Densities, submeasures and partitions of groups

In 1995 in Kourovka notebook the second author asked the following problem: it is true that for each partition $G=A_1\cup\dots\cup A_n$ of a group $G$ there is a cell $A_i$ of the partition such that $G=FA_iA_i^{-1}$ for some set $F\subset G$ of cardinality $|F|\le n$? In this paper we survey several partial solutions of this problem, in particular those involving certain canonical invariant densities and submeasures on groups.Comment: 14 pages (this is an update of the preceding version

Densities, submeasures and partitions of groups

In 1995 in Kourovka notebook the second author asked the following problem: is it true that for each partition G=A₁ ∪ ⋯ ∪ An of a group G there is a cell Ai of the partition such that G = FAiA⁻¹i for some set F ⊂ G of cardinality |F |≤ n? In this paper we survey several partial solutions of this problem, in particular those involving certain canonical invariant densities and submeasures on groups. In particular, we show that for any partition G = A₁ ∪ ⋯ ∪ An of a group G there are cells Ai, Aj of the partition such that G = FAjA⁻¹j for some finite set F ⊂ G of cardinality |F| ≤ max₀<k≤n ∑ⁿ⁻kp₌₀kp ≤ n!; G = F ⋅ ⋃x∈ExAiA⁻¹ix⁻¹ for some finite sets F, E ⊂ G with |F| ≤ n; G = FAiA⁻¹iAi for some finite set F ⊂ G of cardinality |F| ≤ n; the set (AiA⁻¹i)⁴ⁿ⁻¹ is a subgroup of index ≤ n in G. The last three statements are derived from the corresponding density results

Ultracompanions of subsets of a group

Let $G$ be a group, $\beta G$ is the Stone-$\check{C}$ech compactification of $\beta G$ endowed with the structure of a right topological semigroup, $G^*=\beta G\setminus G$. Given any subset $A$ of $G$ and $p\in G^*$, we define the $p$-companion \vt_p(A)=A^*\cap Gp of $A$, and characterize the subsets with finite and discrete ultracompanions

Recent progress in subset combinatorics of groups

We systematize and analyze some results obtained in Subset Combinatorics of $G$ groups after publications the previous surveys [1-4]. The main topics: the dynamical and descriptive characterizations of subsets of a group relatively their combinatorial size, Ramsey-product subsets in connection with some general concept of recurrence in $G$-spaces, new ideals in the Boolean algebra $\mathcal{P}_{G}$ of all subsets of a group $G$ and in the Stone-$\check{C}$ech compactification $\beta G$ of $G$ , the combinatorial derivation.Comment: Large, small, thin, thick, sparse and scattered subsets of groups; descriptive complexity; Boolean algebra of subsets of a group; Stone-$\check{C}$ech compactification; ultracompanion; Ramsey-product subset of a group; recurrence; combinatorial derivation. arXiv admin note: text overlap with arXiv:1704.0249

Fraisse Limits, Ramsey Theory, and Topological Dynamics of Automorphism Groups

We study in this paper some connections between the Fraisse theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures.Comment: 73 pages, LaTeX 2e, to appear in Geom. Funct. Ana

Algebraic and Topological Properties of Unitary Groups of II_1 Factors

The thesis is concerned with group theoretical properties of unitary groups, mainly of II_1 factors. The author gives a new and elementary proof of an result on extreme amenability, defines the bounded normal generation property and invariant automatic continuity property and proves these for various unitary groups of functional analytic types