An Application of Word Combinatorics to Decision Problems in Group Theory

In this talk we develop a graph theoretical test on graphs corresponding to subgroups of one-relator groups with small cancellation condition which, if successful, implies that the subgroup under consideration has solvable membership problem with a simple solution. The proof of the solvability of the membership problem relies on word combinatorics in an essential way. One way to describe a group G is by giving a set S of elements of G and a set of defining relations among them such that S generates G, in the sense that every element of G is the product of elements of S or their inverses and every relation among elements of S is a consequence of the given relations. More formally, let X be a set in one to one correspondence, say θ0, with S and let F = F (X) be the free group, freely generated by X. Then θ0 extends uniquely to a homomorphism θ from F (X) onto G. Therefore, if N = Ker θ, then G ∼ =