Small subsets of groups

Abstract

MSC: 54E15 22A15 20F69 03E05Keywords:κ-large subset κ-small subset κ-thin subset Stone-Čech compactification of a discrete group Ballean Ideal Given an infinite group G and an infinite cardinal κ |G|, we say that a. The subject of the paper is the family S κ of all κ-small subsets. We describe the left ideal of the right topological semigroup βG determined by S κ . We study interrelations between κ-small and other (P κ -small and κ-thin) subsets of groups, and prove that G can be generated by some 2-thin subsets. We partition G in countable many subsets which are κ-small for each κ ω. We show that [G] <κ is dual to S κ provided that either κ is regular and κ = |G|, or G is Abelian and κ is a limit cardinal, or G is a divisible Abelian group.

Preservation and reflection of size properties of balleans

“…The monograph [23] contains another concept of size, namely thickness. Since then the size and various cardinal invariants of balleans related to size have been intensively studied by the Ukrainian school [11,12,14,16,17,20,21,22]. The survey [18] is very helpful to get a better idea on the topic.…”

“…A more general version can be obtained by taking an infinite cardinal κ and the ideal I κ = [G] <κ of subsets of cardinality < κ of G. Obviously, I κ is a group ideal, so gives rise to a group ballean B Iκ . This ballean, and especially the case κ = |G| has been intensively studied [11,12,14,16,17,20,21,22]. (d) It is possible to nicely unify items (b) and (c) in the case of a countably infinite group G. It was proved by Smith [25] that every such group G admits a left invariant proper metric d (i.e., such that d(gx, gy) = d(x, y), for every g, x, y ∈ G, and whose closed balls are compact) and every pair of such metrics are coarsely equivalent (actually asymorphic).…”