Question

Is it true that groups $\langle a,b \mid a^n b^k=b^ka^{n+1}, b^la^s=a^sb^{l+1}\rangle$ are non-trivial for almost all (in any sense:))) $n,k,l,s\in\mathbb N$?

Answer 1

The group is always trivial. Indeed, $a^{-ns}b^{l^n}a^{ns}=b^{(l+1)^n}$. On the other hand, $a^{ns}=b^ka^{(n+1)s}b^{-k}$. Substitute $a^{ns}$ from the second equality to the first. You will get that $a^{(n+1)s}$ conjugates $b^{l^n}$ to $b^{(l+1)^n}$. Since $a^{ns}$ does the same, you get that $a^s$ commutes with $b^{l^n}$. Hence $b^{l^n}=b^{(l+1)^n}$ and $b$ is torsion, $b^p=1$. Similarly, $a$ is torsion, $a^q=1$. Notice also that $p=(l+1)^n-l^n$ is coprime with $l$ and $q$ is coprime with $n$. Now we have $b^{-kp}a^{n^p}b^{kp}=a^{(n+1)^p}$. Hence $a^{n^p}=a^{(n+1)^p}$. We already know that $a^{n^l}=a^{(n+1)^l}$. Suppose that the order of $a$ is $t$. Then $t$ divides both $(n+1)^l-n^l$ and $(n+1)^p-n^p$ and is co-prime with $n$. Then $t$ divides $$(n+1)^p-n^p-(n+1)^p+n^l*(n+1)^{p-l}=-n^p+n^l*(n+1)^{p-l}.$$ Since $t$ and $n$ are co-prime, $t$ divides $(n+1)^{p-l}-n^{p-l}$. Proceed by using the Euclidean algorithm. Since $p$ and $l$ are co-prime, we get that $t$ divides $(n+1)-n=1$, so $a=1$. Similarly $b=1$ and the group is trivial.