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 framenable. We show that this class has some permanence properties, stands in contrast with property (T), and contains, for example, all free groups $F_n$, $\text{Aut}\left(F_2\right)$ and $\text{Aut}\left(F_3\right)$, all (countable) lattices of $SL(2,R)$, the Baumslag–Solitar groups $B{S}_{p,q}$, the braid groups $B_n$, and Thompson’s group $F$.