Related papers: One-relator quotients of Partially Commutative Gro…
In this paper, we generalise Magnus' Freiheitssatz and solution to the word problem for one-relator groups by considering one relator quotients of certain classes of right-angled Artin groups and graph products of locally indicable…
In this version small mistakes are corrected and the exposition is changed as suggested by the referee (to appear in Canadian Journal of Mathematics). The first main result of the paper is a criterion for a partially commutative group $\GG$…
This is an English translation of three articles, originally written in German, by Wilhelm Magnus (1907--1990). The articles are from 1930, 1931, and 1932, respectively, and were the first articles published on one-relator group theory. The…
In this paper we study a group G which is the quotient of a free product of groups by the normal closure of a word that is contained in a in a subgroup which has the form of a generalised triangle group. We use known properties of…
We will say that a group G possesses the Magnus property if for any two elements u,v in G with the same normal closure, u is conjugate to v or v^{-1}. We prove that some one-relator groups, including the fundamental groups of closed…
A one-relator surface group is the quotient of an orientable surface group by the normal closure of a single relator. A Magnus subgroup is the fundamental group of a suitable incompressible sub-surface. A number of results are proved about…
In this paper we prove several results regarding decidability of the membership problem for certain submonoids in amalgamated free products and HNN extensions of groups. These general results are then applied to solve the prefix membership…
In the theory of one-relator groups, Magnus subgroups, which are free subgroups obtained by omitting a generator that occurs in the given relator, play an essential structural role. In a previous article, the author proved that if two…
A subgroup $H$ of a group $G$ is confined if the $G$-orbit of $H$ under conjugation is bounded away from the trivial subgroup in the space $\operatorname{Sub}(G)$ of subgroups of $G$. We prove a commutator lemma for confined subgroups. For…
The main goal of this note is to suggest an algebraic approach to the quasi-isometric classification of partially commutative groups (alias right-angled Artin groups). More precisely, we conjecture that if the partially commutative groups…
Three versions of the Freiheitssatz are proved in the context of one-relator quotients of limit groups, where the latter are equipped with 1-acylindrical splittings over cyclic subgroups. These are natural extensions of previously published…
Let $m,n$ be positive integers and $w$ a multilinear commutator word. Assume that $G$ is a finite group having subgroups $G_1,\ldots,G_m$ whose union contains all $w$-values in $G$. Assume further that all elements of the subgroups…
A one-relator group is a group $G_r$ that admits a presentation $\langle S \mid r \rangle$ with a single relation $r$. One-relator groups form a rich classically studied class of groups in Geometric Group Theory. If $r \in F(S)'$, the…
Given a group $G$, we write $g^G$ for the conjugacy class of $G$ containing the element $g$. A theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the commutator subgroup…
Motivated by its applications to the word problem for one-relator inverse monoids, via results of Ivanov, Margolis, and Meakin (2001), we prove several decidability and undecidability results about the submonoid membership problem in…
The first part of the paper centers in the study of embeddability between partially commutative groups. In [KK], for a finite simplicial graph $\Gamma$, the authors introduce an infinite, locally infinite graph $\Gamma^e$, called the…
The known facts about solvability of equations over groups are considered from a more general point of view. A generalized version of the theorem about solvability of unimodular equations over torsion-free groups is proved. In a special…
A Magnus subgroup of a one-relator group is the free subgroup freely generated by a proper subset of the generators. Two such subgroups can intersect in the obvious way or in a larger, exceptional way. The condition of non-exceptional…
We consider the class of finitely generated groups whose relators are powers of commutators of the generators. This class contains as a small subclass graph groups (also called RAAGs), namely if all powers are one. Graph groups are the only…
Magnus' Freiheitssatz states that if a group is defined by a presentation with $m$ generators and a single relator containing the last generating letter, then the first $m-1$ letters freely generate a free subgroup. We study an analogue of…