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

TL;DR

This paper proves that high rank Lie groups with certain Banach space properties have strong property (T), ensuring fixed points for affine actions, thus supporting a conjecture and extending fixed point theory in geometric group analysis.

Contribution

It establishes strong property (T) for high rank Lie groups relative to specific Banach spaces and confirms a related conjecture about fixed point properties.

Findings

High rank Lie groups have strong property (T) relative to certain Banach spaces.

Affine isometric actions of these groups on such spaces have fixed points.

Special linear groups of large rank have bounded quasi-1-cocycles without relying on strong property (T).

Abstract

We consider certain strengthenings of property (T) relative to Banach spaces that are satisfied by high rank Lie groups. Let X be a Banach space for which, for all k, the Banach--Mazur distance to a Hilbert space of all k-dimensional subspaces is bounded above by a power of k strictly less than one half. We prove that every connected simple Lie group of sufficiently large real rank depending on X has strong property (T) of Lafforgue with respect to X. As a consequence, we obtain that every continuous affine isometric action of such a high rank group (or a lattice in such a group) on X has a fixed point. This result corroborates a conjecture of Bader, Furman, Gelander and Monod. 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. Without appealing to strong property (T), we prove that given a Banach space X as above, every special linear group of sufficiently large rank satisfies the following property: every quasi-1-cocycle with values in an isometric representation on X is bounded.