Fixed point property for universal lattice on Schatten classes

TL;DR

This paper proves that universal lattices and higher rank lattices have the fixed point property for affine isometric actions on p-Schatten class operators, extending previous results from commutative Lp-spaces.

Contribution

It establishes the fixed point property for universal lattices on p-Schatten classes, generalizing prior work from commutative Lp-spaces to non-commutative operator spaces.

Findings

Universal lattices have the fixed point property on p-Schatten classes.

Higher rank lattices also share this fixed point property.

Results extend previous theorems from commutative to non-commutative Lp-settings.

Abstract

The special linear group G=SL_n(Z[x1,...,xk]) (n at least 3 and k finite) is called the universal lattice. Let n be at least 4, p be any real number in (1,\infty). The main result is the following: any finite index subgroup of G has the fixed point property with respect to every affine isometric action on the space of p-Schatten class operators. It is in addition shown that higher rank lattices have the same property. These results are generalization of previous theorems repsectively of the author and of Bader--Furman--Gelander--Monod, which treated commutative Lp-setting.