Group pairs with property (T), from arithmetic lattices

Let Gamma 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 Lambda of finite index in Gamma, and an action of Lambda on Z(N) such that the pair (Lambda proportional to Z(N), Z(N)) has property (T). If G has property (T), then so does Lambda proportional to Z(N). If G is the adjoint group of Sp(n, 1), then Lambda proportional to Z(n) is a property (T) group satisfying the Baum-Connes conjecture. If Gamma is an arithmetic lattice in SO(n, 1), then the associated von Neumann algebra (L(Lambda proportional to Z(N))) is a II1-factor in Popa's class HTs. Elaborating on this result of Popa, we construct a countable family of pairwise nonstably isomorphic group II1-factors in the class HTs, all with trivial fundamental groups and with all L-2-Betti numbers being zero.