Towards Strong Banach property (T) for SL(3,R)

Abstract

We prove that SL(3,R) has Strong Banach property (T) in Lafforgue's sense with respect to the Banach spaces that are $\theta >0$ interpolation spaces (for the complex interpolation method) between an arbitrary Banach space and a Banach space with sufficiently good type and cotype. As a consequence, every action of SL(3,R) or its lattices by affine isometries on such a Banach space X has a fixed point, 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 many superreflexive spaces and some nonreflexive spaces as well. We see no obstruction for this class to be equal to all spaces with nontrivial type.

29 pages, 3 figures. Final version, to appear in Israel journal of math. v3: introduction shortened and small changes according to referee's suggestions. Also, I found a gap in the proof of Lemma 3.5 of v2. This Lemma was not used in the paper and was therefore removed from v3. But this Lemma is true, and the interested reader can find a correct proof due to Pisier in arXiv:1403.6415

References in corpus (4)

• Nonlinear spectral calculus and super-expanders
• Simple Lie groups without the Approximation Property
• Strong Banach Property (T) for Simple Algebraic Groups of Higher Rank
• Approximation properties for noncommutative Lp-spaces associated with lattices in Lie groups

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• Superexpanders from group actions on compact manifolds
• Banach space actions and $L^2$-spectral gap
• Nonpositive curvature is not coarsely universal
• Banach property (T) for $\mathrm{S}{\mathrm{L}}_{\mathrm{n}}(\mathbb{Z})$ and its applications
• Garland's method with Banach coefficients
• Almost invariant CND kernels and proper uniformly Lipschitz actions on subspaces of $L^1$