Let $G= \langle S \mid r \rangle$ be a one-relator presentation for a one-ended hyperbolic group, with $r$ cyclically reduced.
Question: Can there be a nontrivial word $w(S)$ which is trivial in the group $G$ but has length shorter than $r$? What if $r$ is the shortest possible word for a one-relator presentation of $G$?
Note that it follows from Newman's spelling theorem that in the torsion case there are no shorter words, since you can apply Dehn's algorithm. Similarly if $r$ gives a $C'(1/6)$ presentation there are no shorter words.
Generally it is known that subwords of $r$ will not be trivial either. This is proved by Weinbaum in On relators and diagrams for groups with one defining relation.
This question grew out of this question on math.se and my answer to it. One thing to note is that without hyperbolicity you can find that some Baumslaug-Solitar groups provide examples with shorter trivial words.