18.13 (2014)

(D. B. A. Epstein). Is it true that the group
$$H = (\mathbb{Z}_3 \times \mathbb{Z}) \ast (\mathbb{Z}_2 \times \mathbb{Z}) = \langle x, y, z, t \mid x^3 = z^2 = [x, y] = [z, t] = 1 \rangle$$ cannot be defined by three relators in the generators $x, y, z, t$?

It is known that the relation module of the group $H$ has rank 3 (K. W. Gruenberg, P. A. Linnell, J. Group Theory, 11, no. 5 (2008), 587–608). An affirmative answer would give a solution of the relation gap problem.