SUBGROUPS OF FREE PRODUCTS WITH AMALGAMATED

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY

Volume 181, July 1973

SUBGROUPS OF FREE PRODUCTS WITH AMALGAMATED SUBGROUPS: A TOPOLOGICAL APPROACH BY

J. C. CHIPMANÍ1) ABSTRACT. system is shown can be described

The structure of an arbitrary subgroup of the limit of a group to be itself the limit of a group system, the elements of which in terms of subgroups of the original group system.

Introduction.

It is the intent

arbitrary

subgroup

members

of that

lowing

of the limit system.

of this

paper

of a group

to describe

system

The description

that

the structure

in terms

of an

of subgroups

is obtained

is given

of the

by the fol-

theorem:

Theorem.

Let

H be any subgroup

H aß' ' aß: a> ß e A^' factors

Then

of H is the free

and a free the free

group

product

H is itself

product

(which

of the the

limit

limit

of a group

of a conjugate

may be trivial).

of a conjugate

of the group

system.

of a subgroup

Each

system

Each

of one of the

of the

of one of the

of the amalgamating

of a subgroup

ÍGa,

G

groups

is

H ~ and a free

group

(which may be trivial). The general

approach

a complex

K whose

and whose

topological

group

system

determined

will

to this

fundamental

problem

group

composition

parallels

be constructed.

by computations

will

The

of the fundamental

and terminology

§11 will give the construction

of the model

discuss

spaces.

§111 will

the relation

in similar

give

of the approach

algebraic

group

complex

here

system

composition

K with

space

be

of K.

throughout,

and

some properties

theorem,

to results

of that

H may then

of a covering

of the main

is to say,

group

of a subgroup

that will be used

the proof taken

That

of the given

of

and will

previously

obtained

settings.

Received

by the

editors

September

23,

AMS(MOS)subject classifications 55A10. Key ivords

and phrases.

Group

product, HNN group, generalized (1) Part of this material was

Dartmouth

the

structure

§1 will set the notation

its covering

be topological.

is the limit

College

and

supervised

1971

and,

in revised

form,

November

22, 1972.

(1970). Primary 20E30, 20F15; Secondary 55A05,

system,

fundamental

free product. included in the

author's

by Professor

Edward

group,

covering

Ph.D.

dissertation

space,

tree

submitted

to

M. Brown. Copyright © 1973, American Mathematical Society

77

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

78

J. C. CHIPMAN I.

system

The

algebraic

(Ga,

H -, z «: a,

ily of groups

\Ga:

a group

H-

Ga will

be referred

construction

being

ß £ A).

a £ AS; associated

and homomorphisms

the amalgamating

the diagram

with

system

that,

for any group

i aa: H ^

while limit

such

that

Iag=

that

¿a=

such

In other products

â'aß= ¿ga

and

is what

is called

of group

1.2.

G = (/,, /-,,-••

to make

product

G is called

to types

to as

the limit

of

is

a,

=

ß £ A) provided

Ga—» M, / „: H „ —* M

homomorphism

referred

k: G —> /Vf

to as generalized that

[5].

have

case

that

and the diagram

later

in §111.

of subgroups

an HNN group

group that

K be

base

K.

K and as-

of the form

t^^-1

= ^(L.,),

of the covering a space

Let

L . of K into

with

presentation

= c¿,(L,),

of coverings

imposed

[3].

a group

t^.j"1

free

been

One particular

be encountered

of the fundamental

reference

be referred

h - are monomorphisms

will

is groups

ß £ A.

been

of monomorphisms

, gen(K); rel(K),

z'-: ¿a:

for example

L . ,ch.{L .) if G has

For the computation need

a,

the

structure

The group

subgroups

¿a,

The

we mean

That

on the conditions

a tree

\ch \ a collection

(G a, H^,

have

on is where

fam-

Ga, G n there —« C»

will

sense.

