flute surface for all integers $n\geq 0$.

Geometric convergence of flute surfaces
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Here we show that it is possible to define a flute surface as a limit of the finite constructions described in the previous section. We begin with the Fuchsian groups representing the surfaces and show that there is a way to normalize the groups so that one representing a surface $S_n$ contains the group for the surface $S_m$ when $n>m$. This leads to the definition of the surface $S_{\infty}$. Then, on the level of the intrinsic geometry, we prove that it is possible to view an arbitrarily large chunk of $S_{\infty}$ as the isometric images of a large chunk of $S_n,$ for $n$ sufficiently large. This last fact leads to a proof that $S_{\infty}$ is a flute surface. These results will allow us to assume, in the proofs of several of the theorems that follow, that we are working on one of the surfaces $S_n,$ rather than on the more complex surface $S_{\infty}.$

\[convergeL\] There exists a nested sequence of Fuchsian groups $\Gamma_0\subset\Gamma_1\subset\ldots, $ so that for each integer $n\geq 0$, ${\Bbb{H}}^2/ \Gamma_n=S_n.$ Furthermore, the image of the imaginary axis $I\subset {\Bbb{H}}^2$ covers the geodesic $\alpha_0$ on 