I have a homework problem of the following form. We're given presentation of a group $\langle x,y \mid R\rangle$ explicitly, and two matrices $X,Y \in \mathrm{GL}(\mathbb{C},2).$ We know $X$ and $Y$ explicitly. The problem is to show that $\langle x,y \mid R\rangle \cong \langle X,Y\rangle.$

What I've done. Let $\langle x,y \rangle$ denote the group freely generated by $x$ and $y$. Then there is a homomorphism $\varphi : \langle x,y\rangle \rightarrow \langle X,Y\rangle$ satisfying $\varphi(x) = X$ and $\varphi(y) = Y$. I have checked that $\varphi(R) = \{1\}$. Thus $\langle \langle R \rangle \rangle \subseteq \mathrm{ker} \varphi.$ It remains to show that: $$\mathrm{ker} \varphi \subseteq \langle \langle R \rangle \rangle.$$

That is, I need to show that if some word is mapped to the identity matrix in $\mathrm{GL}(\mathbb{C},2),$ then it is an element of the normal closure of $R$ in $\langle x,y\rangle.$

Q. In general, how does one attack this kind of problem? What strategies are available?

Added. Here's an explicit statement of the data, but you please DO NOT SOLVE THE PROBLEM. I just need help getting started.

$$R = \{x^ny^{-2},x^{2n}, y^{-1}xyx\}$$

$$X = \begin{bmatrix}e^{i\pi/n} & 0\\0 & e^{-i\pi/n}\end{bmatrix}, \qquad Y = \begin{bmatrix}0 & -1\\1 & 0\end{bmatrix}$$

Once again, let me just emphasize that solving the problem would be completely inappropriate. I just need some ideas for getting started. Thank you.

  • $\begingroup$ You'll need some information about the subgroup $\langle X,Y \rangle$. For instance, if you know a presentation for $\langle X,Y \rangle$ then you can try to define an inverse to $\varphi$. $\endgroup$ – Lee Mosher Aug 17 '14 at 11:29
  • $\begingroup$ @LeeMosher, I know the matrices $X$ and $Y$ explicitly, but that is all. $\endgroup$ – goblin Aug 17 '14 at 13:24
  • $\begingroup$ It's rather hard to give any suggestions here, not knowing what you know about $GL(2,\mathbb{C})$, not knowing anything about the specific matrices $X,Y$, etc. You'll need to make your question more specific. For example, do you know about the connection between $GL(2,\mathbb{C})$ and hyperbolic geometry? $\endgroup$ – Lee Mosher Aug 17 '14 at 14:24
  • $\begingroup$ @LeeMosher, I've included the data. Please don't solve it though. I just need a hand getting started. No, I do not know about the connections to hyperbolic geometry. $\endgroup$ – goblin Aug 18 '14 at 9:27

In your specific example, you can write every element of $H=\langle x, y; R\rangle$ in the form $x^iy^j$. Then, you also know that every element of $K$ also has the form $X^iY^j$ (because of what you have already shown, that $H\twoheadrightarrow K$).

To prove the result, take a matrix of the form $X^iY^j$ and assume that it is equal to the identity matrix. What values of $i$ and $j$ can you then get? (Note that you have $x^iy^j\mapsto X^iY^j$). Conclude that this implies that the map $H\twoheadrightarrow K$ is an isomorphism. (Note: I am not suggesting that you try to prove $\ker\varphi\subseteq\langle\langle R\rangle\rangle$.)

  • $\begingroup$ Thanks. This should be enough to get me going. $\endgroup$ – goblin Aug 18 '14 at 10:59

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