Geometric Group Theory pp 169-195 | Cite as

# Solution of the Membership Problem for Magnus Subgroups in Certain One-Relator Free Products

## Abstract

Let\(
\mathcal{P} = \left\langle {X|\mathcal{R}} \right\rangle
\) be a presentation of a group *G*.
The most fundamental decision problem for \(
\mathcal{P}
\)is the Word Problem which asks for an algorithm to decide whether a given word on *X* represents the neutral element 1 of the group *G*. This problem is known to be unsolvable in general, however, for important classes of groups, it has been solved (see [L-S]). The membership problem (or generalized word problem) for
\(
\mathcal{P}
\) and a subgroup *H* of *G* asks for an algorithm to decide whether a given word on *X* represents an element of *H*. Thus, the word problem is the membership problem for the trivial subgroup, {1 }, of *G*. One of the classes of groups in which the word problem has been solved is the class of groups presented by a single defining relator *R* (one-relator groups). The word problem for one-relator groups was solved by W. Magnus [M]. An important ingredient of his proof, which is the basis for many classical and more recent results, is focusing on subgroups of *G* which are generated by proper subsets of *X*, which miss at least one letter from *X* that occurs in *R* ^{±1}. These subgroups are called *Magnus Subgroups*. In the course of the solution of the word problem, Magnus solved the Membership Problem for Magnus Subgroups. For more on the Membership Problem see [K-M-W] and references therein. See also [L-S].

## Keywords

Word Problem Free Product Double Point Primary Vertex Secondary Vertex## Preview

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