# Frame Vector Group Representations and Amenability Properties

## TL;DR

This paper introduces a new class called framenable groups, characterized by frame representations with almost invariant vectors, broadening the understanding of amenability and contrasting with property (T).

## Contribution

It defines framenable groups via relaxed frame inequalities, expanding the class of groups known to exhibit amenability-like properties.

## Key findings

- Framenable groups include free groups, automorphism groups of free groups, lattices in SL(2,R), Baumslag-Solitar groups, braid groups, and Thompson's group F.
- The class of framenable groups has permanence properties and differs from property (T).
- Provides a new characterization of amenability based on frame representations with almost invariant vectors.

## 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 $\mathbb{F}_n$, $\textup{Aut}(\mathbb{F}_2)$ and $\textup{Aut}(\mathbb{F}_3)$, all (countable) lattices of $SL(2,\mathbb{R})$, the Baumslag-Solitar groups $BS_{p,q}$, the braid groups $B_n$, and Thompson's group $F$.