On intersections of normal subgroups in groups
The paper is a generalization of [2]. For a group H = ‹A|O›, conditions for the equality Ñ₁ ∩ Ñ₂ = Ñ₁, Ñ₂] are given in terms of pictures, where Ñi is the normal closure of a set R¯ i ⊂ H for i = 1, 2.
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irk-123456789-1565992019-06-19T01:28:42Z On intersections of normal subgroups in groups Kulikova, O.V. The paper is a generalization of [2]. For a group H = ‹A|O›, conditions for the equality Ñ₁ ∩ Ñ₂ = Ñ₁, Ñ₂] are given in terms of pictures, where Ñi is the normal closure of a set R¯ i ⊂ H for i = 1, 2. 2004 Article On intersections of normal subgroups in groups / O.V. Kulikova // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 4. — С. 32–47. — Бібліогр.: 6 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 20F05. http://dspace.nbuv.gov.ua/handle/123456789/156599 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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The paper is a generalization of [2]. For a group
H = ‹A|O›, conditions for the equality Ñ₁ ∩ Ñ₂ = Ñ₁, Ñ₂] are
given in terms of pictures, where Ñi is the normal closure of a set R¯
i ⊂ H for i = 1, 2. |
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Article |
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Kulikova, O.V. |
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Kulikova, O.V. On intersections of normal subgroups in groups Algebra and Discrete Mathematics |
| author_facet |
Kulikova, O.V. |
| author_sort |
Kulikova, O.V. |
| title |
On intersections of normal subgroups in groups |
| title_short |
On intersections of normal subgroups in groups |
| title_full |
On intersections of normal subgroups in groups |
| title_fullStr |
On intersections of normal subgroups in groups |
| title_full_unstemmed |
On intersections of normal subgroups in groups |
| title_sort |
on intersections of normal subgroups in groups |
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Інститут прикладної математики і механіки НАН України |
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2004 |
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http://dspace.nbuv.gov.ua/handle/123456789/156599 |
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On intersections of normal subgroups in groups / O.V. Kulikova // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 4. — С. 32–47. — Бібліогр.: 6 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT kulikovaov onintersectionsofnormalsubgroupsingroups |
| first_indexed |
2025-07-14T08:59:41Z |
| last_indexed |
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1837612224483426304 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 4. (2004). pp. 32 – 47
c© Journal “Algebra and Discrete Mathematics”
On intersections of normal subgroups in groups
O. V. Kulikova
Communicated by A. Yu. Olshanskii
Abstract. The paper is a generalization of [2]. For a group
H = 〈A|O〉, conditions for the equality N̄1 ∩ N̄2 = [N̄1, N̄2] are
given in terms of pictures, where N̄i is the normal closure of a set
R̄i ⊂ H for i = 1, 2.
Introduction
The present paper is a generalization of [2]. So here we will use the
definitions and notation from [2]. Moreover, in most of proofs in this
paper we will refer to the proofs in [2] after necessary remarks.
Let F be a free group generated by an alphabet A.
Let H be a group given by presentation 〈A|O〉, where O is a set of
words on A. By N denote the normal closure of O in F . We will suppose
that O is the set of all words vanishing in H, i.e., O consists of all words
from N .
Consider two sets of elements R̄1, R̄2 ⊂ H such that r̄1 6= tr̄±1
2
t−1 in
H for any r̄1 ∈ R̄1, r̄2 ∈ R̄2, and t ∈ H. By N̄i denote the normal closure
of R̄i in H for i = 1, 2. The aim of this paper is to find out necessary
and sufficient conditions for
N̄1 ∩ N̄2 = [N̄1, N̄2] in H. (1)
These conditions will be expressed in terms of certain geometric objects
called pictures (see, for example [1] or [2]).
The research of the author was partially supported by the RFBR (No. 02-01-
00170).
2000 Mathematics Subject Classification: 20F05.
Key words and phrases: Normal closure of sets of elements in groups, presen-
tations of groups, pictures, mutual commutants, intersection of groups, aspherisity.
O. V. Kulikova 33
One can reformulate the above problem as follows.
Let Φ : F → H be the canonical homomorphism. For each r̄ ∈ R̄i let
r be its reduced lift to the free group F and Ri be the symmetrized set
of {r|r̄ ∈ R̄i} for i = 1, 2. By N1 and N2 denote the normal closures in
F of R1 and R2 respectively. Then equality (1) holds iff
N1N ∩N2N = [N1, N2] ·N. (2)
Generally, we will consider the latter situation and conditions for (1)
will be gotten as consequences. In particular, we will show that if a
presentation 〈A | R1, R2, O〉 is aspherical (the definition of it will be
given below) then equalities (1) and (2) hold.
The paper is divided into two sections, each of which is further sub-
divided. In the first section we give main definitions, formulate the main
result (Theorem 1), prove corollaries of it and consider some examples.
