Strengthening Kazhdan's Property (T) by Bochner Methods

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In this paper, we propose a property which is a natural generalization of Kazhdan's property (T) and prove that many, but not all, groups with property (T) also have this property. Let be a finitely generated group. One definition of having property (T) is that H¹(,π,)=0 where the coefficient module is a Hilbert space and π is a unitary representation of on. Here we allow more general coefficients and say that has property F ⊗ H if H¹(,π₁⊗π₂,F⊗)=0 if (F,π₁) is any representation with (F)<∞ and (,π₂) is a unitary representation. The main result of this paper is that a uniform lattice in a semisimple Lie group has property F ⊗ H if and only if it has property (T). The proof hinges on an extension of a Bochner-type formula due to Matsushima-Murakami and Raghunathan. We give a new and more transparent derivation of this formula as the difference of two classical Weitzenböck formula's for two different structures on the same bundle. Our Bochner-type formula is also used in our work on harmonic maps into continuum products Fisher-Hitchman2,Fisher-Hitchman1. Some further applications of property F ⊗ H in the context of group actions will be given in Fisher-Hitchman3.