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16.68 (2006)

Open

Let $W(x, y)$ be a non-trivial reduced group word, and $G$ one of the groups $\text{PSL}(2, \mathbb{R})$, $\text{PSL}(2, \mathbb{C})$, or $\text{SO}(3, \mathbb{R})$. Are all the maps $W : G \times G \to G$ surjective?

Contributor: J. Mycielski

Progress

For $\text{SL}(2, \mathbb{R})$, $\text{SL}(2, \mathbb{C})$, $\text{GL}(2, \mathbb{C})$, and for the group $S^3$ of quaternions of norm 1 there exist non-surjective words; see (J. Mycielski, Amer. Math. Monthly, 84 (1977), 723–726; 85 (1978), 263–264).

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