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International audience; We show that the Teichmüller space of the triangle groups of type (p,q,∞) in the automorphism group of the two-dimensional complex hyperbolic space contains open sets of 0, 1 and two real dimensions. In particular, we identify the Teichmüller space near embeddings of the modular group preserving a complex geodesic.

We extend the notion of the symmetric signature σ( M ,r)∈L n (R) for a compact n-dimensional manifold M without boundary, a reference map r : M→BG and a homomorphism of rings with involutions β : Z G→R to the case with boundary ∂M, where ( M , ∂M )→(M, ∂M) is the G-covering associated to r. We need the assumption that C ∗ ( ∂M ) ⊗ Z G R is R-chain homotopy equivalent to a R-chain complex D ∗ with trivial mth differential for n=2m resp. n=2m+1. We prove a glueing formula, homotopy invariance and additivity for this new notion. Let Z be a closed oriented manifold with reference map Z→BG. Let F⊂Z be a cutting codimension one submanifold F⊂Z and let F →F be the associated G-covering. Denote by α m ( F ) the mth Novikov–Shubin invariant and by b m (2) ( F ) the mth L2-Betti number. If for the discrete group G the Baum–Connes assembly map is rationally injective, then we use σ( M ,r) to prove the additivity (or cut and paste property) of the higher signatures of Z, if we have α m ( F )=∞ + in the case n=2m and, in the case n=2m+1, if we have α m ( F )=∞ + and b m (2) ( F )=0 . This additivity result had been proved (by a different method) in (On the Homotopy Invariance of Higher Signatures for Mainfolds with Boundary, preprint, 1999, Corollary 0.4) when G is Gromov hyperbolic or virtually nilpotent. We give new examples, where these conditions are not satisfied and additivity fails. We explain at the end of the introduction why our paper is greatly motivated by and partially extends some of the work of Leichtnam et al. (On the Homotopy Invariance of Higher Signatures for Mainfolds with Boundary, preprint, 1999), Lott (Math. Ann., 1999) and Weinberger (Contemporary Mathematics, 1999, p. 231).

We study the connection between the condition that the reduced C*-algebra of a finitely presented group is exact and the Novikov conjecture holding. The main result states that if the group is strongly exact in the sense that the inclusion of the group C*-algebra into the uniform Roe algebra of the group is a nuclear embedding then the Novikov conjecture holds for that group., This is a Latex2e file with 12 pages

Let H =Homeo + S 1 be the discrete group of orientation preserving homeomorphisms of the circle S1, and let G be a subgroup. In this work the Euler Class [e(G)] for discrete G-bundles is studied in order to determine the range of powers for which [e(G)] vanishes. A new invariant is introduced, the orbit class of G, as well as an integer associated to it, its holonomy. The first vanishing power of the Euler Class results from the non-vanishing of the holonomy of the orbit class. The highest non-vanishing power of the Euler Class is a consequence of the vanishing of the holonomy. Applications focus on the Based Mapping Class Groups, M g . These can be represented as subgroups of H which exhibit a certain degree of transitivity of their actions depending on their genus g. This leads to a vanishing/non-vanishing result for the powers of the Euler Class of the M g 's. The vanishing theorem and its application to Mapping Class Groups: the k-th power of the Euler Class [e k ( M g )] is zero for k⩾g, is described in this article. The non-vanishing theorem will appear in a sequel.

Crooked planes are polyhedra used to construct fundamental polyhedra for discrete groups of Lorentz isometries acting properly on Minkowski (2+1)-space. This paper explores intersections of crooked planes. Criteria for the disjointness of crooked planes are developed. These criteria are applied to derive sufficient conditions for affine deformations of a discrete subgroup of SO (2, 1) to act properly on Minkowski space.

A discrete group Γ is said to be of type F (n) if and only if there is a classifying complex BΓ with finite n-skeleton. We show that Γ is of type F (n) if Γ admits a regular cellular action on a CW complex Y such that Y is homotopy equivalent to a complex with finite (n−1)-skeleton, Y Γ is a complex with finite n-skeleton, and for each p-cell σ of Y(0⩽p⩽n) the stabilizer Γσ is of type F (n−p) .

Let G be either a compact Lie group or a discrete group and X a space with a continuous G action. Assume the action has finite stabilizers if G is discrete. For a fixed prime p let Xs be the p singular locus of X, i.e. the set of all points in X which are fixed by some element of order p. In this paper we study the Borel construction EG×G Xs (EG denoting as usual the total space of the universal principal G bundle) by relating it to the Borel constructions EG×CG(E)X where E runs through the non-trivial elementary abelian p subgroups of G, CG(E) denotes the centralizer of E in G and X E is the E fixed point set of X.

