Intersections of Magnus subgroups and embedding theorems for cyclically presented groups

Intersections of conjugates of Magnus subgroups of one-relator groups

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 distinct Magnus subgroups M and N of a one-relator group, with free bases S and T are given, then the intersection of M and N is either the free subgroup P generated by the intersection of S and T or the free product of P with an infinite cyclic group. The main result of this article is that if M and N are Magnus subgroups (not necessarily distinct) of a one-rela…

Magnus subgroups of one-relator surface groups

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 the intersections of such subgroups and their conjugates, analogous to results of Bagherzadeh, Brodskii, and Collins in classical one-relator group theory.

One-relator quotients of Partially Commutative Groups

We generalise a key result of one-relator group theory, namely Magnus's Freiheitssatz, to partially commutative groups, under sufficiently strong conditions on the relator. The main theorem shows that under our conditions, on an element $r$ of a partially commutative group $\mathbb{G}$, certain Magnus subgroups embed in the quotient $G=\mathbb{G}/N(r)$; that if $r=s^n$ has root $s$ in $\mathbb{G}$ then the order of $s$ in $G$ is $n$, and under slightly stronger conditions that the word problem of $G$ is decidable. We also give conditions under which the question of which Magnus subgroups of $\…

Freiheitssatz for amalgamated products of free groups over maximal cyclic subgroups

In 1930, Wilhelm Magnus introduced the so-called Freiheitssatz: Let $F$ be a free group with basis $\mathcal{X}$ and let $r$ be a cyclically reduced element of $F$ which contains a basis element $x \in \mathcal{X}$, then every non-trivial element of the normal closure of $r$ in $F$ contains the basis element $x$. Equivalently, the subgroup freely generated by $\mathcal{X} \backslash \{x\}$ embeds canonically into the quotient group $F / \langle \! \langle r \rangle \! \rangle_{F}$. In this article, we want to introduce a Freiheitssatz for amalgamated products $G=A \ast_{U} B$ of free groups $A…

推广Collins定理

Generalising Collins' Theorem

本文将Collins关于单关系群中Magnus子群共轭交集的结果推广至局部可指示群的单关系积语境中。

双生成一关系群

Hyperbolic one-relator groups

本文提出原始扩展群的概念，通过刻画超几何原始扩展群来解决超几何一关系群的分类问题，并利用Christoffel词和一关系塔机制进行证明。