2011/11/22 by C. R. E. Raja, Raja, C. R. E.
Mathematics · #22D10 #22D40 #22E15 #60G50 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Advanced Topics in Algebra #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Probability (math.PR) #Representation Theory (math.RT) #math.DS #math.GR #math.PR #math.RT #msc:22D10 #msc:22D40 #msc:22E15 #msc:60G50
paper · pdf · doi:10.48550/arxiv.1111.5148
arxiv created 2011/11/22 · openalex publication_date 2011/11/22 · arxiv updated 2011/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider strong relative property (T) for pairs (\Ga, G) where \Ga acts on G. If N is a connected Lie group and \Ga is a group of automorphisms of N, we choose a finite index subgroup \Ga 0 of \Ga and obtain that (\Ga, [\Ga 0, N]) has strong relative property (T) provided Zariski-closure of \Ga has no compact factor of positive dimension. We apply this to obtain the following: G is 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 Snc that have property (T), then we show that (\Ga, R) has strong relative property (T) for a Zariski-dense closed subgroup of Snc if and only if R=[Snc,R]. The case when N is a vector group is discussed separately and some interesting results are proved. We also considered actions on solenoids K and proved that if \Ga acts on a solenoid K, then (\Ga, K) has strong relative property (T) under certain conditions on \Ga. For actions on solenoids we provided some alternatives in terms of amenability and strong relative property (T). 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.