Strong property (T) for higher-rank simple Lie groups: Figure 1. Article Swipe
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· 2015
· Open Access
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· DOI: https://doi.org/10.1112/plms/pdv040
· OA: W1836042012
We prove that connected higher rank simple Lie groups have Lafforgue's strong\nproperty (T) with respect to a certain class of Banach spaces\n$\\mathcal{E}_{10}$ containing many classical superreflexive spaces and some\nnon-reflexive spaces as well. This generalizes the result of Lafforgue\nasserting that $\\mathrm{SL}(3,\\mathbb{R})$ has strong property (T) with respect\nto Hilbert spaces and the more recent result of the second named author\nasserting that $\\mathrm{SL}(3,\\mathbb{R})$ has strong property (T) with respect\nto a certain larger class of Banach spaces. For the generalization to higher\nrank groups, it is sufficient to prove strong property (T) for\n$\\mathrm{Sp}(2,\\mathbb{R})$ and its universal covering group. As consequences\nof our main result, it follows that for $X \\in \\mathcal{E}_{10}$, connected\nhigher rank simple Lie groups and their lattices have property (F$_X$) of\nBader, Furman, Gelander and Monod, and that the expanders contructed from a\nlattice in a connected higher rank simple Lie group do not admit a coarse\nembedding into $X$.\n
