  We study property (T) and the fixed point property for actions on $L^p$ and
other Banach spaces. We show that property (T) holds when $L^2$ is replaced by
$L^p$ (and even a subspace/quotient of $L^p$), and that in fact it is
independent of $1\leq p<\infty$. We show that the fixed point property for
$L^p$ follows from property (T) when $1<p< 2+\e$. For simple Lie groups and
their lattices, we prove that the fixed point property for $L^p$ holds for any
$1< p<\infty$ if and only if the rank is at least two. Finally, we obtain a
superrigidity result for actions of irreducible lattices in products of general
groups on superreflexive Banach spaces.
