The $p\,$-approximation property for simple Lie groups with finite center

TL;DR

This paper demonstrates that certain simple Lie groups with finite center lack the $p$-approximation property for all $p$ in (1,∞), extending previous results from the case $p=2$ to all such $p$.

Contribution

It proves the absence of the $p$-approximation property for a broad class of simple Lie groups with finite center and real rank greater than 1, generalizing earlier results.

Findings

$ ext{SL}(3, ext{R})$ and $ ext{Sp}(2, ext{R})$ lack $p$-approximation property for all $p$ in (1,∞)

Connected simple Lie groups with finite center and higher real rank also lack this property

Lattices in these groups share the same non-approximation property

Abstract

We prove that, for any $1<p<\infty$, the groups $\text{SL}(3,\mathbb{R})$ and $\text{Sp}(2,\mathbb{R})$ do not have the $p\,$-approximation property of An, Lee and Ruan, which implies in particular that they are not $p\,$-weakly amenable. It follows that the same holds for any connected simple Lie group with finite center and real rank greater than 1, as well as for any lattice in it. This extends Haagerup and de Laat's result for the AP, which in this language corresponds to the case $p=2$.