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13.15-14.10, rum 3721, Tsachik Gelander (Hebrew University):
Property (T) and rigidity for actions on Banach spaces

**Abstract.**
I'll discuss a joint work with Bader, Furman and Monod. We studied
Kazhdan property (T) and the fixed point property for actions on Banach
spaces instead of Hilbert spaces.
We showed 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 showed that the fixed point property for $L^p$
follows
from property (T) when $1 < p < 2+\epsilon$. For simple Lie groups and their
lattices, we proved 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
obtained
a superrigidity result for actions of irreducible lattices in products of
general groups on superreflexive Banach spaces.
Some of our results have applications to the study of group actions on
manifolds.