THE SET of (based) homotopy classes of homotopy equivalences from a (pointed) space X to itself forms a group G(X), which is analogous to the group of automorphisms of a discrete group. It is of interest for many reasons and has attracted the attention of many authors. Recently, Sullivan[9] and Wilkerson[l l] have shown that if X is a simply-connected finite CW-complex, then G(X) is finitely-presented. Other authors had considered the structure of this group for various broad classes of spaces ([0], [3], [4], [lo]). In specific cases (see, for example [7], [8]), explicit computations show that G(X) is related to the group of units in a known algebra. In all these examples G(X) is finitely-presented. In this note, we give an example of a finite CW-complex X with infinitely-generated group of self-equivalences. We show that if X = S’ v S’ v S’ (more generally, S’ v S” v SzP-‘, p > I), then G(X) is infinitely-generated. Along the way, we need to analyze the group of self-equivalences of S’ v SZ (more generally, S’ v Sp, p > l), which is in fact finitely-presented. Thus our example is perhaps as simple as possible. Instead of considering spaces with few cells, another approach to finding a reasonable space X, with G(X) infinitely-generated, would be to consider spaces with a finite number of non-zero homotopy groups, each homotopy group finitely-generated and P,(X) finitely-presented. In fact, there is an example of a finitely-presented group 7~ for which Aut(n), the group of automorphisms of ?r, is not finitely-generated (see [S]). Hence if X = K(r, I), then G(X) is not finitely-generated. However, K(r, I) is a rather large complex, and there is no reason to believe that there is a small subcomplex Y of K(r, 1) with G(Y) infinitely-generated. Thus these two approaches are different. We work in the category of connected complexes with base-point, and base-point preserving maps and homotopies. When not necessary, the base-point shall be omitted from the notation.

ONE OF the interesting spaces in algebraic geometry is the quotient space of the Siegel upper half space E, under the action of the discrete group r, = Sp,(Z). In the language of algebraic geometry, this quotient space G,/F, is the coarse moduli space of principal polarized abelian varieties (for definition see [16]). More than thirty years ago, Satake discovered a natural compactification 6,*/r, of this space obtained by attaching lower dimensional Siegel spaces, so that as a set

INTRODUCYrION LET M be an oriented closed n-dimensional P-manifold and (M, 9) a transversely oriented codimension one P-foliation of M. The purpose of this paper is to study foliated cobordism class of (M, S) assuming that 9 is almost without holonomy. In virtue of the works of Haefliger [3,4], Mather [ 12,131 and Thurston [24,25], foliated cobordism groups can be studied in the following lines. Namely there is a universal space Brim, called the Haefliger’s classifying space for r,m-structures, so that any (M, $r) determines an ndimensional’ homology class of Brim, which turns out to be closely related to the cobordism class of (M, 9). On the other hand there is a map BDiffKmlW -+ QBr,m from the classifying space of the discrete group DiffKm[W of all Cm-diffeomorphisms of Iw with compact support to the loop space RBr,m of sr,m, which induces an isomorphism on integral homology. Thus in some sense the study of homology classes of Brim can be reduced to the study of those of DiffKaIw. In our case, these two fundamental results work very well. Our main result is as follows. Let.(M, 9) be as before and assume that 9 is almost without holonomy. Then it is homologous to a disjoint union of finite number of foliated S-bundles over (n I)-dimensional tori. For n = 3, in particular, it follows that (M, 9) is foliated cobordant to a disjoint union of foliated S-bundles over T2. The foliated cobordism classes of foliated S’-bundles over T2 were studied by Tsuboi in [26]. Fukui and Oshikiri proved the nullity of the foliated cobordism classes of certain foliated 3-manifolds ([2,21]). By our method, we can give a wider class of foliated 3-manifolds which are foliated null-cobordant (Corollary to Theorem 2). Also together with results of Wallet [29] and Herman [7], we re-obtain the vanishing of the Godbillon-Vey class of an almost without holonomy foliation (M, 5) which we previously proved in [16]. The main tool of this paper is the notion of foliated J-bundles which we developed in [16] in order to calculate the Godbillon-Vey class. Associated to each (M, S), there is a foliated J-bundle and the original 9 can be “embedded” in it as the graph of 9. Then we can deform the underlying r,” -structure of 9 by simply moving this graph on the total space of the J-bundle. This method is originally due to Haefliger. The foliated J-bundles associated to (M, 9’) are determined by the holonomies of the compact leaves and the Novikov transformations (which depend on the non-compact leaves). These two data are essential. In fact, the structure of the foliated S’-bundles over T”-’ to which (M, 9:) is homologous, are determined by these data. Some of the results in this paper were contained in $5 of our preprint [15] some part of which has been published in [ 161. We would like to refer the reader to [16] for the generalities of the foliation which are almost without holonomy and the construction of the associated foliated J-bundles and other related notions. In this paper, all manifolds, foliations and diffeomorphisms are assumed to be smooth (C=). Moreover. foliations will mean transversely oriented codimension one foliations.

