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8.68 (1982)

Solved

Let $G = \langle a, b \mid r = 1 \rangle$ where $r$ is a cyclically reduced word that is not a proper power of any word in $a, b$. If $G$ is residually finite, is $G_t = \langle a, b \mid r^t = 1 \rangle$, $t > 1$, residually finite?

Contributor: C. Y. Tang

Progress

Editors’ comment: An analogous fact was proved for relative one-relator groups (S. J. Pride, Proc. Amer. Math. Soc., 136, no. 2 (2008), 377–386). Yes, it is (D. T. Wise, The structure of groups with a quasiconvex hierarchy, Annals of Mathematics Studies, 209, Princeton University Press, Princeton, NJ, 2021).

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