$B_{n+1}^c(r)\simeq B_n^c(r).$ The first part of the proposition follows.

Let $\tilde{S}(r)$ be the connected $\pi_{N_r}$-preimage of $S_{N_r}(r)$, which contains $i$ and let $\Gamma\subset \Gamma_{N_r}$ denote the stabilizer of $\tilde{S}(r)$ in $\Gamma_{N_r}$. By what we have shown, it follows that $\Gamma$ is the stabilizer of $\tilde{S}(r)$ in $\Gamma_n$ for all $n\geq N_r$ and therefore in $\Gamma_{\infty}$, as well. The final statement of the proposition follows.

Define $S_{\infty}(r)= \varphi_{n}(S_n(r))$ for some $n>N_r$. The sequence of closed balls $\pi_{\infty}({B^c(i,k)})$, for $k\in{\Bbb{N}}$, is a compact exhaustion of $S_{\infty}.$ Associated to this exhaustion there is an infinite end ${\sf E}$ with ${\sf E}_k=S_{\infty}\setminus  S_{\infty}(k).$ It follows from the proposition that every other end is finite. Thus we have

$S_{\infty}$ is a flute.

To simplify notation, we shall henceforth dispense with superscripts and refer to the geodesic $\alpha_0$ and the point $p$ on the surfaces $S_n$ and $S_{\infty}.$

A special class of critical rays on $S_\infty$  {#number4}
===============================================

Given a point $c \in C,$ let $\sigma_c$ denote the geodesic ray on $S_{\infty}=S(\alpha_0, p, C,\{F_i\})$ beginning at $c,$ orthogonal to $\alpha_0$ and oriented out the infinite end of $S_{\infty}$. If $c$ lies on the boundary of one of the intervals $I_j$ in the complement of $C$, then 