there is systolic rigidity in the one- style. The Filling paper contains numerous other dimensional case. MG’s program has been pur- results and techniques as well as an in-depth vi- sued by various authors, including Babenko, Katz, sion of metric geometry, and not only of Rie- and Suciu, in recent years and has been almost com- mannian manifolds but also of “Finsler mani- pletely finished. The guess is that no manifold will folds”.14 Finsler manifolds are those for which the have any systolic rigidity. For MG’s version of the metric structure is given by a collection of convex history of this subject, see his “systolic reminis- symmetric bodies in the various tangent spaces of cences” at the end of Chapter 4 of [13], Metric a manifold under consideration. The Riemannian Structures for Riemannian and Non-Riemannian case is when they all are ellipsoids. Other new in- Manifolds. variants introduced in the Filling paper are the vi- sual closure and the visual volume, which are await- ing other applications. We turn to Gromov’s contribution to surgery, a Still, the Filling paper is diffusing slowly into var- basic tool in algebraic topology. His work lies in ious fields. For example, MG proves a result that “geometrically controlled surgery”, or “quantitative apparently seems trivial: Consider a Riemannian surgery”. The basic surgery operation works with manifold with the property that every ball is con- two manifolds, M and N, of equal dimension. The tractible inside the ball with same center and twice trivial kind of surgery is the connected sum: one the radius; then the volume of this manifold is in- joins M and N by a small tube after having taken finite. This type of universal contractibility is now a small ball out of each. More sophisticated surgery basic in Riemannian geometry and is related to the is possible in higher dimensions. From Md one topic called controlled topology.15 15See, for example, the survey of Berger “Riemannian geometry during the second half of the twentieth century”, 14For these spaces, see, for example, D. Bao, S.-S. Chern, in the Jahresbericht der DMW in 1998 or reprinted as AMS and Z. Shen, An Introduction to Finsler Geometry, Lec- University Lecture Notes, volume 17, in 2000 (ISBN 0- ture Notes in Math., Springer-Verlag, 2000. 821-82052-4).

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removes a topological sphere Sk and looks at a [11] ———, Asymptotic invariants of infinite groups, Geo- small tubular neighborhood of it inside Md. The metric Group Theory (G. Noble and M. Roller, eds.), London Math. Soc. Lecture Notes Series, vol. 182, boundary of this tube is Sk Sd k 1. One does the Cambridge Univ. Press, Cambridge, 1993, pp. 75–263. same to Nd, but this time with a sphere Sdk1 . [12] ———, Positive curvature, macroscopic dimension, Here we have again a tube, and the boundary this spectral gaps and higher signatures, Functional dk1 k time is S S . It remains to glue the two Analysis on the Eve of the 21st Century (New boundaries by exchanging the order of the fac- Brunswick, NJ, 1993), (S. Gindikin, J. Lepowsky, and tors. The simplest example is two copies of S1 S2, R. Wilson, eds.), vol. II, Birkhäuser, Boston, 1996, pp. with surgery around a circle in each of them. If one 1–213. does not interchange the role of the parallels and [13] ———, Metric Structures for Riemannian and Non-Rie- the meridians on the torus S1 S1, the final result mannian Spaces, Birkhäuser, Boston, 1999. is still S1 S2. But interchanging them yields the sphere S3 (as in the Hopf fibration of S3 ). MG used controlled surgery (with metrics) in a number of cases. The first is in a joint 1980 paper with Lawson, “The classification of simply con- nected manifolds of positive scalar curvature”, which is the basic work about the classification of Riemannian manifolds of positive scalar curva- ture. The authors show that, if one starts with two Riemannian manifolds M and N, both with posi- tive scalar curvature, and if one performs a surgery of codimension at least 3, then the manifold so ob- tained can be endowed with a Riemannian metric that is also of positive scalar curvature. In the Filling paper [5] controlled surgery is used at least twice: first for the optimal systolic in- equality for surfaces of high genus, second to prove the isoperimetric inequality in infinite di- mensions. Controlled surgery is also a basic tool for proving systolic softness for some manifolds. One constructs local boxes that are systolically soft and controls the surgery done when intro- ducing these boxes into the manifolds.

In Part II we shall discuss MG’s contributions to Riemannian geometry and related subjects; some of these have been hinted at in Part I.

References [1] W. BALLMANN, M. GROMOV, and V. SCHROEDER, Manifolds of Nonpositive Curvature, Birkhäuser, Boston, 1985. [2] M. GROMOV, Homotopical effects of dilatation, J. Dif- ferential Geom. 13 (1978), 303–310. [3] ———, Groups of polynomial growth and expanding maps, Publ. Math. IHÉS 53 (1981), 53–73. [4] ———, Curvature, diameter and Betti numbers, Com- ment. Math. Helv. 56 (1981), 179–195. [5] ———, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), 1–147. [6] ———, Volume and bounded cohomology, Publ. Math. IHÉS 56 (1983), 5–99. [7] ———, Pseudoholomorphic curves in symplectic man- ifolds, Invent. Math. 82 (1985), 307–347. [8] ———, Partial Differential Relations, Springer-Verlag, Berlin, 1986. [9] ———, Hyperbolic groups, Essays in Group Theory (S. Gersten, ed.), MSRI Publications, vol. 8, Springer, New York, 1987, pp. 75–263. [10] ———, Foliated Plateau problem, I and II, Geom. Funct. Anal. 1 (1991), 14–79 and 253–320.

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