Abstract of a paper by Tim de Laat and Mikael de la Salle
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 http://arXiv.org/abs/1401.3611
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