Related papers: On compactifications and product-free sets

We prove that a group obtained as a quotient of the free product of finitely many cubulable groups by a finite set of relators satisfying the classical $C'(1/6)$--small cancellation condition is cubulable. This yields a new large class of…

In a previous paper [1] [MR4101040], we initiated a systematic study of semihypergroups and had a thorough discussion about some important analytic and algebraic objects associated to this class of objects. In this paper, we investigate…

We say that a finite group $G$ satisfies the independence property if, for every pair of distinct elements $x$ and $y$ of $G$, either $\{x,y\}$ is contained in a minimal generating set for $G$ or one of $x$ and $y$ is a power of the other.…

For infinite products of compact spaces, Tychonoff's theorem asserts that their product is compact, in the product topology. Tychonoff's theorem is shown to be equivalent to the axiom of choice. In this paper, we show that any countable…

We consider factorizations $G=XY$ where $G$ is a general group, $X$ and $Y$ are normal subsets of $G$ and any $g\in G$ has a unique representation $g=xy$ with $x\in X$ and $y\in Y$. This definition coincides with the customary and…

Let G be a group of the form G_1* ... *G_n, the free product of n subgroups, and let M be a ZG-module of the form $\bigoplus_{i=1}^n M_i \otimes_{\mathbb{Z}G_i} \mathbb{Z}G$. We shall give formulae in various situations for $d_{ZG}(M)$, the…

Given a group $G$ and a subset $X \subset G$, an element $g \in G$ is called quasi-positive if it is equal to a product of conjugates of elements in the semigroup generated by $X$. This notion is important in the context of braid groups,…

In a series of papers starting in [Sel01] and culminating in [Sel07], Z. Sela proved that free groups, and more generally torsion-free hyperbolic groups, have a stable first-order theory. The question of the stability of the free product of…

A set of elements of a finite abelian group is called sum-free if it contains no Schur triple, i.e., no triple of elements $x,y,z$ with $x+y=z$. The study of how large the largest sum-free subset of a given abelian group is had started more…

Let $G$ be a simple algebraic group over an algebraically closed field $\Bbbk$ of positive characteristic. We consider the questions of when the tensor product of two simple $G$-modules is multiplicity free or completely reducible. We…

Let $G$ be a unique product group, i.e., for any two finite subsets $A$ and $B$ of $G$ there exists $x\in G$ which can be uniquely expressed as a product of an element of $A$ and an element of $B$. We prove that, if $C$ is a finite subset…

A classification of the ways in which an element of a free group can be expressed as a product of commutators or as a product of squares is given. This is then applied to some particular classes of elements. Finally, a question about…

We consider the question of when sets definable in first-order expansions of groups contain the product of two infinite sets (we refer to this as the "productset property"). We first show that the productset property holds for any definable…

We give an infinite family of torsion-free groups that do not satisfy the unique product property. For these examples, we also show that each group contains arbitrarily large sets whose square has no uniquely represented element.

A group is properly 3-realizable if it is the fundamental group of a compact polyhedron whose universal covering is proper homotopically equivalent to some 3-manifold. We prove that when such a group is also quasi-simply filtered then it…

Three non-empty subsets $S,T,U$ of a group $G$ are said to satisfy the triple product property (TPP) if, for elements $s,s' \in S$, and $t,t' \in T$, and $u,u' \in U$, the equation $s's^{-1}t't^{-1}u'u^{-1}=1$ holds if and only if $s = s'$,…

We prove that product-free sets of the free group over a finite alphabet have maximum density $1/2$ with respect to the natural measure that assigns total weight one to each set of irreducible words of a given size. This confirms a…

We discuss several questions concerning sum-free sets in groups, raised by Erd\H{o}s in his survey "Extremal problems in number theory" (Proceedings of the Symp. Pure Math. VIII AMS) published in 1965. Among other things, we give a…

Let $G$ be a finite group and let $p$ be a prime. We continue the search for generic constructions of free products and free monoids in the unit group $\mathcal{U}(\mathbb{Z}G)$ of the integral group ring $\mathbb{Z}G$. For a nilpotent…

Let Q be a (non-unital) simple ring. A nonempty subset S of Q is said to have zero product if S^2=0. We classify all maximal zero product subsets of Q. We also describe the relationship between the maximal zero product subsets of Q and the…