a unique

aß ôr a^

or the

Definition sociated

^

structures

iaß,

later

type

and

Iao=

such

of a group

is an indexed

H^ -> G; a, ß £ A) where gj^

exists

H -, the

this

Another a group

there

depending

be of interest

is a tree;

kßiga,

Hsystem

of homomorphisms

amalgamation,

on the groups will

h^:

^'imlt °f tne ¿?rouP system

settings

with

the groups

of the group

in the categorical

M and family

there

of groups

Definition 1.1. (G, ga: Ga^G, ^' aß= S/3z/3a *s tne

every

is the limit

that

pair

By the

of the group

here

we mean

i aa: H „—> Ga and

to as the factors groups.

studied

By this

space

may have;

. . . ). we will

this

is

done via the following: Definition

1.3.

subcomplexes covering

of K.

whose

component

(ii)

consist

interest

conditions

(i)

The completed

of B. (^ B2 where

Definition

nonempty

{Ka: a e A|

members

Of particular ditional

Let

1.4.

sequence

given

2

of the complex

associated

with

A = Ka fot some

K by connected

\Ka'

a. e Aj

completed

covers

which

satisfy

the ad-

by

A„, ...

cover

,\

2

of the complex

of collections

of K by connected

A. is the set of all path

K consists

of subcomplexes subcomplexes

components

of pairwise

of a finite

of K such

A

If A e À., then

(iv)

The

consists

set of path of disjoint

A -

LA.

components

. is connected

intersections

A, n A

of

U^

and nonempty,

Is equal

sets).


that

of K.

where A 1,, A 2. £ A.1 — 1,. (iii)

is the

a or A is an arc

B2 £ X.

be those

An 72-stage

A„ is a cover

cover

of A where

B,,

will

be a covering

to A^ (or equivalently,

79

SUBGROUPS OF FREE PRODUCTS For convenience, II.

we write

In this

Of primary

section

importance

lem under

£ A); then

there

that

and

K^n

Let

is a complex

Let

\La,

L „: a,

va and from

v Q to each

of the

the resulting

complex

K . Note

77~g to va and

For each

let

y0, y/j and

y2 be representative

La,

and

L „ respectively. '

the paths

Let

_1

_

by attaching

the first

stage

it suffices

is homotopic

to

y',

and the arc tion iaß

s.

is the fact

an arc

v a)

complexes

attached. from

v,

s from

are unique

of discs

to

v 0 to

paths

s^

5) is now attached

of generators and

zßaia)

by identifying is done

been

to

for H „, in L^,

boundaries

with

for each

a,

/3 and

y'

ä

that

the inclusion CL, ß £ A,

of the way that

for

was

wilj

discs

a disc

Let

been

(5) O Ka).

not increase

D be such

in L a.

null

a disc loop for

we have Since

homotopic

z'o

that

y

is a

in L a and

*¡ Ga.

of L „, the arc from v^ A) ^ H ~.

induce

for

L „ are

LaU

does

L a

L „.

holds

the

to

attached

'a«(''aa)

already

ffj(Ka) * tt^LJ

maps

from

of the

y be a representative

have

n Kg) ^ n^L

anc* z/3a ôr eacn

by the addition

of such

and let

loop

so

7 -,

Ka to be constructed

(2) O K).

Ka n K „ consists

nliKa

of La, We

of attaching

the addition

the discs

Therefore ß,

of the union for the

of L a U (5) Cl K ), the result

in âd

r ~ in H^

from

consisting introduced

Now because

a representative

Thence

there

from 3) followed

that

i agir aß) - 1 and

a different

arcs

we consider

retract

loops

of the way that

hence in L a U (S) O Ka). For

have

this

discs

to show

to the relation

homomorphism,

nA,La,

discs

for a, i oßia\

of a L „ consists

of null homotopic

rise

which

of the construction. the addition

Because

with of these

1

a deformation

Therefore

rao.

n K „ —> K

appropriate

a £ a, a set

are attached

To see

the class giving

loops

each

Introduce

that

Ka be the complex

discs

the appropriate

constructed,

K

z7 „, 077^

obtained.

