geodesics $\epsilon_{i+1}^j$ and $\beta_{i+1}^j,\, j=1,2 ,$ so that $\epsilon_{i+1}^j(t)$ is identified with $\beta_{i+1}^j(t)$ for $t\in {{\mathbb{R}}}.$ On $S_{i+1}$ the identified pair $\epsilon_{i+1}^j$ and $\beta_{i+1}^j $ shall be denoted by $\beta_{i+1}^j $, which inherits their parameterization. The surface $S_{i+1}$ has a hyperbolic metric defined locally at every point and is easily seen to be complete. Therefore $S_{i+1}$ is a hyperbolic surface. Furthermore, it naturally contains a copy of $S_{i}\setminus E_{i+1}$, as well as a copy of $F_{i+1}^*$.

The set of all the geodesics $\beta_{m}^j $ on $S_n$ where $m=1,\ldots, n$ and $ j=1,2 $ shall be denoted by $B_n$.

\[prop\] The surfaces $S_n$ are all flute surfaces.

First observe that if two planar surfaces are glued together in the plane then the result is a planar surface. Each flute $F_i$, as well as its cut open relative $F_i^*$, is planar. By induction it follows that each of the surfaces $S_n$ is planar

Next we argue by induction that each $S_n$ has a single infinite end. Suppose this is true for $S_i$. Let $K^*_j$ be a compact exhaustion of $S_i$ with the property that $I_{i+1} \in K^*_j$ for each $j$ and similarly choose a compact exhaustion $K^{\#}_j$ of $F_{i+1}$ so that $\alpha_{i+1,0}\subset 