Weak amenability of Lie groups made discrete

TL;DR

This paper characterizes which connected Lie groups have all countable subgroups weakly amenable, provides criteria for semisimple Lie groups, and links weak amenability of Lie groups to their discrete counterparts.

Contribution

It offers a complete characterization of weak amenability in connected Lie groups and relates it to their discrete versions, advancing understanding of harmonic analysis on these groups.

Findings

Characterization of connected Lie groups with all countable subgroups weakly amenable

Criteria for weak amenability in connected semisimple Lie groups

Weak amenability of Lie groups when considered as discrete groups

Abstract

We completely characterize connected Lie groups all of whose countable subgroups are weakly amenable. We also provide a characterization of connected semisimple Lie groups that are weakly amenable. Finally, we show that a connected Lie group is weakly amenable if the group is weakly amenable as a discrete group.