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.