Strengthening Kazhdan's property (T) by Bochner methods
Bochner method, Cohomology of lattices in semi-simple Lie groups, Rigidity
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(Γ,π,H)=0 where the coefficient module H is a Hilbert space and π is a unitary representation of Γ on H. Here we allow more general coefficients and say that Γ has property F ⊗ H if H 1(Γ,π⊗ π, 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 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 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). © 2011 Springer Science+Business Media B.V.
Original Publication Date
DOI of published version
Fisher, David and Hitchman, Theron, "Strengthening Kazhdan's property (T) by Bochner methods" (2012). Faculty Publications. 1741.