of the surface, as described by the special classes of infinite critical and subcritical geodesic rays. These are geodesic rays that head either directly, or almost directly, out the infinite end of the surface. In the theory of Fuchsian groups, these types of rays are related to the existence of Dirichlet and Garnett points in the limit set of a Fuchsian group representing the surface [@haas; @Waterman; @Sullivan].

Our approach is to employ a sequence of cut and paste operations to construct flute surfaces with complex end structure, where the building blocks are the simple untwisted flutes surfaces studied in [@bas1; @haas]. We refer to the surfaces constructed in this way as [*quilted*]{} flute surfaces. The geometry out the infinite end of a quilted flute surface is considerably more complex than the end geometry of an untwisted flute. Nevertheless, we demonstrate that there are certain fundamental similarities.

The boundary at infinity of a Dirichlet polygon for a Fuchsian group may be regarded as one measure of the complexity of the end geometry of the surface represented by the group. Applying our construction of quilted surfaces we show that, up to a sparse set, one can exercise surprising control over the 