Consider the matrices
$$
A=\begin{pmatrix}1&1\\0&1\end{pmatrix},
\qquad
B=\begin{pmatrix}1&0\\1&1\end{pmatrix}
\in\operatorname{SL}_2(\mathbb Z).
$$
Their projective action on the positive real line is
$$
A(x)=x+1\in(1,\infty),
\qquad
B(x)=\frac{x}{x+1}\in(0,1).
$$
These disjoint images give the monoid version of the <ping-pong lemma>: after removing a common initial letter, two different positive words act differently. Thus $A,B$ generate a <free monoid>. There are $2^n$ distinct positive words of length $n$, all lying in the radius-$n$ word ball for any generating set enlarged to contain $A,B$. Therefore the <Exponential growth of SL2 of Z> gives
$$
\boxed{\operatorname{SL}_2(\mathbb Z)\text{ has exponential growth}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 143 / 3 / d / Solution

{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>.

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