Strong relative property (T) and spectral gap of random walks

We consider strong relative property (T) for pairs (Γ, G) where Γ acts on G. If N is a connected nilpotent Lie group and Γ is a group of automorphisms of N, we choose a finite index subgroup Γ ⁰ of Γ and obtain that (Γ , Γ ⁰, N]) has strong relative property (T) provided Zariski-closure of Γ has no compact factor of positive dimension. We apply this to obtain the following: Let G be a connected Lie group with solvable radical R and a semisimple Levi subgroup S. If S _nc denotes the product of noncompact simple factors of S and S _T denotes the product of simple factors in S that have property (T), then we show that (Γ , R) or ${(\Gamma S_{T}, \overline{S_{T}R})}$ has strong relative property (T) for a ’Zariski-dense’ closed subgroup Γ of S _nc if and only if R = S _nc , R]. We also provide some applications to the spectral gap of π (μ) =  ∫ π (g) d μ (g) where π is a certain unitary representation and μ is a probability measure.