Frame Vector Group Representations and Amenability Properties

Abstract

We provide a new characterization of amenability for countable groups, based on frame representations admitting almost invariant vectors. By relaxing the frame inequalities, thereby weakening amenability, we obtain a large class of countable groups which we call {\it framenable}. We show that this class has some permanence properties, stands in contrast with property (T), and contains, for example, all free groups ${\mathbb{F}}_n$, $\text{Aut}({\mathbb{F}}_2)$ and $\text{Aut}({\mathbb{F}}_3)$, all (countable) lattices of $SL(2,\mathbb{R})$, the Baumslag-Solitar groups $B{S}_{p,q}$, the braid groups $B_n$, and Thompson's group $F$.

v3: thanks to Professor Jolissaint for valuable comments