Related papers: Finding non-trivial elements and splittings in gro…
We consider the class of finitely generated groups which have a normal form computable in logspace. We prove that the class of such groups is closed under finite extensions, finite index subgroups, direct products, wreath products, and also…
It is well known that every finite simple group can be generated by two elements and this leads to a wide range of problems that have been the focus of intensive research in recent years. In this survey article we discuss some of the…
We study systematically groups whose marked finite quotients form a recursive set. We give several definitions, and prove basic properties of this class of groups, and in particular emphasize the link between the growth of the depth…
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 show that any group $G$ is contained in some sharply 2-transitive group $\mathcal{G}$ without a non-trivial abelian normal subgroup. This answers a long-standing open question. The involutions in the groups $\mathcal{G}$ that we…
In this paper we prove the following result. Let $G$ be a simply connected simple linear algebraic group of exceptional Lie type over an algebraically closed field $F$ of characteristic $p\geq 0$, and let $u\in G$ be a nonidentity unipotent…
We consider finitely presented,residually finite groups $G$ and finitely generated normal subgroups $A$ such that the inclusion $A\hookrightarrow G$ induces an isomorphism from the profinite completion of $A$ to a direct factor of the…
The aim of this paper is to compare and contrast the class of residually finite groups with the class of equationally Noetherian groups - groups over which every system of coefficient-free equations is equivalent to a finite subsystem. It…
We construct a finitely presented (two-sided) totally orderable group with insoluble word problem.
It is well known that every finite simple group has a generating pair. Moreover, Guralnick and Kantor proved that every finite simple group has the stronger property, known as $\frac{3}{2}$-generation, that every nontrivial element is…
Computer based techniques for recognizing finitely presented groups are quite powerful. Tools available for this purpose are outlined. They are available both in stand-alone programs and in more comprehensive systems. A general…
Semigroups generated by topological operations such as closure, interior or boundary are considered. It is noted that some of these semigroups are in general finite and noncommutative. The problem is formulated whether they are always…
We prove that every non-abelian finite simple group is generated by an involution and an element of prime order.
This paper introduces the concept of a generating set for stochastic matrices -- a subset of matrices whose repeated composition generates the entire set. Understanding such generating sets requires specifying the "indivisible elements" and…
We construct a 2-generated 2-related group without non-trivial finite factors. That answers a question of J. Button.
The celebrated Stallings' decomposition theorem states that the splitting of a finite index subgroup $H$ of a finitely generated group $G$ as an amalgamated free product or an HNN-extension over a finite group implies the same for $G$. We…
The aim of the article is to show that there are many finite extensions of arithmetic groups which are not residually finite. Suppose $G$ is a simple algebraic group over the rational numbers satisfying both strong approximation, and the…
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…
We present novel constructions concerning the homology of finitely generated groups. Each construction draws on ideas of Gilbert Baumslag. There is a finitely presented acyclic group $U$ such that $U$ has no proper subgroups of finite index…
We prove that every finitely generated, residually finite group $G$ embeds into a finitely generated perfect branch group $\Gamma$ such that many properties of $G$ are preserved under this embedding. Among those are the properties of being…