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].

This article originates from the Barcelona conference.