Sep 23, 1971 - and a free group (which may be trivial). Each of the ... free products with amalgamation, depending on the conditions ..... graph with one class.
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
^a'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 ^or 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).
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
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 ^or 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 ^ad
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\ ^e
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
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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
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
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.
in 1-1 correspondence
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).
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
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) ^U
{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
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
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
in 1-1 correspondence
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^Hl^ilaiHlß))
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
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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
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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~
n^iK)
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
(L^o 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
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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
in 1-1 correspondence
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
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