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- W2951788909 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 $G$ 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 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 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}." @default.
- W2951788909 created "2019-06-27" @default.
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- W2951788909 date "2006-09-23" @default.
- W2951788909 modified "2023-09-27" @default.
- W2951788909 title "Strengthening Kazhdan's Property $(T)$ by Bochner Methods" @default.
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