6 A. J. DUNCAN AND J. HOWIE
in G
2
,
and hence in H, using the identities o/
=
(3'
= "/ =
8'
=
1 and the first
defining relation of G
2
.
The other two desired equations also hold in
H
by the
defining relations of
H
as an HNN extension.
Theorems 1 and 3 take care of most possible combinations of trivial and nontrivial
corner labels, but unfortunately not all. To complete the proof we need to introduce
a technique of changing variables. We can do this whenever there are two adjacent
trivial corner labels on the cube, and the result is to change the system to a new one
in which the corner labels are the same elements of
G,
but differently distributed
around the cube.
There are twelve different ways in which we can have two adjacent trivial corner
labels (one for each edge of the cube). For each of these there are two possible
changes of variable (one for each face incident at that edge). There are thus twenty-
four potential ways in which this change of variable can take place. Although the
effects of these twenty-four moves are different, the idea in each case is the same,
so we shall describe only one of them, by way of illustration. We shall then invoke
cubical symmetry to show that all twenty-four possible changes of variable are
permissible.
Let us suppose that 'Y
=
8
=
1 in G, in other words that el-
1
= f =
d- 1i.
In this case we shall replace
z
by a new variable w
=
y
f
z,
noting that the equa-
tion
eyfzgy-
1
hz-
1
=
1 becomes
ewgy-
1
hw-
1
yf
=
1 in terms of this new vari-
able. Similarly, in the presence of the equation
axbycx- 1dy-
1
=
1, the equation
izjxkz-
1
lx-
1 =
1 is equivalent to
axbyel-
1
zk-
1
x-
1
j-
1
(yd-
1
iz)-
1 =
1, or (using
the identities
yel-
1
z
=
w
=
yd-
1
iz)
to
axbwk-
1
x-
1
j-
1
w-
1
=
1.
Hence we may replace our original set of three equations in
x,
y,
z
by a new set
of equations
(also modelled on K) in
x,
y,
w,
with corner labels cyclic conjugates of
{1,
1,(3,a,a',(3',"f',8'}
in place of the labels
{a,(3, 1, 1,a',(3',"f',8'}
of Figure 1. In other words, we have essentially interchanged the edge that had two
trivial corner labels with one of its nearest parallels.
Now let F be any face of the cube and let E
1
and Ez be parallel edges incident at
F.
Suppose that E
1
has trivial corner labels and that we wish to change variables
so as to interchange E
1
and E
2 .
We choose a symmetry a of the cube taking E1
to "(8 and E
2
to (3a. The above argument (adapted to the new labels) at a, (3, "f
and 8 is used to interchange (3a and "(8. The symmetry a-
1
restores the original
labelling with E
1
and E
2
interchanged.
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