Frame Vector Group Representations and Amenability Properties

TL;DR

The paper introduces framenability, a relaxation of amenability defined via frame representations that admit almost invariant vectors. It proves a central equivalence (Theorem 'amen') connecting amenability to frame-type representations and almost invariant vectors, and then defines framenable groups as those admitting a weak-frame realization on a countable subset while preserving the trivial containment $1_G\prec\pi$. It shows framenability is a large, non-subgroup-hereditary class that nonetheless passes to certain constructions (surjections, products, semidirects, finite-index extensions) and can be studied via induced representations. A broad array of examples demonstrates framenability for many non-(T) groups (e.g., free groups, certain automorphism groups, lattices in $SL_2(\mathbb{R})$, BS$_{p,q}$, $B_n$, Thompson's $F$), while highlighting boundaries with property (T) and Haagerup property, and posing several open questions about the precise landscape of framenable groups.

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 $F$.

Theorems & Definitions (64)

• Definition 2.1
• Example 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Lemma 2.7
• proof
• Theorem 2.8
• Lemma 2.9