Property (T) and rigidity for actions on Banach spaces

We study property (T) and the fixed point property for actions on Lp and other Banach spaces. We show that property (T) holds when L2 is replaced by Lp (and even a subspace/quotient of Lp), and that in fact it is independent of 1 ≤ p < ∞. We show that the fixed point property for Lp follows from property (T) when 1 < p < 2+ε. For simple Lie groups and their lattices, we prove that the fixed point property for Lp holds for any 1 < p < ∞ 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.