Let $F$ be a free group and $w\in F$ a cyclically reduced word. Let $v$ be a non-trivial proper subword of $w$. Is it true that $v\notin \langle w^F\rangle$?
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asked Jun 14, 2022 at 20:12
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1$\begingroup$ Actually, for context, there exist examples if, instead, one only assumes that $v$ is an arbitrary reduced word with $0<|v|<|w|$: math.stackexchange.com/a/1958310/35400 . Namely, $xzx^{-1}z^{-1}$ belongs to the normal closure of $x^2zxz$ (indeed $\langle x,z\mid x^2zxz\rangle$ is a disguised presentation of the cyclic group $\langle x,y\mid xy^2\rangle$). $\endgroup$– YCorCommented Jun 15, 2022 at 9:00
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This is true. See Theorem 2 of ON RELATORS AND DIAGRAMS FOR GROUPS WITH ONE DEFINING RELATION BY C. M. WEINBAUM.
answered Jun 14, 2022 at 20:24
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$\begingroup$ Actually this seems to be asked and answered here math.stackexchange.com/questions/1957873/… $\endgroup$ Commented Jun 14, 2022 at 20:27