Geom Dedicata (2012) 160:333–364 DOI 10.1007/s10711-011-9686-9 ORIGINAL PAPER Strengthening Kazhdan’s property (T ) by Bochner methods David Fisher · Theron Hitchman Received: 17 January 2011 / Accepted: 29 November 2011 / Published online: 11 December 2011 © Springer Science+Business Media B.V. 2011 Abstract In this paper, we propose a property which is a natural generalization of Kazh- dan’s property (T ) and prove that many, but not all, groups with property (T ) also have this property. Let Ŵ be a finitely generated group. One definition of Ŵ having property (T ) is that H 1 (Ŵ, π, H) = 0 where the coefficient module H is a Hilbert space and π is a unitary rep- resentation of Ŵ on H. Here we allow more general coefficients and say that Ŵ has property F ⊗ H if H 1 (Ŵ, π 1 ⊗π 2 , F ⊗H) = 0 if ( F,π 1 ) is any representation with dim( F )< ∞ and (H,π 2 ) 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 Rag- hunathan. 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 and Hitchman in preparation; Fisher and Hitchman in Int Math Res Not 72405:1–19, 2006). Some further applications of property F ⊗ H in the context of group actions will be given in Fisher and Hitchman (in preparation). Keywords Rigidity · Bochner method · Cohomology of lattices in semi-simple Lie groups Mathematics Subject Classification (2000) 53C35 · 53C25 1 Introduction and statements of results Property (T ), introduced by Kazhdan in 1966 in [20], plays a fundamental role in the study of discrete subgroups of Lie groups and more general finitely generated groups. In this D. Fisher (B ) Department of Mathematics, Indiana University, Bloomington, IN, USA e-mail: fisherdm@indiana.edu T. Hitchman Department of Mathematics, University of Northern Iowa, Cedar Falls, IA 50614-0506, USA e-mail: theron.hitchman@uni.edu 123