% Ga.

L a is clearly

a covering

and introduce

anci Y a- •s/3a)/2's/3a. ■ This

a £ A, let

rz,(Ko)

v0q.

A family

_1

ß and those

that

Since

i

yQ s oßY \s~öß

Now for each

v a.

that

with

ß £ A and each

Discs

K be the complex

L o for all claim

a,

K has

of complexes

so formed

of the

s aa from each

prob-

H o, z .,: a, ß

771(KaO K„)^ maps

point

of 1-spheres

as follows.

a.

at this

spaces

K

(Ga,

Moreover,

^ Ga,

be a family

and call

and

system

W G.

tr^Kj

remark

not in the

algebraic

respectively.

ß £ A\ ê

for the

by the inlcusion

zo

to be a wedge

v0, v.

ttAK)

that

will be considered.

as follows:

of the group

induced

aß> vaß1 % H aß'

points

each

limit

and

model

is given

K with

in

constructions

topological

\Ka: a £ Ai such

will be considered

v.,

G be the

K „ —» K „ are

Proof.

Select

be the

the homomorphisms

^ ^ 0! n\^

topological

Its formulation

II. 1.

by subcomplexes such

will

discussion.

Theorem

X = U"_0\-

the necessary

Inherent

the desired


to vQ,

in the construchomomorphisms

80

J. C. CHIPMAN To show

tree

that

77,(K) «s G a maximal

is constructed

imal trees imal tree

T a.

a vertex arc

by first

in each

of the

Note

that

taking

L„.

s.

Thus

\JaT

tree

meets

in each

in a maximal

tree

of fundamental of the arc

s

II.2.

The

from

v^

to vQ, and max-

is then

of the completed by [l] to give

n.{K)

have

of any of the

be the associated

tree

L „ to

contain

in K.

generated

the

This

by the

as the limit Since

77,(X) % G as was

to a max-

of 5) must

cover

of the cover.

maximal

extended

from a vertex

a maximal

we then

K

of the system

the fundamental

group

desired.

completed

cover

of {K : a £ AJ

and define

(i) A0 = {/Ca: ae (ii)

K.

in K and is in fact

member

2

tree

for

discs

suffices

Let

above

s, the arcs

Ka this

T a the path

of the members

is trivial

Corollary constructed

which

groups

is chosen

of any of the attached

a is a tree

every

the arc

In each

in L a or the interior

maximal

tree

Aj,

A, = iL „U s U arc from v^g to vQ: a, ß £ A\,

(iii) A2 = iH Then

2

is a 2-stage

ProoL

This

cover

is immediate

For our purposes space tion

of its fundamental

completion

of this

will

also

is that,

points

x,y

the fact

2

path

final

topological

cover.

{L

be lifted

fact

needed Let

covering

. u s U arc from

III. system

In this will

section

be studied.

conditions

down

is a 2-stage

1.3.

a 2-stage

consist

Take

of the

arc

A

iLa„

The

that

x

that

arc

p~

also

been

This

to y . This

is)

as

this

subgroups

given

before


a,

choose and use

p{x ) to piy ). gives

of K.

of

A Q, A ,, and

be made

vQ),

us the

namely:

space

components

of to

is not im-

UA,from

problem,

covering

of various will

have

is).

vaa

To see

in Ka-

in A from

vao to vA), and

and results

only detail

the the

of

of p~

p{x ), p{y ) in Ka fot some

of the

An attempt

that

components

U s U arc from

the algebraic

the structure

by considering

to check

be connected.

{K, p) be an arbitrary

consisting

the computa-

to be the arc components

to find a path

to study

covering

facilitate

is obtained difficult

is).

to points

to a path

will

not

of p~

of p

cover

an arbitrary

the arc components

for A e AQ, A - IjA , will

can then

Corollary

braic

will

arc components

£ A, project

that

This

p~

2,

choice

It is

vag to v A and

be a 2-stage

A , to be the

of K that

An obvious for a £ A.

