the ray $\sigma_c$ will be contained in the scaffolding geodesic on $S_{\infty}$ passing through $c$.

\[sigma\]

For each $c\in C,\,\sigma_c  $ is an infinite critical ray.

We begin by proving several lemmas, which, along with the lemmas of the next section, shall be of use throughout the paper. Theorem \[sigma\] will follow as an immediate corollary of Lemma \[nnew5\] in which we prove something slightly stronger about the ray $\sigma_c,$ namely, subarcs of $\sigma_c$ realize the distance between any point on $\sigma_c\backslash\{\sigma_c(0)\}$, and the curve $\alpha_0.$

Many of the lemmas in this and the next section will share the common setup stated below, which will be invoked repeatedly.

Let $S^*$ be either the closed set $F^*,$ an untwisted flute cut open along its canonical Dirichlet ray $\beta^*,$ or the hyperbolic strip $E^*\subset {\Bbb{H}}^2$ bounded by two parallel, non asymptotic geodesics. Let $\alpha_0$ denote the first, cut-open dividing loop on $F^*,$ or the common orthogonal between the boundary geodesics on $E^*$. The geodesics $\beta_1$ and $\beta_2$ on the boundary of $S^*$ are parameterized so that $\beta_1(0)$ and $\beta_2(0)$ are the end points of $\alpha_0$ and the geodesics $\beta_i(t),$ $i=1,2,$ go out the same end of the surface as $t \to \infty;$ 