Approximation properties of simple Lie groups made discrete

TL;DR

This paper investigates approximation properties of connected simple Lie groups with discrete topology, establishing their equivalence and proving weak amenability for certain groups.

Contribution

It proves the equivalence of key approximation properties for these Lie groups and demonstrates weak amenability of GL(2,K) with constant 1 over any field K.

Findings

Equivalence of Haagerup property, weak amenability, and weak Haagerup property for these groups

Proof that GL(2,K) is weakly amenable with constant 1 for any field K

Advances understanding of approximation properties in discrete simple Lie groups

Abstract

In this paper we consider the class of connected simple Lie groups equipped with the discrete topology. We show that within this class of groups the following approximation properties are equivalent: (1) the Haagerup property; (2) weak amenability; (3) the weak Haagerup property. In order to obtain the above result we prove that the discrete group GL(2,K) is weakly amenable with constant 1 for any field K.