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](https://math.stackexchange.com/q/1957873/29123) and my answer to it. One thing to note is that without hyperbolicity you can find that some [Baumslaug-Solitar groups](https://en.wikipedia.org/wiki/Baumslag%E2%80%93Solitar_group) provide examples with shorter trivial words.



Can a hyperbolic, one ended, one relator group, have a shorter trivial word?
I think I found an example with shorter trivial words using a handy characterizations in a paper by Ivanov and Schupp called [*On hyperbolicity of small cancellation groups and one-relator groups*](http://www.ams.org/journals/tran/1998-350-05/S0002-9947-98-01818-2/).


Consider $\langle a,b,c \mid ab^2ac^{12}\rangle$. By checking Whitehead automorphisms this relation is as short as possible in the $Aut(F\_3)$ orbit and has every generator in the relation, so does not have infinitely many ends(Thanks to ADL for the correction). One can look at the abelianization to rule out zero or two ends. Theorem 3 in the above paper says that this group is hyperbolic since it has exactly two occurrences of $a$, no $a^{-1}$, and $b^2c^{-12}$ is not a proper power in the free group generated by $a,b,c$. Now $$(ab^2ac^{12})c(c^{-12}a^{-1}b^{-2}a^{-1})c^{-1}=ab^2aca^{-1}b^{-2}{a^{-1}}c^{-1}$$ 
which is shorter than the defining relation.


