Strengthening Kazhdan’s property (T) by Bochner methods

Abstract

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 Γ be a finitely generated group. One definition of Γ having property (T) is that \({H^{1}(\Gamma, \pi, {\mathcal{H}}) = 0}\) where the coefficient module \({{\mathcal{H}}}\) is a Hilbert space and π is a unitary representation of Γ on \({{\mathcal{H}}}\). Here we allow more general coefficients and say that Γ has property \({F \otimes {H}}\) if \({H^{1}(\Gamma, \pi_{1}{\otimes}\pi_{2}, F{\otimes} {\mathcal{H}}) = 0}\) if (F, π1) is any representation with dim(F) < ∞ and \({({\mathcal{H}}, \pi_{2})}\) is a unitary representation. The main result of this paper is that a uniform lattice in a semisimple Lie group has property \({F \otimes {H}}\) 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 Weitzenbock 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 and Hitchman in preparation; Fisher and Hitchman in Int Math Res Not 72405:1–19, 2006). Some further applications of property \({F\otimes {H}}\) in the context of group actions will be given in Fisher and Hitchman (in preparation).