Sparse, thin and other subsets of groups. (English) Zbl 1186.20024

Lutsenko, Ie.; Protasov, I. V.

Int. J. Algebra Comput. 19, No. 4, 491-510 (2009).

All groups are assumed to be infinite. The authors consider eleven types of “small” subsets in groups. There are studied the following features: (i) relations between small subsets; (ii) existence of small sets of generators for groups; (iii) decomposition of small subsets; (iv) small subsets in amenable discrete groups.
A subset \(A\) of a group \(G\) is called ‘large’ provided there exists a finite subset \(F\) such that \(G=FA\). A subset \(A\) of a group \(G\) is called ‘small’ provided \(G\setminus FA\) is large for every finite subset \(F\). Furthermore, a subset \(A\) of a group \(G\) is called ‘sparse’ (‘\(k\)-sparse’, \(k\in\mathbb{N}\)) if for every infinite subset \(X\) of \(G\) there exists a finite subset \(\emptyset\neq F\subset X\) (a subset \(\emptyset\neq F\subset X\), \(|F|\leq k\)) such that \(\bigcap_{g\in F}gA\) is finite.
It is proved that every sparse subset of a group is small (Theorem 4.1). It is also proved that every group contains for each \(k\in\mathbb{N}\) a \((k+1)\)-sparse but not \(k\)-sparse subset (Theorem 1.1). Every group contains a sparse subset which is not \(k\)-sparse for each \(k\in\mathbb{N}\) (Theorem 1.2).
A subset \(A\) of a group \(G\) is called ‘thin’ (‘\(k\)-thin’, \(k\in\mathbb{N}\)) if \(gA\cap A\) is finite (\(|gA\cap A|\leq k\)) for every \(g\neq e\). Every group can be generated by some \(2\)-thin subset (Theorem 2.1). Every group contains a thin subset which is not \(k\)-thin for each \(k\in\mathbb{N}\) (Theorem 2.2). For every infinite thin subset \(A\) of a group \(G\), there exists a \(2\)-thin subset \(B\) such that \(A\cup B\) is not \(2\)-sparse (Theorem 3.3). If \(G\) is a non-torsion group or the set of cardinalities of its finite subgroups is unbounded, then \(G\) has a \(2\)-sparse subset which cannot be partitioned into finitely many thin subsets (Theorem 3.4).
In Section 5 the authors study “small” subsets in discrete amenable groups. It is proved that any sparse or weakly \(P\)-small subset \(A\) of an amenable group is an absolute zero-subset, i.e. \(\mu(A)=0\) for any Banach measure (Theorem 5.1). It is given a characterization of small subsets of an amenable group through Banach measures (Theorem 5.2).
Let \(G\) be any group and \(\beta G\) be its Stone-Čech compactification with the structure of a right-topological semigroup (this construction can be found, for instance, in [N. Hindman, Lect. Notes Math. 1401, 97-118 (1989; Zbl 0701.05060)]). It is established a duality between closed left ideals of \(\beta G\) and left invariant Boolean ideals of the Boolean algebra \(\mathcal P_G\) of all subsets of \(G\). The authors show how this duality can be used to translate some results of the paper in \(\beta G\)-language.

Reviewer: Mihail I. Ursul (Oradea)

MSC:

20F05	Generators, relations, and presentations of groups
20A05	Axiomatics and elementary properties of groups
22A05	Structure of general topological groups
54H11	Topological groups (topological aspects)
43A07	Means on groups, semigroups, etc.; amenable groups
06E15	Stone spaces (Boolean spaces) and related structures

Keywords:

large subsets; small subsets; thin subsets; sparse subsets of groups; Stone-Čech compactifications of discrete groups; Boolean group ideals; small sets of generators

Citations:

Zbl 0701.05060

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