be used to to prove Theorem \[C\]. In the last section we look at the finer structure of the set of infinite critical and subcritical geodesics rays. The main result of Section \[close\] is that there is an underlying geodesic scaffolding, heading out the infinite end, which all infinite critical and subcritical geodesic rays must approach asymptotically. This is a broad generalization of a similar result about untwisted flutes, that appeared in [@haas]. In obtaining these results we employ a number of lemmas. Several of these are results about plane hyperbolic geometry which are interesting in their own right.

Basic properties of flute surfaces {#review}
==================================

The main reference for the material in this section is [@haas].

Define an [*end*]{} [E]{} of a manifold $M$ as follows. Let $
K_1 \subset K_{2} \subset ...\subset M$ be a nested sequence of compact subsets of $M$ so that $\bigcup^{\infty}_{i=1} K_{i} = M$. An end [E]{} is a sequence of connected components $\{{\sf E}_i\}$ in the complement of $K_{i}$ so that ${\sf E}_{i+1} \subset {\sf
E}_i$. This definition can be made independent of the given exhaustion $\{ K_i\}$. A ray $\sigma$ is said to [*go out the end*]{} [E]{}, if for each integer 