$i > 0$ all but a compact segment of $\sigma$ belongs to ${\sf E}_i.$

Let $S$ be a hyperbolic surface. An end [E]{} of $S$ is called a [*puncture*]{} if there is a subset $D$ of $S$ which is conformally equivalent to the punctured disc $\{ z \ | \
0<|z|<1\}$ and for $i$ large, $ {\sf E}_i\subset D$. Similarly, we call [E]{} a [*hole*]{} if there is a subset $D$ of $S$ which is conformally equivalent to an annulus $\{ z \ |\ 1<|z|<r\}$ for some $r>1$ and for $i$ large, $ {\sf E}_i\subset D$. [E]{} is a [*finite end*]{} if it is either a puncture or a hole; otherwise it is an [*infinite end*]{}.

An end ${\sf E} = \{{\sf E}_i\}$ is said to be of the [*second kind*]{} if $S$ contains a half-plane $P$ and for all $i>
0$, ${\sf E}_i\bigcap P\not=\emptyset$. If an end is not of the second kind then it is of the [*first kind*]{}. A puncture is of the first kind and a hole is of the second kind.

Let ${\bf S}$ denote the infinite cylinder ${\bf C}={\rm
S}^1\times(0,\infty),$ with the set of points $\{ (1,n)|\,n\in \Bbb{N}
\}$ deleted, and define the space 