Group Pairs with Property (T), from Arithmetic Lattices

Abstract

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 \({\mathbb Z}^{N}\) such that the pair ( \(\Lambda \ltimes {\mathbb Z}^{N}, {\mathbb Z}^{N}\)) has property (T). If G has property (T), then so does \(\Lambda \ltimes {\mathbb Z}^{N}\). If G is the adjoint group of Sp(n, 1), then \(\Lambda \ltimes {\mathbb Z}^{N}\) 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 \((L(\Lambda \ltimes {\mathbb Z}^{N}))\) is a II1-factor in Popa’s class \({\cal HT}_{s}\). Elaborating on this result of Popa, we construct a countable family of pairwise nonstably isomorphic group II1-factors in the class \({\cal HT}_{s}\), all with trivial fundamental groups and with all L2-Betti numbers being zero.