The second section is devoted to the proof of Theorem 1′ which is equiv-
alent to Theorem 1.
I am grateful to my scientific advisor professor A.Yu.Ol’shanskii for
his constant attention to my work.
1. Formulation of theorems and corollaries
1.1. Definitions. Relations between definitions
In the beginning we give some definitions generalizing notions of [2].
By H we denote a group given by 〈A|O〉, by 1 the identity in H. Let
G = 〈A|O ∪R〉
be a presentation of a group, where R is a symmetrized set of cyclically
reduced words on A.
Since the set of defining relations is the union of O and R, a picture P
over G contains vertices of two types. Vertices of the first type correspond
to defining relations of the set O. Such vertices will be called 0-vertices.
Vertices of the second type correspond to the set R. They will be called
R-vertices. Note that this situation is dual to one considered in [4, 6].
Note that in this paper R generally will consist of two sets R1 and
R2.
For two words u and v on A, u ≡ v means that u is equal to v letter
by letter.
A dipole in a picture P over G = 〈A | O,R〉 is two R-vertices V1 and
V2 of P if there is a simple path ψ connecting points p1 and p2 lying on
the circles C1 and C2 around these vertices such that
Lab−1(ψ)Labp1
+(C1)Lab(ψ)Labp2
+(C2) = 1 in H.
34 On intersections of normal subgroups in groups
A picture over G = 〈A | O,R〉 is reduced if it does not contain a
dipole.
A presentation G = 〈A | O,R〉 is aspherical if every connected spher-
ical picture over G = 〈A | O,R〉 contains a dipole.
Further we will need the following analogue of a well-known result
(use Theorem 11.1 of [4] and dualise).
Lemma 1. Let W be a non-empty word on the alphabet A. Then W
belongs to RF ·N if and only if there is a disk picture over the presentation
G = 〈A | O,R〉 with the boundary label W .
Note that if R is empty, the formulation of Lemma 1 is standard. In
particular, W̄ ∈ Φ(R)H (where Φ : F → H is the canonical homomor-
phism) iff there is a disk picture over the presentation G = 〈A | O,R〉
with the boundary label equal to a lift of W̄ to the free group on A.
We will use 0-transformations of a picture P over G defined as follows.
(1) Let ψ be a simple closed path not crossing through vertices of P
and dividing P into two parts one of which does not contain R-vertices
and hence forms a disk subpicture P̃ overH. Assume that another picture
P̄ can be constructed over H with Lab(∂P̄ ) ≡ Lab(∂P̃ ). Then replacing
of P̃ by P̄ in P is a 0-transformation of the first type.
(2) Let γ be a simple path not passing through any vertex of P and
intersecting some edges such that Lab(γ) = 1 in H, i.e. Lab(γ) ∈ N .
We can assume that γ is not closed, otherwise cut out from γ a small
interval not intersecting edges and denote the obtained path by γ. It is
possible to draw a simple closed path ψ by-passing near γ in the both
directions so that ψ intersects the same edges as γ does. Hence Lab(ψ) ≡
Lab(γ)Lab−1(γ) and ψ divides P into two parts one of which does not
contain R-vertices and forms a disk subpicture P̃ over H. To use a 0-
transformation of the first type, let us construct a new disk picture over
H with the boundary label equal to Lab(∂P̃ ).
There are a disk picture P ′ over H with Lab(∂P ′) ≡ Lab(γ) and a
disk picture P ′′ over H with Lab(∂P ′′) ≡ Lab−1(γ) (by Lemma 1 in the
case of empty R). It is clear that it is possible to divide the boundary
of P ′ into two non-trivial arcs ς ′ and ς̄ ′ one of which (say ς ′) is met by
edges and Lab(ς ′) ≡ Lab(∂P ′), and the other one (ς̄ ′) is not met by edges.
Similarly the boundary of P ′′ can be divided into two non trivial arcs ς ′′
and ς̄ ′′ such that Lab(ς ′′) ≡ Lab(∂P ′′) and ς̄ ′′ is not met by edges. Pasting
together the disk pictures P ′ and P ′′ by the arcs ς̄ ′ and ς̄ ′′ gives a disk
picture P̄ with the boundary label equal to Lab(γ)Lab−1(γ). Replacing
of P̃ by P̄ in P (that is the 0-transformation of the first type) gives rise
to "cutting" of the edges intersected by γ. This transformation of P is a
0-transformation of the second type. It is denoted by T 0(γ).
O. V. Kulikova 35
Notice that 0-transformations change neither the number ofR-vertices
in P nor the boundary label of P .