A2 to be the arc components

mediate

has

cover,

U arc from

(Ka)>

and

(Ka)

of K,

in studying

a covering

group.

of p~

iLaßUs

cover p~

from the construction

we will be interested

{K, p) of K and finding

arc components

p~

of K.

p~ \7

Then

{K ), respectively.

of the limit

to consider

K

of a group

various

and to interpret

algetheir

81

SUBGROUPS OF FREE PRODUCTS their

topological

problems

counterparts.

assures

that

of a group

ing space

system

of the type

covering

space

selves

a 2-stage

respect

now.

Suppose we let

that

ponents

Edges

has

limit

are then been

done

of a system

For each

there

corresponding

exists

between

edges

of S

to 77j(A) * FiDiA)) induced

"path"

by the

automorphism

in the

W. from

given

it will not be too cumbersome

here.

The

theorem

that

we recall

that

this

trees

in DiA)

of X which

DÍB)

a

are related from

involved

of the

—* DiA)

corresponding

followed

to the

for an 72-stage

of the 2-stage

naturally

with

A homomorphism

homomorphism

may prove

most

is in 1-1 cor-

The homomorphism

B —»A and

for the case arises

DÍA)

in A,

will be the product

While

as the

is as follows.

of the maximal

B C A.

Max-

W .. Once

of 77.(A) and

The set

is contained

maps

intersection. of the

K is given

to members

that

with the arc com-

description

is the free product DiA).

i = 1, 2, • • • ,72

in 1-1 correspondence

of each

whose

by the word

B to A.

in [l];

For

to nonempty

of the complex

inclusion

cover

structure

according

corresponding suppose

homomorphisms

of the cover-

of its cover.

given

of vertices

of the complements

For example

Determin-

group

K,

of the components

which

to a member

a

are them-

II.1.


of generators

77j(B) * FiDiB)) by the inner

one class

group

is a group

such

which

of the members

of the complex

and homomorphisms

any two groups

by containment.

with

for each

on a set

with those

vertex

the fundamental

a cover-

there

[l].

cover

the fundamental

a free group

respondence

of spaces

theorem

for a given

be studying

noted

of Theorem

are introduced

chosen

we will

consisting

and the other

of groups

A £ 2

FÍDÍA)),

see

which space

Thus

of the cover

groups

of algebraic

theorem

a covering

As was

of Van Kampen's

graph

of A._.

of LJa ..

limit

exist

vA % H.

in §II.

in computing

2, is an 7z-stage

W. be the bipartite

imal trees

of that

cover

For details,

with the members

this

rest

there nAK,

to the fundamental

we will use the version

its formulation

vA

treatments

and the basic

that

constructed

of the members

of H will

with

For this

and a subgroup

spaces

ing the structure ing space

H of nAK,

of model

topological

spaces

v Q £ K such

possesses

covering

most

of covering

for any subgroup

(K, p) of K and basepoint limit

As with

the key tool is that

covers

considered

from the topological

set-

ting is then

Theorem (Ga,

Ha„,

III. 1 • Let

ia„:

of the factors

a,

ß £ A).

Then

is the free product

and a free group free product

H be any subgroup

(which

H is itself

of a conjugate

may be trivial).

of a conjugate

of the limit

of a subgroup

Each

the limit

of the group

of a group

of a subgroup

system.

groups

H' „ and a free

(which may be trivial).


Each

of one of the

of the amalgamating

of one of the

system

G

is the

group

82

J. C. CHIPMAN Proof.

Using

the construction

fundamental

group

is the limit

of that let

complex.

