Let $U$ be a "triangular" region in the unit disc $\mathbb{D}$ bounded by three hyperbolic geodesics $\gamma_{1}, \gamma_{2}, \gamma_{3}$ that do not meet in $\mathbb{D}$ nor on its boundary. Let $J_{k}$ be inversion in $\gamma_{k}$ and set

$$A=J_{2} \circ J_{1} ; \quad B=J_{3} \circ J_{2} .$$

Let $G$ be the group generated by the Möbius transformations $A$ and $B$. Describe briefly a fundamental set for the group $G$ acting on $\mathbb{D}$.

Prove that $G$ is a free group on the two generators $A$ and $B$. Describe the quotient surface $\mathbb{D} / G$.