There is an obvious fundamental domain 
for the action of 
on 
.
Let
be the planes in that meet the sphere at infinity
in the circles
,
let
be the corresponding open half-spaces
that meet the sphere at infinity in the disks
,
and let be the complement of
in .
(See Figure 3.)
Figure 3: Planes in hyperbolic 3-space.
Note that whereas we described F as the
exterior
of
,
we can best describe as the
interior
of
.
The manifold 
is obtained from by gluing
the boundary components
together in pairs.
If we cut apart along the n/2 surfaces along which
it was glued, we recover .
By the cutting law,
any lower bound for 
is also a lower bound for
.
Hence instead of cutting up we will work on cutting up .
We will forget all about the pairings,
and drop the assumption the n is even.
Here's what we will prove:
Theorem.
Let be the manifold (with boundary) corresponding to the finite
collection of circles
in the Riemann sphere.
Then
for some universal constant 
.