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4.45 (1973)

Solved

Let $G$ be a free product amalgamating proper subgroups $H$ and $K$ of $A$ and $B$, respectively.
$\qquad$ a) Suppose that $H, K$ are finite and $|A : H| > 2, |B : K| \geqslant 2$. Is $G$ SQ-universal (see 4.39)?
$\qquad$ b) Suppose that $A, B, H, K$ are free groups of finite ranks. Can $G$ be simple?

Contributor: P. M. Neumann

Progress

a) Yes, it is (K. I. Lossov, Siberian Math. J., 27, no. 6 (1986), 890–899).
b) Yes, it can (M. Burger, S. Mozes, Inst. Hautes Études Sci. Publ. Math., 92 (2000), 151–194). But if one of the indices $|A : H|, |B : K|$ is infinite, then no, it cannot (S. V. Ivanov, P. E. Schupp, in: Algorithmic problems in groups and semigroups, Int. Conf., Lincoln, NE, 1998, Boston MA, Birkhäuser, 2000, 139–142).

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