11.18 (1990)

Let $G(a, b) = \langle x, y \mid x = [x, \phantom{}_a y], \ y = [y, \phantom{}_b x] \rangle$. (see 11.16.) Is $G(a, b)$ finite?

It is easy to show that $G(1, b) = 1$ and one can show that $G(2, 2) = 1$. Nothing is known about $G(2, 3)$. If one could show that every minimal simple group is a quotient of some $G(a, b)$, then this would yield a very nice sequence of words in two variables to characterize soluble groups, see (R. Brandl, J. S. Wilson, J. Algebra, 116 (1988), 334–341).