Volume 17, Issue 1

Let E λ be the Hilbert space spanned by the eigenfunctions of the non-Euclidean Laplacian associated with a positive discrete eigenvalue λ. In this paper, the trace of the Hecke operators acting on the space E λ is computed explicitly for Hecke congruence subgroups of non-square free level.

Let Γ be a group together with a sequence of normal subgroups Γ ⊃ Γ 1 ⊃ Γ 2 ... of finite index [Γ : Γ k ] such that ⋂ k Γ k = {1}. Let ( X , Y ) be a (compact) 4 n -dimensional Poincaré pair and p : ( X ̄, Y ̄) → ( X , Y ) be a Γ-covering, i.e. normal covering with Γ as deck transformation group. We get associated Γ/Γ k -coverings ( X k , Y k ). We prove that where sign or sign (2) is the signature or L 2 -signature, respectively, and the convergence of the right side for any such sequence (Γ k ) k ≥1 is part of the statement. If Γ is amenable, we prove in a similar way an approximation theorem for sign (2) ( X ̄, Y ̄) in terms of the signatures of a regular exhaustion of X ̄. Our results are extensions of Lück's approximation results for L 2 -Betti numbers [10, Theorem 0.1].

In this paper we answer negatively a question posed by Casacuberta, Farjoun, and Libman about the preservation of perfect groups under localization functors. Indeed, we show that a certain P -localization of Berrick and Casacuberta's universal acyclic group is not perfect. We also investigate under which conditions perfectness is preserved: For instance, we show that if the localization of a perfect group is finite then it is perfect.

We describe entire solutions for certain functional and partial differential equations related to the Fermat varieties in ℂ 2 .

We investigate spectral asymptotic properties of a measure geometric Laplacian which is given as the second derivative w.r.t. two atomless finite Borel measures μ and ν with compact supports L ≔ supp μ and K ≔ supp ν , such that L ⊆ K ⊆ IR. This operator is a generalization of the well-known Sturm-Liouville operator as well as of the measure geometric Laplacian given by . In the special case of self similar measures—Hausdor measures or, more general, self similar measures with arbitrary weights living on Cantor like sets—we determine the asymptotic behaviour of the eigenvalue counting function. This increases under both Dirichlet and Neumann boundary conditions like x γ where the spectral exponent γ is given in dependence of the weights of the measures μ and ν .

Let M be a compact closed n -dimensional manifold. Given a Riemannian metric on M , we consider the zeta functions Z ( s ) for the de Rham Laplacian and the Bochner Laplacian on p -forms. The hessian of Z ( s ) with respect to variations of the metric is given by a pseudodi erential operator T s . When the real part of s is less than n /2−1 we compute the principal symbol of T s . This can be used to determine whether a general critical metric for ( d/ds ) k Z ( s ) has finite index, or whether it is an essential saddle point.

In this paper we prove global existence for certain multispeed Dirichlet-wave equations with quadratic nonlinearities outside of obstacles. We assume the natural null condition for systems of quasilinear wave equations with multiple speeds. The null condition only puts restrictions on the self-interactions of each wave family. We use the method of commuting vector fields and weighted space-time L 2 estimates.