Dense subgroups with property \((T)\) in Lie groups (Q2427580)

Let \(G\) be a locally compact group. The subset \(X\) is said to be a Kazhdan subset of \(G\) if \(\exists \varepsilon >0\) such that every continuous unitary representation having \((X, \varepsilon )\)-invariant vectors actually has nonzero invariant vectors. The locally compact group \(G\) has property (T) if it has a compact Kazhdan subset. In the paper, connected Lie groups that have a dense finitely generated subgroup \(\Gamma\) with property (T) are characterized. The main result of the paper: Let \(G\) be a connected Lie group. Then \(G\) has a dense, finitely generated subgroup with property (T) iff \(G\) has property (T), i.e. satisfies: (i) every amenable quotient of \(G\) is compact; (ii) no simple quotient of \(G\) is locally isomorphic to \(\text{SO}(n,1)\) or \(\text{SU}(n,1)\) for some \(n\geq 2\); (iii) \({\mathbb R}/{\mathbb Z}\) is not a quotient of \(G\); (iv) \(\text{SO}_3 ({\mathbb R})\) is not a quotient of \(G\).