Strengthening Kazhdan's Property $(T)$ by Bochner Methods

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 $\G$ be a finitely generated group. One definition of $\G$ having property $(T)$ is that $H^1(\G,\pi,\fh)=0$ where the coefficient module $\fh$ is a Hilbert space and $\pi$ is a unitary representation of $\G$ on $\fh$. Here we allow more general coefficients and say that $\G$ has property $F \otimes H$ if $H^1(\G,\pi_1{\otimes}\pi_2,F{\otimes}\fh)=0$ if $(F,\pi_1)$ is any representation with $\dim(F)<\infty$ and $(\fh,\pi_2)$ 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 Weitzenb\"{o}ck 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 \cite{Fisher-Hitchman2,Fisher-Hitchman1}. Some further applications of property $F \otimes H$ in the context of group actions will be given in \cite{Fisher-Hitchman3}.