there is an number $L>0$ so that if $|\beta|<L$ then $C_S(\beta,R)$ is isometric to the collar $C_A(\alpha,R),$ where $A$ is an annulus with core geodesic of length $|\alpha|=|\beta|.$ More precisely, $C_S(\beta,R)$ is isometric to the collar $C_A(\alpha,R)  $ if $\sinh R= (2\sinh |\beta|/2)^{-1},$ where $|\alpha|=|\beta|,$ [@matelski].

One can analogously define the collars $C_{E_j }(I_j,R)$ and $C_{ F^*_j }(\alpha_{j,0}, R)$ about the geodesic segments on the surfaces $E_j$ and $F^*_j$, respectively. Observe that identifying the boundary geodesics $\epsilon_j^1$ and $\epsilon_j^2$ of $E$ will produce an annulus. The Collar Lemma, applied to this annulus and the flute $F_j$, implies that given $R>0$ there is a $J\in {\Bbb{N}}$ so that if $j> J$ then $C_{E_j}(I_j,R)$ is isometrically equivalent to $C_{F^*_j}(\alpha_{j,0}, R)$ by an isometry taking $\alpha_{j,0}$ to $I_j.$

The proposition is trivial if the number of complementary components, $l$, of $C$ in $\alpha_0$ is finite. We therefore suppose that $l=\infty.$

We normalize so that the surfaces are represented by Fuchsian groups, as in Lemma \[convergeL\]. Following up on the comments preceding the statement of the proposition, given $r>0$ there is an $M_r\in {\Bbb{N}}$ so that if $g\in \Gamma_{n}$ for $M_r\geq n\geq\infty$, and $g(B^c(i,r))\cap B^c(i,r)\not = \emptyset,$ then $g\in \Gamma_{M_r}.$ The closed ball $B^c(i,r)$ 