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We consider absolutely free nonassociative algebras and, more generally, absolutely free algebras with (maybe infinitely) many multilinear operations. Such algebras are described in terms of labeled reduced planar rooted trees. This allows…

We examine the relationship between finitely and infinitely generated relatively hyperbolic groups, in two different contexts. First, we elaborate on a remark from math.GR/0601311, which states that the version of Dehn filling in relatively…

We investigate the possible structures imposed on a finite group by its possession of an automorphism sending a large fraction of the group elements to their cubes, the philosophy being that this should force the group to be, in some sense,…

We study Frobenius extensions which are free-filtered by a totally ordered, finitely generated abelian group, and their free-graded counterparts. First we show that the Frobenius property passes up from a free-graded extension to a…

In [J. Algebra 452 (2016), 372-389], we characterise when the sequence of free subgroup numbers of a finitely generated virtually free group $\Gamma$ is ultimately periodic modulo a given prime power. Here, we show that, in the remaining…

We present a sharp upper bound for the number of generators of a finite group in terms of the ratio between the order and the exponent.

We develop a version of Freiman's theorem for a class of non-abelian groups, which includes finite nilpotent, supersolvable and solvable A-groups. To do this we have to replace the small doubling hypothesis with a stronger relative…

An approach to identify the normal subgroups determined by ideals in free group rings with the help of the derived functors of non-additive functors is explored. A similar approach, i.e., via derived functors, for computing limits of…

A complex number $\lambda$ is said to be non-free if the subgroup of $SL(2,\bc)$ generated by $$X=\begin{pmatrix} 1& 1\\ 0 & 1 \end{pmatrix} \,\, \text{and}\,\,\,Y_{\lambda}=\begin{pmatrix} 1& 0\\ \lambda & 1 \end{pmatrix}$$ is not a free…

The isomorphism problem for infinite finitely presented groups is probably the hardest among standard algorithmic problems in group theory. Classes of groups where it has been completely solved are nilpotent groups, hyperbolic groups, and…

In this note we study the logarithmic derivation module of a non-free arrangement. We prove a generalized addition theorem for all arrangements. This addition theorem allows us to find various relationships between non-free arrangements,…

Given a finitely generated linear group $G$ over $\mathbb{Q}$, we construct a simple group $\Gamma$ that has the same finiteness properties as $G$ and admits $G$ as a quasi-retract. As an application, we construct a simple group of type…

We introduce the concept of quantifying the extent to which a finitely generated group is residually finite. The quantification is carried out for some examples including free groups, the first Grigorchuk group, finitely generated nilpotent…

We like to build Abelian groups (or R-modules) which on the one hand are quite free, say $\aleph_{\omega + 1}$-free, and on the other hand, are complicated in suitable sense. We choose as our test problem having no non-trivial homomorphism…

Let $F$ be an algebraic extension of the rational numbers and $E$ an elliptic curve defined over some number field contained in $F$. The absolute logarithmic Weil height, respectively the N\'eron-Tate height, induces a norm on $F^*$ modulo…

Let $G$ be a finite group, $A$ a finite abelian group. Each homomorphism $\phi:G\to A\wr S_n$ induces a homomorphism $\bar{\phi}:G\to A$ in a natural way. We show that as $\phi$ is chosen randomly, then the distribution of $\bar{\phi}$ is…

All groups have 2 generators. For every prime power q, the Generalized Burnside Theorem (Theorem GB) produces an infinite number of solvable groups, Some, such as groups of a prime power exponent, have only elements of finite order and are…

We construct examples of localizations in the category of groups which take the Mathieu group $M_{11}$ to groups of arbitrarily large cardinality which are ``abelian up to finitely many generators''. The paper is part of a broader study on…

It is well known there is no finitely generated abelian group which has the $R_\infty$ property. We will show that also many non-finitely generated abelian groups do not have the $R_\infty$ property, but this does not hold for all of them.…

A result of the author shows that the behavior of Gowers norms on bounded exponent abelian groups is connected to finite nilspaces. Motivated by this, we investigate the structure of finite nilspaces. As an application we prove inverse…