2
A. J. DUNCAN AND J. HOWIE
Neumann shows that Property A holds for words w of the form gxk, where g
E
G
and k
=f.
0, and hence that any group G can be embedded in a group H in which
every element is a k-th power, for all
k
0. Levin [L] generalizes Neumann's results
to show that Property A holds for words
w
of the form g1xg2x ... gtx and proves a
corresponding result on embeddings.
Clearly Property A does not hold in general (e.g.
w1
=
x,w2
=
x-
1g, 1
=f.
g
E
G) and the problem is to find necessary and sufficient conditions under which it
holds. Conditions can be imposed on the group
G
or on the words Wi· One such
is that
{w1
,···
,wm}
be non-singular: that is that then-tuples (di,
1
,···
,di,n),
i
=
1, · · · , m, are linearly independent, where di,j is the exponent sum of Xj in Wi·
The Kervaire-Laudenbach Conjecture is
Kervaire-Laudenbach Conjecture.
If {
w1
, · · · ,
wm}
is non-singular then Prop-
erty A holds.
(The attribution to Kervaire originates from a special case of the problem which
arises from consideration of higher dimensional knot groups [K], whilst Serre has
attributed the same special case to Laudenbach [B, Problem Section, p. 734].)
The set {
w1, · · · , Wm}
~
G * F is called K ervaire if Property A holds. The
following result of Gerstenhaber and Rothaus gives strong support to the Kervaire-
Laudenbach conjecture.
Theorem 1
[GR,R].
Let G be
a
locally residually finite group. Then {
w1
,
· · · ,
wm}
is Kervaire
if
it is non-singular.
(A group H is residually finite if, given any h
E
H there exists a finite group K
and a homomorphism ¢ : H
----
K with ¢(h)
=f.
1. A group is locally
P
if every
non-trivial finitely generated subgroup is
P
(where
P
is some property.)) Moreover
Howie [H1] has shown
Theorem 2.
If
G is locally indicable then { w1
, · · · ,
Wm} is Kervaire
if
it is non-
singular.
(A group H is indicable if there exists an epimorphism from H to Z.)
If w
E
G * F(x
1
, · · · ,
Xn) then the x-length of w is
2::7=
1
Ej where Ej is the number of
occurrences of
xj
1
in
w.
Gersten shows [G1] that to prove the Kervaire-Laudenbach
conjecture it is sufficient to prove it under the additional assumption that the x-
length of w is at most 3, fori
=
1, · · · , m. Furthermore he shows (loc. cit.) that
any set {
w1
, · · · ,
Wm}
of non-singular words of x-length at most 2 is Kervaire. In
addition Howie [H2] has shown that any non-singular set of two words of x-length
at most 3 is Kervaire.
In addition to the results of Neumann and Levin mentioned above there are
several special cases of Kervaire words with
n
=
m
=
1. For instance Edjvet
[EM1,EM2] shows that axkdx-l is Kervaire, provided that k
=f.
l and
(lal, ldl) =f.
(2, 3). Edjvet and Howie [EH] show that if
w
is a non-singular wordinG* F(x) of
x-length at most 4 then w is Kervaire.
The Kervaire conjecture remains unsettled. However even if it proves to be true
it doesn't tell the whole story. For example let H
=
G, x: x- 1ax
=
b
,
where a
and bare elements of G of equal order. Then G embeds in H in spite of the word
x- 1axb-
1
being non-singular. (cj. Edjvet's results above.)
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