On strong property (T) and fixed point properties for Lie groups

Abstract

We consider certain strengthenings of property (T) relative to Banach spaces. Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> be a Banach space for which the Banach–Mazur distance to a Hilbert space of all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> -dimensional subspaces grows as a power of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> strictly less than one half. We prove that every connected simple Lie group of sufficiently large real rank has strong property (T) of Lafforgue with respect to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> . As a consequence, every continuous affine isometric action of such a high rank group (or a lattice in such a group) on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> has a fixed point. For the special linear Lie groups, we also present a more direct approach to fixed point properties, or, more precisely, to the boundedness of quasi-cocycles. We prove that every special linear group of sufficiently large rank satisfies the following property: every quasi- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> </mml:math> -cocycle with values in an isometric representation on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> is bounded.