Related papers: About the difficulty to prove the Baum Connes conj…

We prove that cocompact (and more generally: undistorted) lattices on $\tilde{A}_2$-buildings satisfy Lafforgue's strong property (T), thus exhibiting the first examples that are not related to algebraic groups over local fields. Our…

We prove several cases of Zimmer's conjecture for actions of higher-rank cocompact lattices on low dimensional manifolds. For example, if $\Gamma$ is a cocompact lattice in $\mathrm{Sl}(n, \mathbb R)$, $M$ is a compact manifold, and…

Let $\Gamma$ be a weakly irreducible higher rank lattice. In this paper, we will prove various rigidity results for the $\Gamma$-action following a philosophy of the Zimmer program. We provide new rigidity results including local and global…

Let $\Gamma \stackrel{i}{\hookrightarrow} L$ be a lattice in the real simple Lie group $L$. If $L$ is of rank at least 2 (respectively locally isomorphic to $Sp(n,1)$) any unbounded morphism $\rho: \Gamma \longrightarrow G$ into a simple…

Let $G$ be the symplectic group $Sp_4$ over a non Archimedean local field of any characteristic. It is proved in this paper that for $p\in[1,4/3)\cup (4,\infty]$ neither the group $G$ nor its lattices have the property of approximation by…

We prove that every lattice in a product of higher rank simple Lie groups or higher rank simple algebraic groups over local fields has Vincent Lafforgue's strong property (T). Over non-archimedean local fields, we also prove that they have…

Let \Gamma be a non-cocompact lattice on a locally finite regular right-angled building X. We prove that if \Gamma has a strict fundamental domain then \Gamma is not finitely generated. We use the separation properties of subcomplexes of X…

For an extension $1\rightarrow N \rightarrow \Gamma \xrightarrow{q} \Gamma / N \rightarrow 1$ of discrete countable groups, it is known that the Baum-Connes conjecture with coefficients holds for $\Gamma$ if it holds for $\Gamma / N$ and…

Let $X=G/K$ be an irreducible Hermitian symmetric space of the non-compact type and let $S\in G^\mbb{C}$ be the associated compression semi-group. Let $\Gamma$ be a discrete subgroup of $G$. We give a sufficient condition for…

We prove Zimmer's conjecture for co-compact lattices in ${\rm SL}(n, \mathbb C)$: for any co-compact lattice in ${\rm SL}(n, \mathbb C)$, $n \geq 3$, any $\Gamma$-action on a compact manifold $M$ with dimension: (I) less than $2n-2$ if $n…

Let G be a complete Kac-Moody group of rank n \geq 2 over the finite field of order q, with Weyl group W and building \Delta. We first show that if W is right-angled, then for all q \neq 1 mod 4 the group G admits a cocompact lattice \Gamma…

Let $G$ be a connected semisimple real algebraic group and $\Gamma<G$ be its Zariski dense discrete subgroup. We prove that if $\Gamma\backslash G$ admits any finite Bowen-Margulis-Sullivan measure, then $\Gamma$ is virtually a product of…

Given a (not necessarily discrete) proper metric space $M$ with bounded geometry, we define a groupoid $G(M)$. We show that the coarse Baum--Connes conjecture with coefficients, which states that the assembly map with coefficients for G(M)…

We establish the Hasse principle (local-global principle) in the context of the Baum-Connes conjecture with coefficients. We illustrate this principle with the discrete group $GL(2,F)$ where $F$ is any global field.

Let $\Gamma$ be a finitely generated cocompact lattice of a totally disconnected locally compact group $G$, and $C$ a dense subgroup of $G$ that contains and commensurates $\Gamma$. We study the problem of describing all finitely generated…

A planar semimodular lattice $K$ is \emph{slim} if $\mathsf{M}_{3}$ is not a sublattice of~$K$. In a recent paper, G. Cz\'edli found four new properties of congruence lattices of slim, planar, semimodular lattices, including the \emph{No…

We develop further basic tools in the theory of continuous bounded cohomology of locally compact groups. We apply this tools to establish a Milnor-Wood type inequality in a very general context and to prove a global rigidity result which…

Let k be a global field and let k_v be the completion of k with respect to v, a non-archimedean place of k. Let \mathbf{G} be a connected, simply-connected algebraic group over k, which is absolutely almost simple of k_v-rank 1. Let…

A planar semimodular lattice $K$ is \emph{slim} if $\mathsf{M}_3$ is not a sublattice of~$K$. In a recent paper, G. Cz\'edli found four new properties of congruence lattices of slim, planar, semimodular lattices, including the \emph{No…

Owing to a renewed interest in flat space higher spin gauge theories, in this note we provide further details and clarifications on the results presented in arXiv:1107.5843 and arXiv: 1209.5755, which investigated their locality properties.…