In [7], Raghunathan proved a vanishing theorem for H’(r, V,) when I is a uniform (cocompact) discrete subgroup of a semi-simple Lie group G and V,, is an irreducible real representation of G. In the case G is simple, Raghunathan’s theorem implies H’ (I, I’,,) = 0 unless G is locally isomorphic to SO@, 1) and V, = sj V where V is the standard representation of SO (n, 1) and .@‘j Vdenotes the harmonic (annihilated by the wave operator A 8 /at2) polynomials on Vof degreej or G is locally isomorphic to S U (n, 1) and V, = Sj I’, the symmetric power of the standard representation V or its dual. Here j is a non-negative integer, possibly zero. The purpose of this paper is to prove a complement to Raghunathan’s theorem in the case G = SO (n, 1). Before stating our theorem we establish some notation and terminology. By a two sided hypersurface we mean a hypersurface with trivial normal bundle. Any orientable hypersurface of an orientable manifold is two-sided. D will denote the symmetric space attached to G, however, the symbol H” will be used to denote hyperbolic n-space. If ,VO is a representation of G and I is a uniform, discrete, torsion free subgroup of G then I’, will denote the corresponding flat bundle (local system) over the locally-symmetric space M = r\D. Since M is a space of type K(r, 1) we have

Topology Vol. IO pp. 177-183. Pergamon Press, 1971. Printed in Cc-at Britain. ANOSOV MAPS, POLYCYCLIC GROUPS AND HOMOLOGY MORRIS W. HIRSCH~ (Receioed 15 Septenlber 1970) INTRODUCTION LET M DENOTE a compact Riemannian manifold. A C’ map f: A4 + M is called Anosoo if there exists a continuous splitting TM = E”@ E” of the tangent bundle of M, and constants C > 0, 1. > 1, such that Tf(E”) = E”, TJ(E’) c ES, and furthermore for all positive integers m and tangent vectors X E TM: This condition is independent ITf”(X) I 2 CJ. IXl if XEEU. of the Riemannian metric. Anosov maps have been studied in [I, 2, 7, 1 I], and elsewhere. Examples of Anosov maps can be obtained from a Lie group G, a discrete subgroup I with G/T compact, and an endomorphism 4: G ---t G such that #(I) c I; see [8]. If the derivative of q5 at the identity of G has no eigenvalues of absolute value one, then the map G/I -+ G/lY induced by 4 is an Anosov map. All known Anosov maps are intimately related to maps of this type. As an example take G = R” and r = Z”, the integer lattice; then $I is defined by an n x n integer matrix A and G/T is the n-torus T”. In this case we can identify A with the linear transformation f*: induced by f on the first homology H,(T”; R) -+ H,(T”; R) group with real coefficients. John Franks [2] has proved that iff: T” -+ T” is any Anosov has no root of unity among its eigenvalues. diffeomorphism, then I;, Theorem 1 extends Franks’ result to Anosov maps on a wider class of manifolds which includes all nilmanifolds. As an application. many manifolds that do not admit Anosov diffeomorphisms are constructed. For example: the Cartesian product of the Klein bottle and a torus. 7 Supported in part by NSF Grant GP 22723.

IN THIS paper, we study the space of volume forms on a given smooth, compact, oriented manifold. In particular, it is shown that if M is of dimension 22, then the cohomology of smooth differential forms on Vol(M) invariant under the action of Diff(M) is freely generated by one form in dimension n + 1. This form is then related by a van Est-type map, as in the paper of Dupont[3], to the Godbillon-Vey invariant for flat M-bundles. The geometric nature of this identification is exploited to give a new view of an example, due to Thurston, of a family of foliations in which the Godbillon-Vey class varies continuously. This work comprises part of the author’s Ph.D. thesis written under under the direction of Raoul Bott. Special thanks are also due to Dusa McDuff, who made several helpful suggestions. When one replaces Vol(M) by R(M), the space of Riemannian metrics on M, there is again a van Est-type map from invariant forms on R(M) to the group cohomology of the discrete group Diff,(M). For results in this situation, see[6].