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

TL;DR

This paper establishes that SL(3,R) possesses a strong form of property (T) for a broad class of Banach spaces, leading to fixed point properties and non-embeddability results for expanders.

Contribution

It proves SL(3,R) has Strong Banach property (T) for interpolation spaces between arbitrary Banach spaces and those with good type and cotype, extending the class of spaces with fixed point properties.

Findings

SL(3,R) has Strong Banach property (T) for a wide class of Banach spaces.

Actions of SL(3,R) or its lattices on these spaces have fixed points.

Expanders from SL(3,Z) do not embed coarsely into these Banach spaces.

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.