Markoff's isometry conjecture for the modular torus

Let $\mathbf{M}$ be the modular torus, the hyperbolic once-punctured torus described in the paper. A pair of simple closed geodesics is a pair of simple geodesics on $\mathbf{M}$ with no self-intersections. Markoff's isometry conjecture. If $\alpha$ and $\beta$ are simple closed geodesics on $\mathbf{M}$ of the same length, then an isometry of $\mathbf{M}$ takes $\alpha$ to $\beta$. This is presented as an equivalent geometric formulation of the Markoff conjecture.