The author surveys Connes’ results on the longitudinal Laplace operator along a (regular) foliation and its spectrum, and discusses their generalization to any singular foliation on a compact manifold. Namely, it is proved that the Laplacian of a singular foliation is an essentially self-adjoint operator (unbounded) and has the same spectrum in every (faithful) representation, in particular, in L 2 of the manifold and L 2 of a leaf. The author also discusses briefly the relation of the Baum-Connes assembly map with the calculation of the spectrum.
The author constructs unitary intertwiners for degenerate C*-algebraic universal principal series of SL(n + 1) over a local field by explicitely normalizing standard intertwining integrals at the level of Hilbert modules.
The authors examine the quantization commutes with reduction phenomenon for Hamiltonian actions of compact Lie groups on closed symplectic manifolds from the point of view of topological K-theory and K-homology. They develop the machinery of K-theory wrong-way maps in the context of orbifolds and use it to relate the quantization commutes with reduction phenomenon to Bott periodicity and the K-theory formulation of the Weyl character formula.
In this paper, the author studies the coarse embedding into uniformly convex Banach spaces. The author proves that the property of coarse embedding into Banach spaces can be preserved under taking the union of the metric spaces under certain conditions. As an application, for a group G strongly relatively hyperbolic to a subgroup H, the author proves that B(n) = {g ∈ G | |g| S∪ℋ ≤ n} admits a coarse embedding into a uniformly convex Banach space if H does.
Let H be an extension of a finite group Q by a finite group G. Inspired by the results of duality theorems for étale gerbes on orbifolds, the authors describe the number of conjugacy classes of H that map to the same conjugacy class of Q. Furthermore, a generalization of the orthogonality relation between characters of G is proved.
The notions of metric sparsification property and finite decomposition complexity are recently introduced in metric geometry to study the coarse Novikov conjecture and the stable Borel conjecture. In this paper, it is proved that a metric space X has finite decomposition complexity with respect to metric sparsification property if and only if X itself has metric sparsification property. As a consequence, the authors obtain an alternative proof of a very recent result by Guentner, Tessera and Yu that all countable linear groups have the metric sparsification property and hence the operator norm localization property.
This paper discusses “geometric property (T)”. This is a property of metric spaces introduced in earlier works of the authors for its applications to K-theory. Geometric property (T) is a strong form of “expansion property”, in particular, for a sequence (X n) of bounded degree finite graphs, it is strictly stronger than (X n) being an expander in the sense that the Cheeger constants h(X n) are bounded below.
In this paper, the authors show that geometric property (T) is a coarse invariant, i.e., it depends only on the large-scale geometry of a metric space X. The authors also discuss how geometric property (T) interacts with amenability, property (T) for groups, and coarse geometric notions of a-T-menability. In particular, it is shown that property (T) for a residually finite group is characterised by geometric property (T) for its finite quotients.
The authors use geometric techniques to prove that the restricted wreath product F ≀ ℂ is a quasi-isometrically embedded subgroup of Thompson’s group F.
The author uses the unitary representation theory of SL 2(ℝ) to understand the Rankin-Cohen brackets for modular forms. Then this interpretation is used to study the corresponding deformation problems that Paula Cohen, Yuri Manin and Don Zagier initiated. Two uniqueness results are established.