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S Dec 31, 2020 at 23:04 history bounty ended CommunityBot
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Sep 2, 2019 at 18:14 comment added user1729 If you are willing to assume that the word $w$ is a proper power, so there exists some word $v$ and some integer $n>1$ s.t. $w\equiv v^n$, then there are some pretty strong restrictions on the shape of possible diagrams. The starting point is B.B.Newman's Spelling Theorem, which says (roughly) that any disc diagram must contain a cyclic shift of $W^{n-1}$ on the boundary (or its inverse). Hruska and Wise went much further in their paper "Towers, ladders and the B. B. Newman Spelling Theorem", J. Aust. Math. Soc. (2001). "Towers" is the method of proof, "ladder" is the shape of the diagrams.
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