infinity by ${\Bbb{N}}$, so that $d(\Delta_i , \Delta_{\infty})\geq d(\Delta_{i+1} , \Delta_{\infty}).$ Let $\overline{\alpha}_1$ be a simple closed curve on $S$ that divides ${{\mathbb{C}}}$ into two pieces, one of which contains only the two complementary components $\Delta_1$ and the component containing the point at infinity. Let $\alpha_1$ be the geodesic in the free homotopy class of $\overline{\alpha}_1$ on $S$. Suppose the geodesics $\alpha_1,\ldots,\alpha_n$ have been defined. Let $\overline{\alpha}_{n+1}$ be a simple closed curve, disjoint from $\alpha_n$, so that the region of ${{\mathbb{C}}}$ bounded by $\alpha_n$ and $\overline{\alpha}_{n+1}$ contains the single complementary component $\Delta_{n+1}$. Let $\alpha_{n+1}$ be the geodesic freely homotopic to $\overline{\alpha}_{n+1}$ on $S$.

Let $Q$ be the set of geodesics $\alpha_i,\, i\in{\Bbb{N}}$ defined above. Let $P_i$ denote the connected component of $S\setminus Q$ whose boundary meets the boundary of $\Delta_i$. Each of the $P_i$ is a pair of pants. Now $S$ can be reconstructed from the sets $P_i$ by gluing’ $P_i$ to $P_{i+1}$ along their common geodesic boundary $\alpha_i$ to get a flute surface, as in [@bas1].

To prove the converse, we simply observe that every closed curve on a flute surface $F$ divides. It is known, (see [@ahlfors]), that $F$ is then conformally equivalent to a domain in the 