As was

noted

{K, p) be the covering

By Corollary

II.2,

of the inverse themselves groups The

correspond

corresponding

will

member

for A2 will

fundamental

group

the

cover

them

of the group

be trivial

2 which

cover

onto

groups

the group

H.

of the arc components

of 2

their

are

fundamental

of the members

of the subgroups

Since

each

of 2. of the

of the fundamental

s) the computation

described

Now

subgroup

The members

conjugates below.

whose

be 2-stage.

and as such

A? is the arc

yield

will to the

consists

of 2. of 2

system

(recall

K be the complex

\Ka: a £ AS be the covering

corresponds

of the fundamental

carry

of K will

let

X of this

K which

of the members

to subgroups p

II.1

and let

p of the members

spaces

monomorphism

groups

of

a 2-stage

under

covering

there,

space

K has

images

of Theorem

of the system,

of the

in the statement

of the

theorem. We would

now like to consider

the particular setting.

types

Since

phisms

in these

of the group

conditions

hold

following

by inclusion

be a monomorphism.

required

that

case

we have

A, B are covering

are monomorphisms;

Note that the same

group theorem

The

system

first

of A and

then

of reasoning

applies

setting

the

considerations

places

some

restriction

induced

by inclusion

diagram:

B the homomorphisms

{p | A)

,

follows.

to z'^: 77,(A) —> 77,(K) for

have

divided

themselves

on the structure

the amalgamating

or requiring

a single

is typified

i¿ 77,(A) —>

A.

as requiring

second

The

77,(B)

such

The

if these

for any two corresponding

commutative

of Kurosh)

in [3] and [6]).

homomor-

that

(p | B)t

the proposition

type

of

77,(B)

z'^: 77,(A) —> 77,(K) for A over

In the algebraic two lines.

the various

hold for H as well.

Then

the following

spaces

for some

in an algebraic

to check

A C B the homomorphism

77,00 -£-»

A £ AQ and

will

theorem

end.

(pM)* Since

that

we need

of G they to that

77,(A) Û

{p I ß)

considered

is a monomorphism.

will

In this

of this

been

Let A, B e 2 such that the homomorphism

A, B of 2 such

Proof.

forms

have

it is usually

is devoted

members also

that

are monomorphisms

in the representation

III.2.

induced

the various

systems

cases

system

proposition

Proposition 77,(ß)

of group

groups

amalgamating in the original


to be trivial subgroup

along

of the original

(the

original

(as considered

work of H.

Neumann

[5]

83

SUBGROUPS OF FREE PRODUCTS which

makes

restriction

no restriction

of restriction

Let

vertices

system.

Then

product

should

edges

expect

Since

that

with

This

it will

yield

is a tree group

III.3.

final kind

on the original

that

is to say a graph

and amalgamating

groups

the homomorphisms

is what

Karass

is basically

a simplification

in some the first

of the

of the

and Solitar

a geometric

with

call

a

simplification

in the topological

we

setting.

It

is

// the diagram

then there

system.

setting

condition

with

situation

but results

In this

system;

the factors

the condition

and the form it takes Lemma

of the group


D is a tree.

in [3].

system

as the following

in 1-1 correspondence and with

group

may be described.

D be the diagram

system

does,

that

can be summarized

structure.

tree

on the original

on the subgroups

of the group

is a complex

K possesses

K with

a 1-stage

system

nAK)

ÍGa,

Haß,

isomorphic

covering

which

ia„:

a,

ß £ A)

to the limit

satisfies

of the

the properties

of Theorem II. 1. Proof. However, without arcs

The

construction

since disturbing

discs

as before

is of course

the appropriate contend

final Before

will

permits

has

almost

this

will

results

yield

tree

is obtained

this

for free

the necessary

to those

the neces-

cover

of the fundamental

to the Kurosh

products

cover.

theorem

here

to the

in [4].

a single

amalgamated

explicitly

It should

contained

of

cover

by Massey

with

to be group

a 1-stage

let us examine

from a 1-stage

disjoint

to attach

to the one given

products,

similar

s, introduce

Proceed

one only has

approach

identical

the arc

II.1.

may be made

be 1-stage.

the computation to be made,

of applying

what

presentation

what

be noted

by studying

type that

such

explicitly.

