Introduction
Let
F
be a finitely generated, free group, let Aut(F) denote the group of automor-
phisms ofF, and let End(
F)
denote the set of endomorphisms of
F.
The
rank ofF,
denoted r(F), is the cardinal of a free generating set of
F.
The
reduced rank ofF,
denoted r(F), is max{r(F) - 1, 0}, that is, one less than
the rank, except for the trivial group where the reduced rank coincides with the
rank, which is zero. By an inert subgroup of
F,
we mean a subgroup
H
such that
r(H
n
K)
$
r(K) for every subgroup K of F. This is quite a restrictive condition,
since the rank of intersections of subgroups of
F
in general can behave like the order
of the
product
of the ranks. In particular, if
H
is inert in
F
then r(H)
$
r(F).
If S
~
End(F), the
fixed subgroup of S,
denoted Fix(S), is the subgroup of
F
consisting of all those elements of
F
which are fixed by all elements of
S,
so
Fix(S)
=
{x
E
F
I
(x){J
=
x for all
{3
E
S}. For any
{3
E
End(F), we abbreviate
Fix(
{{3})
to Fix({J).
Building on work of Dyer-Scott [8], Gersten
[10],
Jaco-Shalen[15], Cohen-Lustig
[5], and other authors, Bestvina-Handel [2] proved the conjecture of G. P. Scott
that, if
{3 E
Aut(F), then r(Fix({J))
$
r(F).
Thomas
[24]
had used results of Gersten
[10],
Stallings
[22],
and McCool
[17],
to show that if B
~
Aut(F) then r(Fix(B)) is finite. In this paper we show that
if B
~
Aut(F) then r(Fix(B))
$
r(F). Our proof uses the main result proved
by Bestvina-Handel [2] (which is actually stronger than the Scott conjecture), in
a simplified form which was brought to light by Gaboriau-Levitt-Lustig [9]. We
combine this result with graph pullback techniques of Stallings
[21],
to show that
if
{3
E
Aut(F), then Fix({J) is an inert subgroup of
F.
A classical result, recalled as the Inverse Limit Theorem 1.4.12 below, bounds
the rank of an intersection of subgroups of bounded rank, and this then implies
that if
B
~
Aut(F) then Fix(B) is inert in
F,
so r(Fix(B))
$
r(F).
The surjectivity property of automorphisms is used nowhere in our proofs, so for
any set
B
of injective endomorphisms ofF, Fix(B) is inert in
F,
and in particular,
r(Fix(B))
$
r(F).
Concerning general endomorphisms, we can provide the following information
about what is known. Imrich-Thrner
[14]
had used the Bestvina-Handel result to
show that, if
{3
E
End(F), then r(Fix({J))
$
r(F). Recently, G.M. Bergman
[1)
has
used our result that r(Fix(B))
$
r(F) for any set
B
of injective endomorphisms of
F,
to show that the same inequality holds for any set
B
of endomorphisms ofF,
thus proving a conjecture we made in an earlier version of this paper. We conjecture
further that, for any set
B
of endomorphisms ofF, Fix(
B)
is inert in
F;
by recent
results of Thrner [25] it suffices to show that for any idempotent endomorphism
http://dx.doi.org/10.1090/conm/195
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