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