Suppose {L^ „: a,

that

a maximal

K has

a 1-stage

ß £ AS where

of Ka n K ß.

have

when

be a proof

Serre [7] has obtained

groups

now, space

as in Theorem

modification

vn.

cover

and the more general

of group

of using va and

which

that

considering

subgroup

Instead

but now the final

In the case

will

essentially

the following

of a construction

covering

with.

result

result.

vag to the corresponding

The importance 1-stage

proceed

is now a tree

the final

from each

sary

of K will

the diagram

Suppose

tree

the group

(1)

has

L^ „ over that

been

we use

chosen

cover

X with

y is used G a fot

for

W

AQ = \Ka:

to denote

n^Kj

the

and

a £ A! and

the various H^_ ß for

fundamental

A, =

arc components

77j(L£

group

of

„).

Once

K will

presentation:

(Ui\,heniGa)UUeliGa)},Ul^HlîlaiHlß))

where

Ga has

of the

maximal

the group tree

presentation

in W

and the

(gen(Ga): homomorphisms

rel(Ga)), i^n-

\t \ is the complement H „ —> Ga and


84

J. C. CHIPMAN

aß :

a. ß —' G a are either

phism

followed

by conjugation

will be of the general

(2)

an inclusion

by one of the

/..

or an inclusion

So the presentation

homomor-

of n.{K)

form:

(Ui.},{genGj:irelGj,{H^=H^},{£.//|/8Ç1=ff^}). In [3] Karass

and Solitar

study

G - {A * B; U) the free product is then

(3)

the problem

of A and

determined

to be an HNN group

H = {tv t2,-..

,S;telS,t1U*lt~l

where

S is a tree

product

H and associated

be the group

only been version

using

the fact that

Its statement

Theorem HNN group

subgroups jugates

III.4.

also

If we compare

while

limit

of a group

product.

of subgroups

presentation

that

this

III.5.

(a, : r2) respectively.

from the group

H.

groups

has

the appropriate

rather

(restricted

which

subgroups

groups.

system

III.4

is not given

than

an algebraic

case

the

as a tree

the subgroup

to a simple

we notice

is itself

of the group anomaly.

as a tree

to avoid

complex-

may be made. G,

—» G,

a presentation

a2, /:r,,

a2, t, yx:r,,

of

are con-

have and

0

group

presentations

(a,:r.)

: HQ —> G2 be homomorphisms

(x : r„).

Then

the presentation

r2, Ö,(x) = /t92(x)/_1)

to the presentation

H = (a,,

H is an

The associated

group

product

Then

are conjugates

groups

of G.

that will give

Let

0. : H.

G.

which

but it is only a result

suggested

G.,

has

S) to

we have

of any tree

III. 1 and Theorem

of a tree

Let groups

which

vertices

of an arbitrary

proposition

ß = (a,, is equivalent

of a tree product

is disappointing

how the adjustment

Proposition

with

(.S^rel

Since

is a tree

amalgamating

group presentation

The following

ity) indicates

H

Comparing

if we take

for any subgroup

of Theorem

the subgroup

the topology

is an alternate

product.

hold

H.

= Hy-g \).

of the amalgamating

of the limit

glance

SßZ-

with

of G and with

system,

U.

with

of the system

will

product

the statements

At first

Ga\,

of the amalgamating

the subgroup

subgroup

be

is a tree

be conjugates

that,

There

theorem

of the vertices

H of

of A or B intersected

what will occur

!rel

H be any subgroup

base

of subgroups

Ga\:

subgroup

= U°2,...)

of U intersected

the diagram

would Let

whose

conjugates

is precisely

(Igen

of the Karass-Solitar

product.

