Take a look at the domain 
.
It lies above the (x,y)-plane,
between the planes y=0 and y=1,
and on or above the hemispheres
.
(From now on, all geometrical terms used will be Euclidean by default.)
How shall we cut this space into tubes?
The obvious thing is to start by cutting the space apart along the
cylinders that lie above the circles
.
This yields
n-2 domains that are congruent from the hyperbolic point of view,
along with a residual piece.
The residual piece can be cut into vertical tubes,
so all we have to do is show how to cut the hyperbolically congruent domains
into tubes that grow more or less exponentially.
Unfortunately, this can't be done;
the bottom of the spectrum of one of these domains is 0.
The problem is that the intersection of one of these domains
with the plane 
has Euclidean area on the order of
,
hence hyperbolic area on the order of
.
(See Figure 6.)
Since there isn't an exponential room at infinity,
there is no way to cut the domain into tubes that grow exponentially.
Figure 6: No room at infinity.
Notice that in higher dimension,
the difficulty we have just encountered disappears.
In the upper half-space model of 
,
the Euclidean measure of the intersection of an analogous domain with
the hyperplane at height 
is still
on the order of ,
only now the hyperbolic measure is
.
Thus there is plenty of room at infinity,
and it is a simple matter to construct an appropriate cutting into tubes.
Back in 
,
we recognize that because of the difficulty
we have just discussed,
we must allow at least some of the tubes that begin
over a given hemisphere to spread out beyond the corresponding circle.
In so doing,
they will most likely stray out over other hemispheres,
and confusion is liable to result.
Our task will be to avoid this confusion.