Related papers: Limit groups and groups acting freely on $\bbR^n$-…

We study the positive theory of groups acting on trees and show that under the presence of weak small cancellation elements, the positive theory of the group is trivial, i.e. coincides with the positive theory of a non-abelian free group.…

We study natural linear representations of self-similar groups over finite fields. In particular, we show that if the group is generated by a finite automaton, then obtained matrices are automatic. This shows a new relation between two…

We provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA). The additional sufficient conditions are all met by finite…

We show that every finite abelian group $G$ occurs as the group of rational points of an ordinary abelian variety over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_5$. We produce partial results for abelian varieties over a general finite…

Let G be a finite group. It has recently been proved that every nontrivial element of G is contained in a generating set of minimal size if and only if all proper quotients of G require fewer generators than G. It is natural to ask which…

We introduce the property of having good subgroups for actions of countable discrete groups on compact metrizable spaces, and show that it implies comparison when the acting group is amenable. As a consequence, free actions on…

We prove that any finitely generated subgroup of the plane Cremona group consisting only of algebraic elements is of bounded degree. This follows from a more general result on `decent' actions on infinite direct sums. We apply our results…

We prove that a self-similar free abelian group has finite rank. We apply the result to self-similar wreath products of abelian groups $G=BwrX$. We show that if $X$ is torsion-free, then $B$ is torsion of finite exponent. Furthemore, we…

Suppose a group $G$ is quasi-isometric to a free product of a finite set $S$ of finitely generated abelian groups; let $S'$ denote the set of ranks of the free abelian parts of the groups in $S$. Then $G$ is commensurable with the free…

Elementarily free groups are the finitely generated groups with the same elementary theory as free groups. We prove that elementarily free groups are subgroup separable, answering a question of Zlil Sela.

Finitely generated (non-abelian) free metabelian pro-p groups, and wreath products of f.g. free abelian pro-p groups, are all finitely axiomatizable in the class of all profinite groups.

Balogh, Liu, Sharifzadeh and Treglown [Journal of the European Mathematical Society, 2018] recently gave a sharp count on the number of maximal sum-free subsets of $\{1, \dots, n\}$, thereby answering a question of Cameron and Erd\H{o}s. In…

We show that certain orderable groups admit no isolated left orders. The groups we consider are cyclic amalgamations of a free group with a general orderable group, the HNN extensions of free groups over cyclic subgroups, and a particular…

The vertices of the Cayley graph of a finitely generated semigroup form a set of sites which can be labeled by elements of a finite alphabet in a manner governed by a nonnegative real interaction matrix, respecting nearest neighbor…

We give two examples of a finitely generated subgroup of a free group and a subset, closed in the profinite topology of a free group, such that their product is not closed in the profinite topology of a free group.

The existence of an infinite simple boundedly generated 2-generated group and the existence of a boundedly simple 2-generated group containing a free non-cyclic subgroup are proved.

We prove that a variety of algebras whose finitely generated members are free must be definitionally equivalent to the variety of sets, the variety of pointed sets, a variety of vector spaces over a division ring, or a variety of affine…

We classify finite groups that can act by automorphisms and birational automorphisms on non-trivial Severi-Brauer surfaces over fields of characteristic zero.

Bestvina, Feighn and Handel proved that every subgroup of the outer automorphism group, $\textrm{Out}(F_n)$, of the free group of rank $n$ is either virtually finitely generated abelian or contains a nonabelian free group. In this note we…

We study the automorphism groups attached to a free algebra with multiple, possibly infinitely many, composition laws. As an application, we prove that the automorphism group of finitely generated vertex algebras over noetherian rings are…