The isomorphism problem for cyclically pinched one-relator groups

Abstract

Let G = (ai ,..., ap,bl ,..., b,;wv = l), 1 5 p, 1 5 q, where 1 # w = w(al,... ,ap) and 1 # u = v(bl,. . .,bq). The group G is of great interest both for group theory and for topology (see [ 2,7,17] ). We are concerned with the one-relator presentations of G and the solution of the isomorphism problem for G. In [lo] we showed that if p = q = 2 and neither w nor v is a power of a primitive element in the free group (al, az;) or (bl, b2; ), respectively, then for each minimal generating system {xi, . . . , x4} of G there is a presentation of G with one defining relation, and G has only finitely many Nielsen equivalence classes of minimal generating systems. The proof in [lo] is based on an investigation of the subgroups of G of rank _< 4 and on the group-theoretical lemma (Lemma 2.1) in [lo] for which an extension to more than two generators is not available. Here we extend the result of [ lo] for minimal generating systems of G. We cannot adapt the arguments in [lo] in general but the knowledge that a system in G is a minimal generating system of G gives us some information about G which allows us to extend the results in [lo] to the following one. If w or u is not a power of a primitive element in the free group (al,. . . , a,; ) or (h,...,b,;), respectively, then for each generating system {xi,. . . ,x,+,} of G there is a presentation of G with one defining relation, and G has only finitely many Nielsen equivalence classes of minimal generating systems, and, hence, we can decide algorithmically in finitely many steps whether an arbitrary one-relator group is or is not isomorphic to G (see Theorem 3.4). This result stands in some contrast to the corresponding results in [4] and [ 161.