  The action of the mapping class group of the thrice-punctured projective
plane on its $\mathrm{GL}(2,\mathbb{C})$ character variety produces an
algorithm for generating the simple length spectra of quasi-Fuchsian
thrice-punctured projective planes. We apply this algorithm to quasi-Fuchsian
representations of the corresponding fundamental group to prove: a sharp
upper-bound for the length of its shortest geodesic, a McShane identity and the
surprising result of non-polynomial growth for the number of simple closed
geodesic lengths.
