The canonical trace on the reduced C^*-algebra of a discrete group gives rise to a homomorphism from theK-theory of this C^*-algebra to the real numbers. Thi s paper studies the range of this homomorphism. For torsion-free groups, the Baum-Connes conjecture, together with Atiyah's L²-index theorem, implies that the range consists of the integers.

We give a direct and elementary proof that if G acts on a tree and admits a homomorphism α to another group H whose restriction α|_Gv to every stabilizer group of a vertex is injective, then

tr_GG(K(C^*_r H))⊂tr_H(K(C^*_rH)).

This follows from a general relative Fredholm module technique.

Examples are in particular HNN-extensions of H where the stable letter acts by conjugation with an element of H, or amalgamated-free products G=H_*UH of two copies of the same groups along a subgroup U.

We establish a version of "level-lowering" for mod p Galois representations arising from the reductions of representations associated to Hilbert modular forms. In particular, we show that level-lowering can be easily achieved if one replaces the base field with a suitable solvable extension. This is often enough for applications to proving the modularity of p-adic representations.

We determine optimal L^p-properties for the solutions of the general nonlinear elliptic system in the plane of the form

f_{$\overline{z}$}=H(z, f_z), h∈ L^p(C),

where H is a measurable function satisfying |H(z,w₁)−H(z,w₂)|≤ k|w₁−w₂| and k is a constant k<1.

We also establish the precise invertibility and spectral properties in L^p(C) for the operators

I−Tμ, I−μT, and T−μ,

where T is the Beurling transform. These operators are basic in the theory of quasi-conformal mappings and in linear and nonlinear elliptic partial differential equations (PDEs) in two dimensions. In particular, we prove invertibility in L^p(C) whenever 1+||μ||_∞ <p<1+1/||μ||_∞.

We also prove related results with applications to the regularity of weakly quasiconformal mappings.

We consider the Cauchy problem for the wave equation in the whole space ℝⁿ, with initial data that are distributions supported on finite sets. The main result is a precise description of the geometry of the sets of stationary points of the solutions to the wave equation.

In this paper we prove a generalization of conjectures of Bogomolov and Lang in terms of an arithmetic Néron-Tate height pairing over a finitely generated field.

In a series of recent papers, G. Shimura obtained an exact formula for the mass of a maximal lattice in a quadratic or hermitian space over a totally real number field. Using Bruhat-Tits theory, we obtain a quick and more conceptual proof of his formula when the form is totally definite.

We establish two topological results.

(A) If M is a 1-connected compact n-manifold and q≥2, then the minimal number of generators for the qth homotopy group π_q(M), MNG(π_q(M)), is bounded above by a number depending only on MNG(H_* (M,ℤ)) and q, where H_* (M,ℤ) is the homology group.

(C) Let $\mathscr {M}$(n,Y) be the collection of compact orientable n-manifolds whose oriented bundles admit SO(n)-invariant fibrations over Y with fiber compact nilpotent manifolds such that the induced SO(n)-actions on Y are equivalent. Then {π_q(M) finitely generated, M∈$\mathscr {M}$(n,Y)} contains only finite isomorphism classes depending only on n,Y,q.

Together with the results of [CG] and [Gr1], from (A) we conclude that (i) if M is a complete n-manifold of nonnegative curvature, then MNG(π_q(M)) is bounded above by a number depending only on n and q≥2. Together with the results of [Ch] and [CFG], from (C) we conclude that (ii) if M is a compact n-manifold whose sectional curvature and diameter satisfy λ≤ sec_M≤Λ and diam_M≤d, then π_q(M) has a finite number of possible isomorphism classes depending on n, λ, Λ, d, q≥2, provided π_q(M) is finitely generated.

We also show that (B) if M is a compact n-manifold with λ≤ sec_M≤Λ and diam(M)≤d, then the cohomology ring, H^* (M,ℚ), may have infinitely many isomorphism classes. In particular, (B) answers some questions raised by K. Grove [Gro].

We extend a result of E. Hrushovski and A. Pillay as follows. Let G be a finite subgroup of GL(n,$\mathbb{F}$) where $\mathbb{F}$ is a field of characteristic p such that p is sufficiently large compared to n. Assume that G is generated by p-elements. Then G is a product of 25 of its Sylow p-subgroups.

If G is a simple group of Lie type in characteristic p, the analogous result holds without any restriction on the Lie rank of G.

We also give an application of the Hrushovski-Pillay result showing that finitely generated adelic profinite groups are boundedly generated (i.e., such a group is a product of finitely many closed procyclic subgroups). This confirms a conjecture of V. Platonov and B. Sury which was motivated by characterizations of the congruence subgroup property for arithmetic groups.

Let f be a (germ of ) holomorphic self-map of ℂ² such that the origin is an isolated fixed point and such that df_O=id. Let v(f) be the degree of the first nonvanishing term in the homogeneous expansion of f−id. We generalize to ℂ² the classical Leau-Fatou flower theorem proving that there exist v(f)−1 holomorphic curves f-invariant, with the origin in their boundary, and attracted by O under the action of f.