Quantifying metric approximations of discrete groups

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 = {(Gα, dα, kα, εα)}α∈I, where each group Gα is equipped with a bi-invariant metric dα and a dimension kα, for strictly positive real numbers εα such that infα εα > 0. Through the notion of a residually amenable profile that we introduce, our approach generalizes classical isoperimetric (aka Følner) 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.

Controlling LEF growth in some group extensions

“…For the sake of comparison with Theorem 1.3, an upper bound on the LEF growth of a regular wreath product of two LEF groups was proved in [1], and in [5] a complementary lower bound was given in a special case.…”

“…The lower bound in Theorem 1.4 uses different methods from the present paper, being based on the fact that one can recognize locally whether a family of finite centreless subgroups generates their direct product. We do not treat the case of the wreath product ∆ ≀ Ω Γ in detail here, although the methods of [1] and [5] would allow one to generalize the bounds of Theorem 1.4 to these groups for non-regular LEF actions, at least when ∆ is finite centreless.…”

“…This Section summarises much of the basic material which was treated in greater detail in [5][ Section 2]. There is also signicicant overlap with the relevant parts of [1].…”

“…The LEF growth function L S Γ of Γ is defined such that L S Γ (n) = |Q|, where Q is of minimal order among finite groups in which the (partial) multiplication on B S (n) may be modelled. LEF growth was introduced in [1] and independently in [4] (under a different name). In this paper we continue our study of the LEF growth function initiated in [5] and continued in [6].…”

Quantifying local embeddings into finite groups

“…There is now a well-established literature on quantitative aspects of residual finiteness [3,5,4], and a growing understanding of quantitative conjugacy stability [16,19] and subgroup separability [10]. The LEF growth function that is the subject of this paper was first defined in [1] (a closely related invariant was independently introduced in [7]), but our understanding of its behaviour has been hitherto undeveloped.…”