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8.11 (1982)

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Consider the group
$$M = \langle x, y, z, t \mid [x, y] = [y, z] = [z, x] = (x, t) = (y, t) = (z, t) = 1 \rangle$$ where $[x, y] = x^{-1}y^{-1}xy$ and $(x, t) = x^{-1}t^{-1}x^{-1}txt$. The subgroup $H = \langle x, t, y \rangle$ is isomorphic to the braid group $\mathfrak{B}_4$, and is normally complemented by $N = \langle (zx^{-1})^M \rangle$. Is $N$ a free group (?) of countably infinite rank (?) on which $H$ acts faithfully by conjugation?

Contributor: D. L. Johnson

Progress

Editors’ comment: As pointed by Y. Antolín, the group $M$ is the Artin group of type $D_4$; it is proved in (B. Perron, J. P. Vannier, Math. Ann., 306, no. 2 (1996), 231–246) that $N$ is a free group of rank 3 and $H$ acts on $N$ faithfully by conjugation.

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