Slant products on the Higson-Roe exact sequence

We construct a slant product $/ \colon \mathrm{S}_p(X \times Y) \times \mathrm{K}_{1-q}(\mathfrak{c}^{\mathrm{red}}Y) \to \mathrm{S}_{p-q}(X)$ on the analytic structure group of Higson and Roe and the K-theory of the stable Higson corona of Emerson and Meyer. The latter is the domain of the co-assembly map $\mu^\ast \colon \mathrm{K}_{1-\ast}(\mathfrak{c}^{\mathrm{red}}Y) \to \mathrm{K}^\ast(Y)$. We obtain such products on the entire Higson--Roe sequence. They imply injectivity results for external product maps. Our results apply to products with aspherical manifolds whose fundamental groups admit coarse embeddings into Hilbert space. To conceptualize the class of manifolds where this method applies, we say that a complete $\mathrm{spin}^{\mathrm{c}}$-manifold is Higson-essential if its fundamental class is detected by the co-assembly map. We prove that coarsely hypereuclidean manifolds are Higson-essential. We draw conclusions for positive scalar curvature metrics on product spaces, particularly on non-compact manifolds. We also obtain equivariant versions of our constructions and discuss related problems of exactness and amenability of the stable Higson corona.Comment: 82 pages; v2: Minor improvements. To appear in Ann. Inst. Fourie

A walk in the noncommutative garden

This text is written for the volume of the school/conference "Noncommutative Geometry 2005" held at IPM Tehran. It gives a survey of methods and results in noncommutative geometry, based on a discussion of significant examples of noncommutative spaces in geometry, number theory, and physics. The paper also contains an outline (the ``Tehran program'') of ongoing joint work with Consani on the noncommutative geometry of the adeles class space and its relation to number theoretic questions.Comment: 106 pages, LaTeX, 23 figure

Endomorphisms of spaces of virtual vectors fixed by a discrete group

Consider a unitary representation $\pi$ of a discrete group $G$, which, when restricted to an almost normal subgroup $\Gamma\subseteq G$, is of type II. We analyze the associated unitary representation $\overline{\pi}^{\rm{p}}$ of $G$ on the Hilbert space of "virtual" $\Gamma_0$-invariant vectors, where $\Gamma_0$ runs over a suitable class of finite index subgroups of $\Gamma$. The unitary representation $\overline{\pi}^{\rm{p}}$ of $G$ is uniquely determined by the requirement that the Hecke operators, for all $\Gamma_0$, are the "block matrix coefficients" of $\overline{\pi}^{\rm{p}}$. If $\pi|_\Gamma$ is an integer multiple of the regular representation, there exists a subspace $L$ of the Hilbert space of the representation $\pi$, acting as a fundamental domain for $\Gamma$. In this case, the space of $\Gamma$-invariant vectors is identified with $L$. When $\pi|_\Gamma$ is not an integer multiple of the regular representation, (e.g. if $G=PGL(2,\mathbb Z[\frac{1}{p}])$, $\Gamma$ is the modular group, $\pi$ belongs to the discrete series of representations of $PSL(2,\mathbb R)$, and the $\Gamma$-invariant vectors are the cusp forms) we assume that $\pi$ is the restriction to a subspace $H_0$ of a larger unitary representation having a subspace $L$ as above. The operator angle between the projection $P_L$ onto $L$ (typically the characteristic function of the fundamental domain) and the projection $P_0$ onto the subspace $H_0$ (typically a Bergman projection onto a space of analytic functions), is the analogue of the space of $\Gamma$- invariant vectors. We prove that the character of the unitary representation $\overline{\pi}^{\rm{p}}$ is uniquely determined by the character of the representation $\pi$.Comment: The exposition has been improved and a normalization constant has been addressed. The result allows a direct computation for the characters of the unitary representation on spaces of invariant vectors (for example automorphic forms) in terms of the characters of the representation to which the fixed vectors are associated (e.g discrete series of PSL(2, R) for automorphic forms