  The aim of this paper is to extend the framework of the spectral method for
proving property (T) to the class of reflexive Banach spaces and present a
condition implying that every affine isometric action of a given group $G$ on a
reflexive Banach space $X$ has a fixed point. This last property is a strong
version of Kazhdan's property (T) and is equivalent to the fact that
$H^1(G,\pi)=0$ for every isometric representation $\pi$ of $G$ on $X$. The
condition is expressed in terms of $p$-Poincar\'{e} constants and we provide
examples of groups, which satisfy such conditions and for which $H^1(G,\pi)$
vanishes for every isometric representation $\pi$ on an $L_p$ space for some
$p>2$. Our methods allow to estimate such a $p$ explicitly and yield several
interesting applications. In particular, we obtain quantitative estimates for
vanishing of 1-cohomology with coefficients in uniformly bounded
representations on a Hilbert space. We also give lower bounds on the conformal
dimension of the boundary of a hyperbolic group in the Gromov density model.