Definition 1. Let R1 and R2 be two sets of words on A. We say that a
presentation G = 〈A | R1∪R2∪O〉 is (R1, R2)O-separable or satisfies the
condition of (R1, R2)O-separability if for every reduced spherical picture P
containing both R1-vertices and R2-vertices, there is a simple closed path
γ dividing the sphere into two disks such that the following conditions
hold:
1) the both disks contain R-vertices;
2) Lab(γ) = 1 in H.
Remark. In this paper the words "to contain R-vertices" mean "to
contain R1-vertices or R2-vertices or both R1- and R2- vertices". The
words "to contain only R1-vertices" mean "to contain no R2-vertex". In
any case there may be 0-vertices in P or may not be.
Assertion 1. If in Definition 1 "every reduced spherical picture" is re-
placed by "every spherical picture", then the set of presentations satisfying
(R1, R2)O-separability is not changed.
Proof. Let P be a non-reduced spherical picture over G = 〈A | R1∪R2∪
O〉 containing both R1-vertices and R2-vertices. Since P is not reduced,
there is a dipole, i.e., there are two R-vertices V1 and V2 and a simple
path ψ connecting points p1 and p2 lying on the circles C1 and C2 around
these vertices such that Lab−1(ψ)Labp1
+(C1)Lab(ψ)Labp2
+(C2) = 1 in
H. It is easily seen that a simple closed path γ from Definition 1 may
be obtained going around V1 and V2 and by-passing near ψ in the both
directions.
Assertion 2. If every spherical picture over a presentation G = 〈A |
R1 ∪ R2 ∪ O〉 containing both R1-vertices and R2-vertices is not reduced
(this condition will be called (R1, R2)O-asphericity), then the presentation
is (R1, R2)O-separable.
Proof. By Assertion 1 the presentation G = 〈A | R1 ∪ R2 ∪ O〉 is
(R1, R2)O-separable if for every spherical picture P containing both R1-
vertices and R2-vertices, there is a simple closed path γ dividing the
sphere into two disks such that the following conditions hold:
1) the both disks contain R-vertices;
2) Lab(γ) = 1 in H.
36 On intersections of normal subgroups in groups
So if every spherical picture over the presentation G = 〈A | R1 ∪R2 ∪O〉
containing both R1-vertices and R2-vertices is not reduced, then such
path can be found similar as in the proof of Assertion 1. Hence (R1, R2)O-
aspherical presentations are (R1, R2)O-separable.
Definition 2. Let R1 and R2 be two sets of words on A. We say that
a presentation G = 〈A | R1 ∪ R2 ∪ O〉 is weakly (R1, R2)O-separable or
satisfies the condition of weak (R1, R2)O-separability if for every reduced
spherical picture P containing both R1-vertices and R2-vertices, there is
a simple closed path γ dividing the sphere into two disks such that the
following three conditions hold:
1) the both disks contain R-vertices;
2) Lab(γ) ∈ [N1, N2] ·N ;
3) if one of the disks does not contain R2-vertices, then the other one
does not contain R1-vertices.
Remark. If the condition 3) in Definition 2 is omitted, then the set of
presentations satisfying only 1) and 2) of Definition 2 is wider as is shown
in [2] for H = F (A).
Remark. Weak (R1, R2)O-separability is not equivalent to the condition
of (R1, R2)O-separability as is shown in [2] for H = F (A).
Definition 3. Let R1 and R2 be two sets of words in F (A). We say that
a presentation G = 〈A | R1 ∪ R2 ∪ O〉 is strictly (R1, R2)O-separable or
satisfies the condition of strict (R1, R2)O-separability if for every spherical
picture P containing both R1-vertices and R2-vertices, there is a simple
closed path γ dividing the sphere into two disks such that the following
three conditions hold:
1) the both disks contain R-vertices;
2) Lab(γ) ∈ [N1, N2] ·N ;
3) one of the disks does not contain R1-vertices and the other one does
not contain R2-vertices.
1.2. Formulation of theorems and corollaries
Now consider the normal closures N̄1 and N̄2 of sets R̄1 and R̄2 in a group
H = 〈A|O〉 and the canonical homomorphism Φ : F = F (A) → H. We
assume that r̄1 6= tr̄±1
2
t−1 in H for any r̄1 ∈ R̄1, r̄2 ∈ R̄2, and t ∈ H. For
each r̄ ∈ R̄i let r be any of its reduced lifts to the free group F and Ri
be the symmetrized set of {r|r̄ ∈ R̄i} for i = 1, 2.
O. V. Kulikova 37
Theorem 1. A presentation G = 〈A | R1∪R2∪O〉 is weakly (R1, R2)O-
separable if and only if N̄1 ∩ N̄2 = [N̄1, N̄2] in H.
Let Ni be the normal closure of Ri in F for i = 1, 2. Since N1N ∩
N2N = [N1, N2] · N if and only if N̄1 ∩ N̄2 = [N̄1, N̄2], Theorem 1 is
equivalent to the following Theorem 1′ which will be proved in Section 2.
