4 A. J. DUNCAN AND J. HOWIE
a'
FIGURE
1.
Corollary. If five
or
more of the eight corner
labels are nontrivial elements of G,
then the system of equations has a solution in
an overgroup of G.
Proof. If
five corners of a cube are coloured, then it is possible to choose a pair of
opposite faces, each of which has at least two coloured corners (try it!).
Proof of Theorem 1.
We
construct the solution group
H
of Theorem
1
in a series
of HNN extensions beginning from G.
Before beginning let us fix once and for all a
pair of opposite faces of the cube.
Without loss of generality, we choose the top and
bottom face (as in Figure 1). Thus we are assuming that
at least two of a, (3, /,
8
are nontrivial elements of G, and similarly for a', (3', 1', 8'.
Let Go
=
G
*
x
,
the HNN extension of
G
with
trivial associated subgroup. Note that each of the
elements axb, cx-1d has infinite order in G0
,
so we may form the HNN extension
G1
=
Go, y
I
(axb)y(cx- 1d)y-
1
of Go.
Lemma 2. If at least two
of a, (3, /,
8 are nontrivial elements of G, and similarly
for a',(3',!',8',
then the elements eyf,lx-1 i form
a
basis
for a
free
subgroup of
rank 2 in
G
1
,
and similarly
for
gy-1h,jxk.
We then form the HNN extension
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