B with amalgamated

= U^l,t2U°l,t-l

conjugates

that this

presentation

of an arbitrary

of the form

with vertices

subgroups

(3) with (2) we see

will

homomorphism

r2, r , y , x —>¡y f

along

y ) to the above

denote

y

presentation

with

will yield

= 02(x)).

r, in r , 6Ar)

its conjugate

relation

of relations

:rl ,r2,ry,6li\)

with

,yx

for any relation,

of r2 and may be added

operations.

to this

yx = ö2(x)).

:r1? r2, f>\(x) = ty%t"1

6L is a homomorphism,

consequence pressed

r2, r>\(x) = td2ix)t~l,

set

via a series

so added

= ty%t

the presentation

will

be in the

by t (both

ex-

of Tietze

I

to obtain

l, ys = 02(x)).

(y , t : r ), then

the functions

induce homomorphisms 0,: Hfí —t H, and 0,: HQ —»H, by

the way we constructed

r . This

gives

H then

as the limit

of the group

system

described. One may ask as well amalgamated

a single question problem

subgroup

the subgroup

is answered

in the affirmative

yields

a description subgroup,

is generated

group.

a different

covering

of a single

of this

a covering components

components

will

give

Ordman

obtains.

topologically

iL)\

We now turn

such

For the

and

The interest

would

the construction

W. being

product

here

is that

treatment with

of the

single of sub-

amalgamating

amalgamating arise

group

by choosing

of K.

In the

of the topological

model

Ka O K„ = L, with 77.(L) being the

covering

the complete

of free

with

to the subgroup

tree

in K and let

has

one could

K = T

Ka n K~= T U iarc

bipartite

graph

amalgamated

the differences

Kf ..

subgroup

to the second

type

in the description

of algebraic


situation

This

that

to algebraically.

our attention

with

This

by conjugates

of the fundamental

this

product

and whose

of the original

T be a maximal

Then

This product

system,

(K, p) corresponding

Let

(Ka)S.

a single

as a free

is generated

a description

with

or tree product.

in [6].

as the free

factors

subgroup,

as the type

as opposed

by Ordman

of subgroups

paper

as follows.

of p~

nîK)

be described

an HNN group

I Ka: a £ A| with each

of p~

of a free product

of the original

for the computation

subgroup.

U iarc

of whose

factor

amalgamated

choose

than

of the subgroup

each

K would have a cover amalgamating

rather

by conjugates

In the setting

case

can also

subgroup

of the corresponding

group

in the case

amalgamated

amalgamated

groups

whether

as

arise

86

J. C. CHIPMAN

typified will

by the original

be made

subgroup

upon

H which

factors

were

type

which

group

with

was

either

of the original

obtained

group

conjugates

also

the condition

inal

group

was

itself

of factor

as the limit

certain

free groups,

jugates

of subgroups

Recall

system

products

that

with

ditions

that have

in terms

of the amalgamating

a 2-stage

cover

been

of W2 be trees

2, A 2 consists

of the arc components

p~

of vertices

in

(Lô U arc from

C were

that

graph,

where

edge

e.

iLag

follows

an arc component

W2 is a bipartite

p~

correspondence

is

an

u arc from

back

to another

each

such

with

that

a may be factored from

vag

a component

to vA) and

component

component

assume

of W2 which

of p~

such

that

is).

the

K while

group

piy ) is a loop in (L

H.

Therefore

into a product

a corresponds

of elements

algebraic

point

a nontrivial

is)

of with

of K is

van

an arc

to a member

from the amalgamating

y. in

y . is a path

corresponds

is not a member

to an of the

of ß (a loop in K) which groups

none

of which

factors

belong

to H. In summary

which

then

do not contain Proposition

individual

¡actors

III.6.

we have

proved

the basepoint // ß

contains

of any confugate

(taking

into account

components

of W2

of K) the corresponding in H of a product


in

2

to v A), y.

which

of component

17

subgroups

Since

e ., et

same

e ., e' . Since

is).

