We formulate and study a new coarse (co-)assembly map. It involves a modification of the Higson corona construction and produces a map dual in an appropriate sense to the standard coarse assembly map. The new assembly map is shown to be an isomorphism in many cases. For the underlying metric space of a group, the coarse co-assembly map is closely related to the existence of a dual Dirac morphism and thus to the Dirac dual Dirac method of attacking the Novikov conjecture.

In this paper we study a Haagerup inequality in the general case of discrete groupoids. We develop two geometrical tools, pinching and tetrahedral change of faces, based on deformation of triangles, to prove it. We show how to use these tools to find all the already known results just by manipulating triangles. We use these tools for groups acting freely and by isometries on the set of vertices of any affine building and give a first reduction of this inequality to its verification on some special triangles and prove the inequality when the building is of type $\tilde{A}_{k_1}\times\cdots\times\tilde{A}_{k_n}$, where $k_i\in\{1,2\}$, $i=1,\dots,n$.

In this paper we study the reduction to characteristic $p$ of the Shimura variety associated to a unitary group which has signature $(n-1,1)$ at its real place. We describe the Newton polygon, the Ekedahl–Oort, and the final stratification. In addition we examine the moduli space of $p$-isogenies using a variant of the local model for Shimura varieties. We apply our results to obtain a proof of the Eichler–Shimura congruence relation for the case that $n$ is even.

The relative trace formula is a tool in the theory of automorphic forms which was invented by Jacquet in order to study period integrals and relate them to Langlands functoriality. In this paper we give an analogue of Arthur’s spectral expansion of the trace formula to the relative setup in the context of $\text{GL}_n$. This is an important step toward application of the relative trace formula and it extends earlier work by several authors to higher rank. Our method is new and based on complex analysis and majorization of Eisenstein series. To that end we use recent lower bounds of Brumley for Rankin–Selberg $L$-functions at the edge of the critical strip.

For each group $G$ having an infinite normal subgroup with the relative property (T) (e.g. $G=H\times K$, with $H$ infinite with property (T) and $K$ arbitrary) and each countable abelian group $\varLambda$ we construct free ergodic measure-preserving actions $\sigma_\varLambda$ of $G$ on the probability space such that the first cohomology group of $\sigma_\varLambda$, $\ssm{H}^1(\sigma_\varLambda,G)$, is equal to $\text{Char}(G)\times\varLambda$. We deduce that $G$ has uncountably many non-stably orbit-equivalent actions. We also calculate 1-cohomology groups and show existence of ‘many’ non-stably orbit-equivalent actions for free products of groups as above.

Soient $p$ un nombre premier et $\mathcal{C}$ la courbe de Hecke $p$-adique de $\mathrm{GL}_2$ de niveau moderé $1$ introduite par Coleman et Mazur. Nous montrons que $\mathcal{C}$ est lisse en ses points Eisenstein critiques et nous donnons une condition nécessaire et suffisante pour que l’application poids soit étale en un tel point en terme de certaines valeurs de la fonction $\zeta$$p$-adique. Un point important est la détermination en ces points du lieu de réductibilité schématique du pseudo-caractère Galoisien porté par $\mathcal{C}$, et ce restreint à un groupe de décomposition en $p$. L’assertion de lissité découle alors de ce que les fonctions $L$ de Dirichlet ont au plus des zéros simples aux entiers.

Let $p$ be a prime number and $\mathcal{C}$ be the $p$-adic tame level $1$ eigencurve introduced by Coleman and Mazur. We prove that $\mathcal{C}$ is smooth at the evil Eisenstein points and we give necessary and sufficient conditions for etaleness of the map to the weight space at these points in terms of $p$-adic zeta values. A key step is the determination at these points of the schematic reducibility locus of the pseudo-character carried by $\mathcal{C}$ restricted to a decomposition group at $p$. Then, the smoothness appears to be a consequence of the fact that the Dirichlet $L$-functions only have simple zeros at integers.