SOLUTION OF THE MEMBERSHIP PROBLEM FOR CERTAIN RATIONAL SUBSETS OF ONE-RELATOR GROUPS WITH A SMALL CANCELLATION CONDITION
Arye Juhász · International Journal of Algebra and Computation · 2012
Let n ≥ 2 be a natural number, X = { x1,…, xn} and let F be the free group, freely generated by X. Let R be a cyclically reduced word in F such that its symmetric closure [Formula: see text] in F satisfies the small cancellation condition C′(1/5) & T(4). Let G be the group presented by [Formula: see text]. A Magnus subsemigroup of G is any subsemigroup of G generated by at most 2n - 1 elements of [Formula: see text]. In this paper we solve the Membership Problem for rational subsets of G which are contained in a Magnus subsemigroup of G, provided that [Formula: see text] satisfies certain combinatorial conditions. We use small cancellation theory with word combinatorics.