Subgroup of GL(2,Z) generated by two matrices. What is the subgroup of $GL_2(Z)$ generated by the matrices:
$\left(
\begin{smallmatrix}
1&1\\
0&1
\end{smallmatrix}
\right)
$
and 
$\left(
\begin{smallmatrix}
0&1\\
1&0
\end{smallmatrix}
\right)?
$
I would like to know the name(s) of this group, references to it, as well as its well-known actions. I have found it to be difficult to look up online, and not discussed in texts. 
Thank you!
 A: The name of the group is $\text{GL}_2(\mathbf{Z})$.
Indeed, note that
$$
\left(
\begin{smallmatrix}
1&0\\
1&1
\end{smallmatrix}
\right)=
\left(
\begin{smallmatrix}
0&1\\
1&0
\end{smallmatrix}
\right)
\left(
\begin{smallmatrix}
1&1\\
0&1
\end{smallmatrix}
\right)
\left(
\begin{smallmatrix}
0&1\\
1&0
\end{smallmatrix}
\right),$$
and thus your group contains $\text{SL}_2(\mathbf{Z})$ (see, e.g., Corollary 2.6 in http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/SL%282,Z%29.pdf). On the other hand, your group is not contained in $\text{SL}_2(\mathbf{Z})$, and since $[\text{GL}_2(\mathbf{Z}):\text{SL}_2(\mathbf{Z})]=2$ your group must be $\text{GL}_2(\mathbf{Z})$.
A: In addition to Sean Eberhard's answer. The group $SL_2(\mathbb{Z})$ is generated by $T=\begin{pmatrix} 1 & 1 \cr 0 & 1 \end{pmatrix}$ and $U=\begin{pmatrix} 1 & 0 \cr 1 & 1 \end{pmatrix}$. Now $\langle A,T\rangle\simeq GL_2(\mathbb{Z})$ for
$A=\begin{pmatrix} 0 & 1 \cr 1 & 0 \end{pmatrix}$, and
$\langle B,T\rangle\simeq SL_2(\mathbb{Z})$
for $B=\begin{pmatrix} 0 & -1 \cr 1 & 0 \end{pmatrix}$. This was discussed in an earlier answer (with reference to 
www.math.uconn.edu/~kconrad/blurbs/grouptheory/SL(2,Z).pdf ).
