This is an announcement for the paper "Towards Banach space strong property (T) for SL(3,R)" by Mikael de la Salle.
Abstract: We prove that SL(3,R) has strong property (T) in Lafforgue's sense with respect to the Banach spaces that are \theta>0 interpolation spaces (for the Lions-Calder'on complex interpolation method) between an arbitrary Banach space and a Banach space with sufficiently good type and cotype. As a consequence, for such a Banach space X, SL(3,R) and its lattices have the fixed point property (F_X) of Bader--Furman--Gelander--Monod, and the expanders contructed from SL(3,Z) do not admit a coarse embedding into X. We also prove a quantitative decay of matrix coefficients (Howe-Moore property) for representations with small exponential growth of SL(3,R) on X. This class of Banach spaces contains the classical superreflexive spaces and some nonreflexive spaces as well. We see no obstruction for this class to be equal to all spaces with nontrivial type.
Archive classification: math.GR math.FA math.MG
Remarks: 31 pages, 3 figures. Comments welcome!
Submitted from: delasall@phare.normalesup.org
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1307.2475
or