Densities, submeasures and partitions of groups

TL;DR

This paper surveys partial solutions to a problem about group partitions, focusing on invariant densities and submeasures, and whether one cell in a partition can generate the entire group through conjugation with a small set.

Contribution

It introduces and discusses various approaches using invariant densities and submeasures to address a longstanding group partition problem.

Findings

Partial solutions identified for specific cases

Invariant densities help analyze group partitions

Conditions under which the problem holds are explored

Abstract

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.