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Let $F$ be a non-Abelian free group with basis $A$, $M$ and $N$ be the normal closures of sets $R_M$ and $R_N$ of words in the alphabet $A^{\pm 1}$. As is known, the group $F/[N, N]$ is torsion-free, but, in general, torsion in $F/[M, N]$…

Given a free product of groups $G = {\large *}_{j \in J} A_j$ and a natural number $n$, what is the minimal possible commutator length of an element $g^n \in G$ not conjugate to elements of the free factors? We give an exhaustive answer to…

For each $n$ we construct examples of finitely presented $C'(1/6)$ small cancellation groups that do not act properly on any $n$-dimensional CAT(0) cube complex.

The notion of simple compact quantum group is introduced. As non-trivial (noncommutative and noncocommutative) examples, the following families of compact quantum groups are shown to be simple: (a) The universal quantum groups $B_u(Q)$ for…

Let $A$ be a $C^{*}$ algebra and $T: A\rightarrow A$ be a linear map which satisfies the functional equation $\begin{cases}T(x)T(y)=T^{2}(xy)\\T(x^{*})=T(x)^{*} \end{cases}$ We prove that under each of the following conditions, $T$ must be…

The set of automorphisms of a one-dimensional \shift $(X, \sigma)$ forms a countable, but often very complicated, group. For zero entropy shifts, it has recently been shown that the automorphism group is more tame. We provide the first…

For a cardinal lambda<lambda_{omega_1} we give a ccc forcing notion P which forces that for some Borel subset B of the Cantor space (1) there a sequence (eta_alpha:alpha<lambda) of distinct elements such that |(eta_alpha+B) cap…

We associate a graph to a possible non-zero zero-divisor in the group algebra of a torsion-free group.

We say that a countable discrete group $\Gamma$ satisfies the invariant von Neumann subalgebras rigidity (ISR) property if every $\Gamma$- invariant von Neumann subalgebra $\mathcal{M}$ in $L(\Gamma)$ is of the form $L(\Lambda)$ for some…

Using a suitably noncommutative flat matrix model, it is shown that the quantum permutation group has free orbitals: that is, a monomial in the generators of the algebra of functions can be zero for trivial reasons only. It is shown that…

Let $H$ be a closed normal subgroup of a compact Lie group $G$ such that $G/H$ is connected. This paper provides a necessary and sufficient condition for every complex representation of $H$ to be extendible to $G$, and also for every…

Let R be any ring (with 1), \Gamma a group and R\Gamma the corresponding group ring. Let Ext_{R\Gamma}^{*}(M,M) be the cohomology ring associated to the R\Gamma-module M. Let H be a subgroup of finite index of \Gamma. The following is a…

Let $A\subset B$ be an extension of commutative reduced rings and $M\subset N$ an extension of positive commutative cancellative torsion-free monoids. We prove that $A$ is subintegrally closed in $B$ and $M$ is subintegrally closed in $N$…

In this article we give a sufficient and necessary condition to determine wether or not an element of the free group induces a non-trivial element of the free Burnside group of sufficiently large odd exponent. This criterion can be stated…

Given a group $X$ we study the algebraic structure of its superextension $\lambda(X)$. This is a right-topological semigroup consisting of all maximal linked systems on $X$ endowed with the operation $$\mathcal A\circ\mathcal B=\{C\subset…

Let G be an abelian group and let lambda be the smallest rank of any group whose direct sum with a free group is isomorphic to G. If lambda is uncountable, then G has lambda pairwise disjoint, non-free subgroups. There is an example where…

Let $G$ be a generalized Baumslag-Solitar group and $\mathcal{C}$ be a class of groups containing at least one non-unit group and closed under taking subgroups, extensions, and Cartesian products of the form $\prod_{y \in Y}X_{y}$, where…

The concept of a k-translatable groupoid is explored in depth. Some properties of idempotent k-translatable groupoids, left cancellative k-translatable groupoids and left unitary k-translatable groupoids are proved. Necessary and sufficient…

Let $A$ be a separable simple exact ${\cal Z}$-stable $C^*$-algebra. We show that the unitay group of ${\tilde A}$ has the cancellation property. If $A$ has continuous scale, the Cuntz semigroup of $\tilde A$ has the strict comparison…

It is proved that the derived subgroup of a finite group is nilpotent if and only if $|ab|\ge |a||b|$ for all primary commutators $a$ and $b$ of coprime orders.