Quantifying metric approximations of discrete groups

2020/08/29 by Arzhantseva, Goulnara, Cherix, Pierre-Alain
#03C20 #20C99 #20E26 #20F69 #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO)

paper · doi:10.48550/arxiv.2008.12954

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 F=\(G_α, d_α, k_α, ε_α)\_{α∈ I,} where each group G_α is equipped with a bi-invariant metric d_α and a dimension k_α, for strictly positive real numbers ε_α such that inf_αε_α>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.

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