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

Abstract

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 Snc denotes the product of noncompact simple factors of S and ST 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 Snc if and only if R = [Snc, R]. We also provide some applications to the spectral gap of π (μ) = ∫ π (g) dμ (g) where π is a certain unitary representation and μ is a probability measure.