A local characterization of Kazhdan projections and applications
Résumé
We give a local characterization of the existence of Kazhdan projections for arbitary families of Banach space representations of a compactly generated locally compact group $G$ and give several applications in terms of stability of rigidity under perturbations. Among them, we show that for many classes E of Banach spaces, property (FE), which asserts that every affine isometric action of $G$ on a space in E has a fixed point, is equivalent to a form introduced by Oppenheim of strong property (T) with respect to spaces to $E$, or to spaces close to $E$. This has to be compared with the well-known fact that, unlike the case of Hilbert spaces, (FE) is in general not equivalent to property (T) with respect to E. Another kind of applications is that many forms of Banach strong property (T) are open in the space of marked groups, and more generally every group with such a property is a quotient of a compactly presented group with the same property.
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