Let $S_4 = \left(\begin{array}{cc}0&-1 \\\ 1&0 \end{array}\right) \textrm{ and } S_6 = \left(\begin{array}{cc} 1&-1 \\\ 1&0\end{array}\right)$. Serre proves in his book on trees that $SL_2(\mathbb{Z}) \cong \mathbb{Z}/4 *_{\mathbb{Z}/2} \mathbb{Z}/6$, and $S_4$ and $S_6$ are the elements corresponding to the generators of $\mathbb Z/4$ and $\mathbb Z/6$ (I'm not sure if this is related to my question). Then let $a = S_4 S_6$ and $b = S_4 S_6^2$. I believe every element of $SL_2(\mathbb Z)$ can be written as $S_6^d w S_6^e$, where $w$ is a word in $a$ and $b$ but not $a^{-1}$ or $b^{-1}$.

I wrote a program (for other purposes) that seems to show that there aren't any relations between $a$ and $b$ that have length 15 or less and don't involve $a^{-1}$ or $b^{-1}$. I'm not certain that the program is right, but if it is, one might make a naive guess that these two elements generate a free group. This makes me suspicious.

1. Does $SL_2(\mathbb Z)$ contain a free group (of rank > 1)? If it does, is there an easy way to determine whether the subgroup generated by $a$ and $b$ is free?

2. A slightly less naive guess is that $a$ and $b$ generate a free monoid in $SL_2(\mathbb Z)$. Is there a reason why $SL_2(\mathbb Z)$ can't contain a free monoid, or an example showing that it does?

Let $S_4 = \left(\begin{array}{cc}0&-1 \\\ 1&0 \end{array}\right) \textrm{ and } S_6 = \left(\begin{array}{cc} 1&-1 \\\ 1&0\end{array}\right)$. Serre proves in his book on trees that $SL_2(\mathbb{Z}) \cong \mathbb{Z}/4 *_{\mathbb{Z}/2} \mathbb{Z}/6$, and $S_4$ and $S_6$ are the elements corresponding to the generators of $\mathbb Z/4$ and $\mathbb Z/6$ (I'm not sure if this is related to my question). Then let $a = S_4 S_6$ and $b = S_4 S_6^2$. I believe every element of $SL_2(\mathbb Z)$ can be written as $S_6^d w S_6^e$, where $w$ is a word in $a$ and $b$ but not $a^{-1}$ or $b^{-1}$.

I wrote a program (for other purposes) that seems to show that there aren't any relations between $a$ and $b$ that have length 15 or less and don't involve $a^{-1}$ or $b^{-1}$. I'm not certain that the program is right, but if it is, one might make a naive guess that these two elements generate a free group. This makes me suspicious.

1. Does $SL_2(\mathbb Z)$ contain a free group (of rank > 1)? If it does, is there an easy way to determine whether the subgroup generated by $a$ and $b$ is free?

2. A slightly less naive guess is that $a$ and $b$ generate a free monoid in $SL_2(\mathbb Z)$. Is there a reason why $SL_2(\mathbb Z)$ can't contain a free monoid, or an example showing that it does?

EDIT: Thanks for the quick replies. As Robin and Jack pointed out, $a$ and $b$ generate SL(2,Z), so clearly don't generate a free group. Also, there are free subgroups that are easy to write down. I'm still curious about #2, though.

Let $S_4 = \left(\begin{array}{cc}0&-1 \\ 1&0 \end{array}\right) \textrm{ and } S_6 = \left(\begin{array}{cc} 1&-1 \\ 1&0\end{array}\right)$ (in the preview these 2x2 matrices are displaying as 1 row). Serre proves in his book on trees that $SL_2(\mathbb{Z}) \cong \mathbb{Z}/4 *_{\mathbb{Z}/2} \mathbb{Z}/6$, and S_4 and S_6 are the elements corresponding to the generators of Z/4 and Z/6 (I'm not sure if this is related to my question). Then let $a = S_4 S_6$ and $b = S_4 S_6^2$. I believe every element of SL(2,Z) can be written as $S_6^d w S_6^e$, where $w$ is a word in $a$ and $b$ but not $a^{-1}$ or $b^{-1}$.

I wrote a program (for other purposes) that seems to show that there aren't any relations between $a$ and $b$ that have length 15 or less and don't involve $a^{-1}$ or $b^{-1}$. I'm not certain that the program is right, but if it is, one might make a naive guess that these two elements generate a free group. This makes me suspicious.

1. Does SL(2,Z) contain a free group (of rank > 1)? If it does, is there an easy way to determine whether the subgroup generated by $a$ and $b$ is free?

2. A slightly less naive guess is that $a$ and $b$ generate a free monoid in SL_2(Z). Is there a reason why SL(2,Z) can't contain a free monoid, or an example showing that it does?

Let $S_4 = \left(\begin{array}{cc}0&-1 \\\ 1&0 \end{array}\right) \textrm{ and } S_6 = \left(\begin{array}{cc} 1&-1 \\\ 1&0\end{array}\right)$. Serre proves in his book on trees that   $SL_2(\mathbb{Z}) \cong \mathbb{Z}/4 *_{\mathbb{Z}/2} \mathbb{Z}/6$, and $S_4$ and $S_6$ are the elements corresponding to the generators of $\mathbb Z/4$ and $\mathbb Z/6$ (I'm not sure if this is related to my question). Then let $a = S_4 S_6$ and $b = S_4 S_6^2$. I believe every element of $SL_2(\mathbb Z)$ can be written as $S_6^d w S_6^e$, where $w$ is a word in $a$ and $b$ but not $a^{-1}$ or $b^{-1}$.

I wrote a program (for other purposes) that seems to show that there aren't any relations between $a$ and   $b$ that have length 15 or less and don't involve $a^{-1}$ or $b^{-1}$. I'm not certain that the program is right, but if it is, one might make a naive guess that these two elements generate a free group. This makes me suspicious.

1. Does $SL_2(\mathbb Z)$ contain a free group (of rank > 1)? If it does, is there an easy way to determine whether the subgroup generated by $a$ and $b$ is free?

2. A slightly less naive guess is that $a$ and $b$ generate a free monoid in $SL_2(\mathbb Z)$. Is there a reason why $SL_2(\mathbb Z)$ can't contain a free monoid, or an example showing that it does?