Abstract: Let Fg 1, g> 1, be the mapping class group consisting of all isotopy classes of base-point and orientation preserving homeomorphisms of a closed, oriented surface F of genus g. Let )~(~1) be its Euler characteristic in the sense of Wall, that is Z(F~I)= [Fgl: F] l z(E/F), where F is any torsion free subgroup of finite index in F~ 1 and E is a contractible space on which F acts freely and properly discontinuously. An example of such a space is the Teichmiiller space ~-~1, and g(F~ ~) can be interpreted as the orbifold Euler characteristic of ~-~'/F~I =JOin, the moduli space of curves of genus g with base point. The purpose of this paper is to prove the following formula for )~(F~I):

Abstract: This paper represents the beginning of an a t tempt to transfer, to the study of outer au tomorphisms of free groups, the powerful geometric techniques that were invented by Thurs ton to study mapping classes of surfaces. Let F, denote the free group of rank n. We will study the g roup Out(F,) of outer au tomorphisms of F, by studying its act ion on a space X, which is analogous to the Teichmtiller space of hyperbol ic metrics on a surface; the points of X, are metric structures on graphs with fundamental group F,. We begin by making this not ion precise. By a graph we shall mean a connected 1-dimensional CW-complex. The 0cells will be called nodes and the l-cells edges. The valence of a node x is the number of oriented edges which terminate at x, i.e. the min imum number of components of an arbitrarily small deleted ne ighborhood of x. An N-graph is a graph endowed with a metric such that each edge is locally isometric to an interval in l l and such that the distance between two points is the length of the shortest edge-path joining them. An N-graph is said to be minimal if it is not homotopy equivalent to any proper subgraph. Th roughou t this paper we will consider only ~,-graphs which are minimal and have no nodes of valence 2. (A minimal N-g raph cannot have nodes of valence 1). Fix a (topological) graph R o with one node and n edges, and choose an identification F--~rl(Ro). If G is an N-graph, then a homotopy equivalence g : R o ~ G is called a marking on G. We define two markings g l :Ro,G1 and g 2 : R o ~ G 2 to be equivalent if there exists an isometry i: GI~G 2 making the following diagram commute up to (free) homotopy :

Abstract: Si π est le groupe fondamental d'une surface orientee fermee S et G est un groupe de Lie satisfaisant des conditions tres generales, alors l'espace Hom (π,G)/G des classes de conjugaison de representation π→G a une structure symplectique naturelle. On etudie la geometrie de cette structure symplectique a l'aide d'une famille naturelle de fonctions sur Hom(π,G)/G

Abstract: Let F = F ~ r be the mapping class group of a surface F of genus g with s punctures and r boundary components. The purpose of this paper is to establish cohomology properties of F parallel to those of the arithmetic groups. If G is a linear algebraic group defined over Q, X is the symmetric space associated to G and A is an arithmetic subgroup of G, then A is virtually torsion free and acts properly discontinuously on X. The rational homology of A is the same as that of X/A. Furthermore there is a "bordification" of X ([BS]) to a manifold with corners 3~ and an extension of the action of A to a properly discontinuous action on Jf so that the quotient X / A is compact. The boundary of Jf is homotopy equivalent to a wedge of spheres, say of dimension d, and the virtual cohomological dimension of A is n d + 1, where n is the dimension of X. In the case of the mapping class group there is no analog for G, Ivanov (unpublished) has proven that F is not arithmetic. Nevertheless, F acts properly discontinuously on Teichmiiller space r which is homeomorphic to Euclidean space (of dimension 6 g 6 + 2 s ) ; 3"will play the role of the symmetric space. The quotient of .9by F is the moduli space of curves whose rational homology is then identified with that of E Harvey [Har] has constructed a Borel-Serre bordification J of 3-by analytic methods. In the case where F has punctures, we will build g by a different, combinatorial method. In addition, we will explicitly describe inside Y a cell complex Y of dimension 4 g 4 + s onto which J may be F-equivariantly retracted, thus establishing an analog of the constructions for SL n by Serre, Soul6 ([Sol) and Ash ([A]). This complex will be of the lowest possible dimension because we will use J= to prove our main result:

Abstract: In this paper we introduce techniques for handling the degeneration of linear series on smooth curves as the curves degenerate to a certain type of reducible curves, curves of compact type. The technically much simpler special case of 1-dimensional series was developed by Beauville [2], Knudsen [21–23], Harris and Mumford [17], in the guise of “admissible covers”. It has proved very useful for studying the Moduli space of curves (the above papers and Harris [16]) and the simplest sorts of Weierstrass points (Diaz [4]). With our extended tools we are able to prove, for example, that: In this paper we present the basic theory of “limit linear series” necessary for proving these results. The results themselves will be taken up in our forthcoming papers [8-12]. Simpler applications, not requiring the tools developed in this paper but perhaps clarified by them, have already been given in our papers [5-7].

