Quantifying metric approximations of discrete groups

TL;DR

This paper introduces a new profile function to quantify metric approximations of finitely generated groups, generalizing classical profiles and encompassing various modern approximation concepts like hyperlinear and sofic groups.

Contribution

It defines a residually amenable profile that unifies and extends existing notions of group approximation profiles, covering a broad range of approximation types.

Findings

Introduces a systematic profile function for metric group approximations.

Generalizes classical isoperimetric profiles to broader approximation contexts.

Encompasses hyperlinear, sofic, and other modern group approximation frameworks.

Abstract

We introduce and systematically study a profile function whose asymptotic behavior quantifies the dimension or the size of a metric approximation of a finitely generated group $G$ by a family of groups $\mathcal{F}=\{(G_{\alpha}, d_{\alpha}, k_{\alpha}, \varepsilon_{\alpha })\}_{\alpha\in I, }$ where each group $G_\alpha$ is equipped with a bi-invariant metric $d_{\alpha}$ and a dimension $k_{\alpha}$, for strictly positive real numbers $\varepsilon_{\alpha }$ such that $\inf_{\alpha }\varepsilon_{\alpha }>0$. Through the notion of a residually amenable profile that we introduce, our approach generalizes classical isoperimetric (aka Folner) profiles of amenable groups and recently introduced functions quantifying residually finite groups. Our viewpoint is much more general and covers hyperlinear and sofic approximations as well as many other metric approximations such as weakly sofic, weakly hyperlinear, and linear sofic approximations.