A CLASS OF ONE-RELATOR GROUPS WITH CENTER

Let the group H have presentation <a1, ..., a(m); a1p1 = a2p1, ..., a(m-1)p(m-1) = a(m)p(m-1)> where m greater-than-or-equal-to 3, pi greater-than-or-equal-to 2 and (pi, pj = 1 if i not-equal-to j. We show that H is a one-relator group precisely if H can be obtained from a suitable group <a, b; a(p) = b(p)> by repeated applications of a (two-stage) procedure consisting of applying central Nielsen transformations followed by adjoining a root of a generator. We conjecture that any one-relator group G with non-trivial centre and G/G' not free abelian of rank two can be obtained in the same way from a suitable group <a, b; a(p) = b(q)>.