Exponential growth of SL2 of Z
= Exponential growth of SL2 of Z
{c}
{title2=Exponential growth of $\operatorname{SL}_2(\mathbb Z)$}
The matrices
$$
A=\begin{pmatrix}1&1\\0&1\end{pmatrix},
\qquad
B=\begin{pmatrix}1&0\\1&1\end{pmatrix}
$$
generate a <free monoid> in $\operatorname{SL}_2(\mathbb Z)$. Hence the $2^n$ positive words of length $n$ are distinct and $\operatorname{SL}_2(\mathbb Z)$ has <exponential growth of a group>.