The Quotient of a Free Product of Groups by a Single High-Powered Relator
James Howie · Proceedings of the London Mathematical Society · 1993
Let A, B be groups, and r ∈ A * B a cyclically reduced word of the form xUyU−l for some word U and letters x, y, where x2 = y3 = l. We call this form exceptional. If m = 4 or m = 5, and G is the one-relator product group (A * B)/N(rm) (where N(·) denotes normal closure), then it was stated in [2,3] (Theorem E) that the cohomology of G is linked to that of A * B and of the finite triangle group G(2, 3, m) = 〈a, b| a2 = b3 = (ab)m = 1〉 by a certain Mayer–Vietoris sequence. The proof of this theorem involves so-called spherical pictures over G that are induced from the finite subgroup G(2, 3, m). This proof fails if r is multiply exceptional, in the sense that both r and some cyclic permutation of r have exceptional forms, because then more spherical pictures arise than are dealt with in [2, 3]. An example of such a word r is ababab−1 where A = 〈 a| a2 = 1 〉 and B = 〈 b| b3 = 1 〉, since both r and its cyclic permutation ab−labab are exceptional. Theorem E of [2, 3] is correct only under the stronger hypothesis that r is uniquely exceptional (i.e. not multiply exceptional). This problem does not affect the other results of [2,3]. A more detailed discussion is given in [1].