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17.14 (2010)

Solved

Can the braid group $B_n$ for $n \geqslant 4$ be embedded into the automorphism group $\text{Aut}(F_{n-2})$ of a free group $F_{n-2}$ of rank $n - 2$?
Artin's theorem implies that $B_n$ can be embedded into $\text{Aut}(F_n)$, and an embedding of $B_n$, $n \geqslant 3$, into $\text{Aut}(F_{n-1})$ was constructed in (B. Perron, J. P. Vannier, Math. Ann., 306 (1996), 231–245).

Contributor: V. G. Bardakov

Progress

No, it cannot for $n = 4$ (V. G. Bardakov, P. Bellingeri, in Knot theory and its applications. ICTS program knot theory and its applications (KTH-2013), IISER Mohali, India, December 10–20, 2013, Amer. Math. Soc., Contemporary Mathematics, 670 (2016), 285–298).

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