I'm given two matrices in $SL_2(\mathbb{Z})$

$$ A = \left(\begin{array}{cc} 2 & 3\\ 3 & 5 \end{array}\right), \ \ B = \left(\begin{array}{cc} 5 & 3\\ 3 & 2 \end{array}\right). $$

Then the group $\langle A, B \rangle$ is free of rank 2. Now my problem is to prove that the following matrix:

$$ \left(\begin{array}{cc} 1 & n\\ 0 & 1 \end{array}\right),n\in \mathbb{Z^*} $$

does not belong to $\langle A, B \rangle$. That is, it cannot be obtained from products of $A$ and $B$ and their inverses.

I started by writing $$ \left(\begin{array}{cc} 0 & 1\\ - 1 & 0 \end{array}\right) \longrightarrow x, \left(\begin{array}{cc} 0 & 1\\ - 1 & 1 \end{array}\right) \longrightarrow y, A \longrightarrow (y x y^{- 1} x^{-1})^2, B \longrightarrow (y^2 x y^2 x^{- 1})^2, $$ but then did not get any further.

Any help is highly appreciated.

I'm given two matrices in $SL_2(\mathbb{Z})$

$$ A = \left(\begin{array}{cc} 2 & 3\\ 3 & 5 \end{array}\right), \ \ B = \left(\begin{array}{cc} 5 & 3\\ 3 & 2 \end{array}\right). $$

Then the group $\langle A, B \rangle$ is free of rank 2. Now my problem is to prove that $\langle A, B \rangle$ does not contain any matrix of the form

$$ \left(\begin{array}{cc} 1 & n\\ 0 & 1 \end{array}\right) $$

with $n\in\mathbb Z$ nonzero. That is, no such matrix can be obtained from products of $A$ and $B$ and their inverses.

I started by writing $$ \left(\begin{array}{cc} 0 & 1\\ - 1 & 0 \end{array}\right) \longrightarrow x, \left(\begin{array}{cc} 0 & 1\\ - 1 & 1 \end{array}\right) \longrightarrow y, A \longrightarrow (y x y^{- 1} x^{-1})^2, B \longrightarrow (y^2 x y^2 x^{- 1})^2, $$ but then did not get any further.

Any help is highly appreciated.

I'm given two matrices in $SL_2(\mathbb{Z})$

$ A = \left(\begin{array}{cc} 2 & 3\\ 3 & 5 \end{array}\right), B = \left(\begin{array}{cc} 5 & 3\\ 3 & 2 \end{array}\right) $

Then the group $<A,B>$ is free of rank 2. Now my problem is to prove that the following matrix:

$\left(\begin{array}{cc} 1 & n\\ 0 & 1 \end{array}\right),n\in \mathbb{Z^*}$

does not belong to $<A,B>$. That is, it cannot be obtained from products of $A$ and $B$ and their inverses.

I started by writing $ \left(\begin{array}{cc} 0 & 1\\ - 1 & 0 \end{array}\right) \longrightarrow x, \left(\begin{array}{cc} 0 & 1\\ - 1 & 1 \end{array}\right) \longrightarrow y, A \longrightarrow (y x y^{- 1} x^{-1})^2, B \longrightarrow (y^2 x y^2 x^{- 1})^2 $ But then did not get any further.

Any help is highly appreciated.

I'm given two matrices in $SL_2(\mathbb{Z})$

$$ A = \left(\begin{array}{cc} 2 & 3\\ 3 & 5 \end{array}\right), \ \ B = \left(\begin{array}{cc} 5 & 3\\ 3 & 2 \end{array}\right). $$

Then the group $\langle A, B \rangle$ is free of rank 2. Now my problem is to prove that the following matrix:

$$ \left(\begin{array}{cc} 1 & n\\ 0 & 1 \end{array}\right),n\in \mathbb{Z^*} $$

does not belong to $\langle A, B \rangle$. That is, it cannot be obtained from products of $A$ and $B$ and their inverses.

I started by writing $$ \left(\begin{array}{cc} 0 & 1\\ - 1 & 0 \end{array}\right) \longrightarrow x, \left(\begin{array}{cc} 0 & 1\\ - 1 & 1 \end{array}\right) \longrightarrow y, A \longrightarrow (y x y^{- 1} x^{-1})^2, B \longrightarrow (y^2 x y^2 x^{- 1})^2, $$ but then did not get any further.

Any help is highly appreciated.

I'm given two matrices in $SL_2(\mathbb{Z})$

$ A = \left(\begin{array}{cc} 2 & 3\\ 3 & 5 \end{array}\right), B = \left(\begin{array}{cc} 5 & 3\\ 3 & 2 \end{array}\right) $

Then the group $<A,B>$ is free of rank 2. Now my problem is to prove that the following matrix:

$\left(\begin{array}{cc} 1 & n\\ 0 & 1 \end{array}\right),n\in \mathbb{Z^*}$

does not belong to $<A,B>$. That is cannot be obtained from products of $A$ and $B$ and their inverses.

I started by writing $ \left(\begin{array}{cc} 0 & 1\\ - 1 & 0 \end{array}\right) \longrightarrow x, \left(\begin{array}{cc} 0 & 1\\ - 1 & 1 \end{array}\right) \longrightarrow y, A \longrightarrow (y x y^{- 1} x^{-1})^2, B \longrightarrow (y^2 x y^2 x^{- 1})^2 $ But then did not get any further.

Any help is highly appreciated.

I'm given two matrices in $SL_2(\mathbb{Z})$

$ A = \left(\begin{array}{cc} 2 & 3\\ 3 & 5 \end{array}\right), B = \left(\begin{array}{cc} 5 & 3\\ 3 & 2 \end{array}\right) $

Then the group $<A,B>$ is free of rank 2. Now my problem is to prove that the following matrix:

$\left(\begin{array}{cc} 1 & n\\ 0 & 1 \end{array}\right),n\in \mathbb{Z^*}$

does not belong to $<A,B>$. That is, it cannot be obtained from products of $A$ and $B$ and their inverses.

I started by writing $ \left(\begin{array}{cc} 0 & 1\\ - 1 & 0 \end{array}\right) \longrightarrow x, \left(\begin{array}{cc} 0 & 1\\ - 1 & 1 \end{array}\right) \longrightarrow y, A \longrightarrow (y x y^{- 1} x^{-1})^2, B \longrightarrow (y^2 x y^2 x^{- 1})^2 $ But then did not get any further.

Any help is highly appreciated.