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Abstract: which shrink towards the center of the initial sphere in finite time. It was shown in [3], that this behaviour is very typical: If the initial hypersurface M o o R , + 1 is uniformly convex, then the surfaces M t contract smoothly to a single point in finite time and the shape of the surfaces becomes spherical at the end of the contraction. If the ambient space N is a general Riemannian manifold, the curvature of N will interfere with the mot ion of the surfaces M r We want to show here that the contract ion first to a small sphere and then to a single point is still

Abstract: On presente une etude systematique des varietes caracteristiques des systemes holonomes a singularites regulieres. Ce sont des sous-varietes lagrangiennes coniques dans le fibre cotangent T * X

Abstract: Pour une variete hyperbolique orientee fermee X et ρ une representation orthogonale de Π 1 X, on definit une fonction zeta de type Ruelle Rρ(s) pour Re(s) grand comme le produit sur les geodesiques closes premieres γ des facteurs det [I−ρ(γ) exp(−sl(γ))], ou l(γ) est la longueur de γ. On continue analytiquement Rρ(s) et on calcule son terme principal en s=0

Abstract: On demontre un theoreme d'existence general pour des metriques d'Einstein homogenes et on presente des espaces homogenes compacts simplement connexes qui ne portent pas une metrique d'Einstein homogene

Abstract: Let 0: M-,S" be a minimal immersion of a compact surface into a unit sphere. Then, the linear functions of 0 are eigenfunctions for the Laplacian of M corresponding to the eigenvalue 2=2 . The main purpose of this paper is to study those minimal surfaces for which 2 is exactly the first non-zero eigenvalue of its Laplacian. This kind of immersions have a peculiar behaviour among all compact minimal surfaces of the sphere and they appear naturally when one considers different geometric problems, as Li and Yau have shown in [6]. The methods that we use in this paper are based, for the most part, on [6]. It is known that the only metric on a 2-dimensional sphere admitting a minimal immersion into S" by the first eigenfunctions is the standard one (this follows, for example, from the fact that the multiplicity of the first eigenvalue for such a metric is at most three, see the Cheng work [3]). Our first result shows that it is possible to extend this property for an arbitrary compact surface, in the following way:

Abstract: Introduction. Let ffi be an adjoint, simple algebraic group of type E,(n = 6, 7, 8), F 4 or G2 over an algebraically closed field K of characteristic p > 0. We assume that (5 has a fixed Fq-rational structure for a finite subfield Fq of K. The purpose of this paper is to determine explicitly the values of the irreducible characters of G = ffi(Fq) at the unipotent elements when p is good [38; I, 4.3]. In this Part I, we shall do this only for large p, and the full result will be proved in the forthcoming Part II. (After submitting the first version of the present paper, the author was informed by Lusztig that he solved analogous problems for classical groups using his theory of character sheaves, and that he can also recover the result of this paper mentioned above.) Our method depends upon: (1) a parametrization of the irreducible characters of G and the determination of their multiplicities in Deligne-Lusztig virtual representations (Lusztig [25]); (2) the computation of the Green functions of G in large characteristic (Springer [36; 7.1@ Shoji [33], Beynon and Spaltenstein [3, 4]); (3) the determination of the values of generalized Gelfand-Graev characters [20] of G in good characteristic (see Section 3). By (1) and (2), the "uniform parts" of the irreducible characters are already known explicitly in large characteristic. This implies that, in large characteristic, our problem for groups of type E 6 o r E 7 has already been solved. (It should be mentioned here that the whole character table of the G2-group is also known by Chang and Ree [8]. Hence, in the G2-case, this paper gives just another approach to (a part of) the result of Chang and Ree.) If G is of type G2, F4 or E 8, the restriction of an irreducible character r of G to the set of unipotent elements can be written as a sum of its uniform part with a function of the form

Abstract: We study the relations between the de Rham cohomology class of a closed differential 1-form ω with Morse singularities on a manifoldM of dimensionn≧3, and the ergodic properties of the foliationFω it defines. We show by examples that, if the fundamental group is “large” enough, very different behaviours can occur in the same class. In contrast, if the fundamental group admits no surjective homomorphism onto the free group on 3 generators, thenFω is always uniquely ergodic provided it has no compact leaf; if the natural homomorphism π1(M)→H1(M,Z)/torsion does not factor through a free group, then the same result is true in almost every cohomology class. We also give results about the existence of noncompact leaves and the number of ergodic measures.

Abstract: It has been ten years since V.G. Drinfeld introduced in [3] the concept of elliptic A-module which we now call Drinfeld A-module. These modules are analogues of elliptic curves for global function field k and we have witnessed a beautiful algebraic theory gradually taking shape. In [13], we have asked a few natural t ranscendence questions (conjectures) concerning these modules, having in mind the analogue of the classical work of Siegel-Schneider about elliptic integrals, and based on evidences provided by the old papers of Carlitz and Wade (cf. [1, 11]). Here in this article we will prove those conjectures. We will show that indeed there is an interesting transcendence theory for each algebraic curve over a finite field. Fol lowing Drinfeld, we let M be a lattice in / ~ , i.e. a finitely generated discrete A-module in k~ and form

Abstract: f ( F r A ) = F r A and ~ f " ( U c ~ c l A ) = F r A (i.e. FrA repels to the side of A). n=O (Assume that U is always sufficiently small so that f has no singularities in U c~ A.) Let R: D Z ~ A be a Riemann map (a conformal homeomorphism) of the unit disc onto A. Then there exists a holomorphic extension g of R l o f o R to a neighbourhood of S 1 and gls, happens to be an expanding map, i.e. there exists n > 0 such that for every z ~ S l , l (g") ' (z)[>l. (Although the proofs are straightforward, we include them for sceptics in the last section, Sect. 7, together with a remark about examples of A and f satisfying the above assumptions.) Denote by _gthe non-tangential limit of R and by ~ K = S 1 the domain where it exists. Denote by ~IJl (g) the space of all Borel, probability g-invariant, ergodic, positive entropy measures on S 1. It is known (see [P]) that for every/~ EgJI (g), K exists ~almost everywhere, so the f-invariant measure R. (/~) on FrA can be considered. The functions log[g'[, log lf ' [ are /z-, respectively R,(/0-integrable (see 1.5). Denote these integrals by )~u (g), g~.(~)(f)(In this paper we consider derivatives with respect to the Riemannian metric on $2.) Define the Hausdorff dimension of any probability measure v: