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.