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I have been thinking about this question for quite some time but now this question by Denis Serre revived some hope.

Question. Let $x,y$ be invertible matrices (say, over $\mathbb C$) and $[x,y,y]=x$ where $[a,b]=a^{-1}b^{-1}ab$, $[a,b,c]=[[a,b],c]$. Does it follow that some power of $x$ is unipotent?

The motivation is this. Consider the one-relator group $\langle x,y \mid [x,y,y]=x\rangle$. It is hyperbolic (proved by A. Minasyan) and residually finite (that is proved in my paper with A. Borisov). If the answer to the above question is "yes", then that group would be non-linear which would provide an explicit example of non-linear hyperbolic group.

Update 1. Can $x$ in the above be a diagonal matrix and not a root of 1?

Update 2. The group is residually finite(proved by A. Borisov and myself), so it has many representations by matrices such that $x, y$ have finite orders (hence their powers are unipotents).

Update 3. The group has presentation as an ascending HNN extension of the free group: $\langle a,b,t \mid a^t=ab, b^t=ba\rangle$. So it is related to the Morse-Thue map. Properties of that map may have something to do with the question. See two quasi-motivations of the question as my comments below.

I have been thinking about this question for quite some time but now this question by Denis Serre revived some hope.

Question. Let $x,y$ be invertible matrices (say, over $\mathbb C$) and $[x,y,y]=x$ where $[a,b]=a^{-1}b^{-1}ab$, $[a,b,c]=[[a,b],c]$. Does it follow that some power of $x$ is unipotent?

The motivation is this. Consider the one-relator group $\langle x,y \mid [x,y,y]=x\rangle$. It is hyperbolic (proved by A. Minasyan) and residually finite (that is proved in my paper with A. Borisov). If the answer to the above question is "yes", then that group would be non-linear which would provide an explicit example of non-linear hyperbolic group.

Update 1. Can $x$ in the above be a diagonal matrix and not a root of 1?

Update 2. The group is residually finite (proved by A. Borisov and myself), so it has many representations by matrices such that $x, y$ have finite orders (hence their powers are unipotents).

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I have been thinking about this question for quite some time but now this question by Denis Serre revived some hope.

Question. Let $x,y$ be invertible matrices (say, over $\mathbb C$) and $[x,y,y]=x$ where $[a,b]=a^{-1}b^{-1}ab$, $[a,b,c]=[[a,b],c]$. Does it follow that some power of $x$ is unipotent?

The motivation is this. Consider the one-relator group $\langle x,y \mid [x,y,y]=x\rangle$. It is hyperbolic (proved by A. Minasyan) and residually finite (that is proved in my paper with A. Borisov). If the answer to the above question is "yes", then that group would be non-linear which would provide an explicit example of non-linear hyperbolic group.

Update1. Can $x$ in the above be a diagonal matrix and not a root of 1?

Update 2. Can $x=\left(\begin{array}{lll} 2 & 0 & 0\\ 0& 1 & 0\\ 0 & 0 & \frac12\end{array}\right)$ and the matrices of size $3$ with $\det=1$? In other word let $x$ be this matrix. Is there a $3\times 3$ matrix $y$ with $\det(y)=1$ such that $[x,y,y]=x$? Perhaps somebody with a good Groebner basis software can check it.

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