Group Pairs with Property (T), from Arithmetic Lattices

Let Γ be an arithmetic lattice in an absolutely simple Lie group G with trivial centre. We prove that there exists an integer N ≥ 2, a subgroup Λ of finite index in Γ, and an action of Λ on such that the pair () has property (T). If G has property (T), then so does. If G is the adjoint group of Sp(n, 1), then is a property (T) group satisfying the Baum–Connes conjecture. If Γ is an arithmetic lattice in SO(n, 1), then the associated von Neumann algebra is a II₁-factor in Popa’s class. Elaborating on this result of Popa, we construct a countable family of pairwise nonstably isomorphic group II₁-factors in the class, all with trivial fundamental groups and with all L²-Betti numbers being zero.Mathematics Subject Classiffications (2000). 22E40, 22E47, 46L80, 37A20