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

Consider the matricesTheir projective action on the positive real line isThese 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