Let me define the following groups $$G(k,l,n):=\langle a,b\mid a^k=1=b^l, (ab)^n=(ba)^n\rangle$$
Fixed $k$ and $l$ (WLOG we can assume those are prime), I would like to know, whether the groups are mutually unequal. More precisely, prove that $(ab)^m\neq (ba)^m$ for $m<n$.
For $k=l=2$ those are the dihedral groups, so the answer is yes. For any other $k$ and $l$, I do not know.
I am not a group theorist, so I have no feeling for such a question, so I do not know, whether it is obvious, or it is probably true but hard to prove, or whether it is totally unclear. I would be happy even for any comment on this.
I was trying to find similar groups in the literature. I encountered the von Dyck groups defined as $$D(k,l,n)=\langle a,b\mid a^k=b^l=(ab)^n=1\rangle.$$ (Again, for $k=l=2$, those are dihedral groups.) Is some similar result known for those von Dyck groups?