Property (T) and rigidity for actions on Banach spaces

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 ≤ p < ∞.
We show that the fixed point property for L^{p} follows from
property (T) when 1 < p < 2+ε.
For simple Lie groups and their lattices, we prove that the fixed point property for
L^{p} 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.

Bibliographical: Acta Math.

Download: pdf | ArXiv:math.GR/0506361

[Papers] [Alex Furman]