Suppose

a.

to a component

we may choose

to

p~

v„.

path

of paths

from that

Accordingly

cover

of one

components

contains

p~

con-

W2 consists

the base

is an edge

~ u arc from

of one of the amalgamating

H. be

of W, may be arbitrary.

thus arc

7

element

for

we will

The


y. corresponds

r\j

groups

system

into a collection of

e'

and

on the 2-stage

is),

ß

only con-

condition

of p~

on the

subgroups system

of 77.(K).

the components

22 - to vA) and the other

For the discussion

that

1-1

while

group

type

describes

as amalgamating

the topological

the components

our attention

of ß that permits

on the original

For

amalgamating of the other

of the original

in the computation

set place

the same

was

of the orig-

and amalgamating

groups

no restriction

There

subgroup

IV. 1 already

let us now try for a description

sub-

groups

groups.

so we will focus

of factors

as a normal

the origin

Theorem

The

or another

amalgamating

with seen

of subgroups

that

class

system

already

Since

of conjugates

now with

dealing

H.

factor

The

of the

system.

system

amalgamating

of the subgroup,

on the subgroup

of a group

are free

we have

description

of a group

in the smallest

of a group

III.5

in the description placed

subgroup.

now no restrictions

The

of an original

normal

of the original

a limit

that

of the original

H not be contained

In Proposition

condition

Recall system.

of the limit

of a subgroup

Z by this

that

[5]. group

of subgroups

of subgroups

that

subgroups.

was that

conjugates

had a conjugate factor

were

that

work of H. Neumann

the type

conjugates of elements

of the from the

87

SUBGROUPS OF FREE PRODUCTS amalgamating conjugates

groups

In the case matches

differs

using

for

that

K as was done

construction where

cannot before.

of K which

it was absolutely

here

It would

probably

minimize

necessary

Either

subgroups

but

all subgroups

groups.

can

using

In the latter

differ.

choosing

We would

a different

be necessary

to permit

approach

may be described

condition

by merely

the use

of

of the original

amalgamating

and the algebraic

only

of the subgroup

of subgroups

or some

be made

would

the topological

groups.

of the original

condition

a match

then,

of H consist

of G.

to the factors

of conjugates

and a free group,

the topological

jecture

system,

of the amalgamating

of subgroups

groups

groups

with regard

free products

groups

only conjugates

group

treatment

in the description

amalgamating

the amalgamating

of the amalgamating

of the arbitrary

the algebraic

be described

case

of G, then

of subgroups

to give

of the arc

an alternate

s to those

the construction

con-

covering

situations

of the desired

complex.

REFERENCES 1. (to

J. C. Chipman,

Van Kampen's

appear). 2. R. H. Crowell

1963. 3.

theorem

and R. H. Fox,

for n-stage

Introduction

to knot

covers.

Trans.

theory,

Ginn,

and D. Solitar,

W. S. Massey,

New York, 1967. 5.

Algebraic

The subgroups topology:

Generalized

E. T. Ordman,

free

Mass.,

of a free

product

of two groups

J. Serre,

An introduction,

Harcourt,

with

an

MR 41 #5499. Brace

& World,

Groupes

products

with

amalgamated

subgroups.

II, Amer.

MR 11, 8.

On subgroups

Philos. Soc. 69 (1971), 13-23. 7.

Boston,

Soc.

MR 35 #2271.

H. Neumann,

J. Math. 71 (1949), 491-540. 6.

Math.

MR 26 #4348. A. Karrass

amalgamated subgroup, Trans. Amer. Math. Soc. 150 (1970), 227-255. 4.

Amer.

discrets,

of amalgamated

free

products,

Proc.

Cambridge

MR 43 #2102. Notes

written

at College

de France,

1968/69.

DEPARTMENT OF MATHEMATICS, OAKLAND UNIVERSITY, ROCHESTER, MICHIGAN 48063
