This is an announcement for the paper "Strong property (T) for higher rank simple Lie groups" by Tim de Laat and Mikael de la Salle.
Abstract: We prove that connected higher rank simple Lie groups have Lafforgue's strong property (T) with respect to a certain class of Banach spaces $\mathcal{E}_{10}$ containing many classical superreflexive spaces and some non-reflexive spaces as well. This generalizes the result of Lafforgue asserting that $\mathrm{SL}(3,\mathbb{R})$ has strong property (T) with respect to Hilbert spaces and the more recent result of the second named author asserting that $\mathrm{SL}(3,\mathbb{R})$ has strong property (T) with respect to a certain larger class of Banach spaces. For the generalization to higher rank groups, it is sufficient to prove strong property (T) for $\mathrm{Sp}(2,\mathbb{R})$ and its universal covering group. As consequences of our main result, it follows that for $X \in \mathcal{E}_{10}$, connected higher rank simple Lie groups and their lattices have property (F$_X$) of Bader, Furman, Gelander and Monod, and the expanders contructed from a lattice in such a group do not admit a coarse embedding into $X$.
Archive classification: math.GR math.FA math.MG
Report Number: CPH-SYM-DNRF92
Remarks: 30 pages, 1 figure
Submitted from: tim.delaat@wis.kuleuven.be
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1401.3611
or