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Given a family $F$ of subsets of a group $G$ we describe the structure of its thin-completion $tau^*(F)$, which is the smallest thin-complete family that contains $I$. A family $F$ of subsets of $G$ is called thin-complete if each $F$-thin subset of $G$ belongs to $F$. A subset $A$ of $G$ is called $F$-thin if for any distinct points $x,y$ of $G$ the intersection $xAcap yA$ belongs to the family $F$. We prove that the thin-completion of an ideal in an ideal. If $G$ is a countable non-torsion group, then the thin-completion $tau^*(F_G)$ of the ideal $F_G$ of finite subsets of $G$ is coanalytic but not Borel in the power-set $P_G$ of $G$.
A subset $D$ of an Abelian group is $decomposable$ if $emptyset e Dsubset D+D$. In the paper we give partial answer to an open problem asking whether every finite decomposable subset $D$ of an Abelian group contains a non-empty subset $Zsubset D$ wit
In this paper we study the generic, i.e., typical, behavior of finitely generated subgroups of hyperbolic groups and also the generic behavior of the word problem for amenable groups. We show that a random set of elements of a nonelementary word hype
A finite group $G$ is called a Schur group, if any Schur ring over $G$ is associated in a natural way with a subgroup of $Sym(G)$ that contains all right translations. Recently, the authors have completely identified the cyclic Schur groups. In this
We study group algebras for compact groups in the category of real and complex weakly complete vector spaces. We also show that the group algebra is a quotient of the weakly complete universal enveloping algebra of the Lie algebra of the compact grou
We show that for every subset X of a closed surface M^2 and every basepoint x_0, the natural homomorphism from the fundamental group to the first shape homotopy group, is injective. In particular, if X is a proper compact subset of M^2, then pi_1(X,x