Theorem 1
′. A presentation G = 〈A | R1∪R2∪O〉 is weakly (R1, R2)O-
separable if and only if N1N ∩N2N = [N1, N2] ·N in F .
In particular, all conditions sufficient for N1N ∩N2N = [N1, N2] ·N are
sufficient for N̄1∩ N̄2 = [N̄1, N̄2]. Thus we have the following statements.
Corollary 1. The conditions of weak (R1, R2)O-separability and strict
(R1, R2)O-separability are equivalent.
Proof. If a presentation is strictly (R1, R2)O-separable, then it is weakly
(R1, R2)O-separable since the only difference is in the third condition of
Definitions and it is clear that the third condition of weak (R1, R2)O-
separability follows from one of strict (R1, R2)O-separability.
It remains to prove the converse statement. Let P be a spherical
picture over a weakly (R1, R2)O-separable presentation G = 〈A | R1 ∪
R2 ∪ O〉 containing both R1-vertices and R2-vertices. It is evident that
there is a simple closed path γ not passing through any vertex of P
and dividing the sphere into two disks one of which does not contain
R1-vertices and the other one does not contain R2-vertices. Hence by
Lemma 1, Lab(γ) ∈ N1N ∩ N2N . Since G = 〈A | R1 ∪ R2 ∪ O〉 is
weakly (R1, R2)O-separable, Theorem 1′ leads to Lab(γ) ∈ [N1, N2] · N .
Consequently γ is desired.
Corollary 2. The conditions of weak (R1, R2)O-separability and weak
(R2, R1)-separability are equivalent. Moreover, we get the equivalent con-
dition if in Definition 2 of weak (R1, R2)O-separability the item 3) is
replaced by
3′) if one disk contains both R1- and R2-vertices, then the other one con-
tains also both R1- and R2-vertices.
Corollary 3. Let a presentation G = 〈A | R1 ∪ R2 ∪ O〉 satisfy one of
the following conditions:
(i) (R1, R2)O-separability;
(ii) (R1, R2)O-asphericity;
(iii) asphericity.
38 On intersections of normal subgroups in groups
Then N̄1 ∩ N̄2 = [N̄1, N̄2] in H and N1N ∩N2N = [N1, N2] ·N in F .
Proof.
(i) The proof is similar to the proof of Theorem 1 (Theorem 1′).
(ii) It follows directly from (i) and Assertion 2.
(iii) It follows from (ii).
Remark. Theorem 1 and Corollary 3 lead that each condition of Corol-
lary 3 implies weak (R1, R2)O-separability.
Remark. It is clear that there is no difference which preimage of r̄ ∈ R̄i
we took forming Ri, since the conditions of weak (R1, R2)O-separability,
(R1, R2)O-separability, (R1, R2)O-asphericity, asphericity are defined up
to elements from N .
1.3. Some applications
(1) In [1], W.A.Bogley and S.J.Pride consider the following situation.
In the notation of [1], let H be a group; adjoin a set of generators x
to H; then factoring of the resulting free product H ∗ 〈x〉 by the normal
closure N of a set r of cyclically reduced elements of H ∗ 〈x〉 \ H gives
rise to a new group G. It is be said that G is defined by the relative
presentation P = 〈H,x; r〉.
W.A.Bogley and S.J.Pride introduce pictures over relative presenta-
tions, give the definitions of a dipole, asphericity and weak asphericity
for relative presentations. Then they give a "wight test" for asphericity
and introduce small cancellation conditions C(p), T (q) for relative pre-
sentations in a slightly non-standard way (part of the definition makes
use of star complexes). They prove that a relative presentation satisfying
C(p), T (q) with 1/p+ 1/q = 1/2 is aspherical (Theorem 2.2 [1]).
In addition, W.A.Bogley and S.J.Pride discuss the interplay between
pictures over P and pictures over an ordinary presentation P̃ defining the
same group G which is considered in the present paper.
Let Q = 〈a; s〉 be an ordinary presentation of H. For each R ∈ r, let
R̃ be its reduced lift to the free group on a ∪ x. Then P̃ = 〈a,x; s, r̃〉
where r̃ = {R̃ : R ∈ r}. It was proved in [1] that if P is orientable (that
is, P is slender and no element of r is a cyclic permutation of its inverse)
and aspherical, then every picture over P̃ having at least one r̃-disk and
having no x-arcs meeting the boundary of the picture, contains an r̃-
dipole (Lemma 1.5 [1]). Also it was shown that without the orientability
assumption, this result is false; and that there is a passage from pictures
over P̃ to pictures over P and a reverse passage. It is easy to see that
weak asphericity of relative presentation P = 〈H,x; r〉 considered in [1]
O. V. Kulikova 39
is equivalent to asphericity (that is a particular case of asphericity in the
sense of the present paper when two r̃-vertices of a dipole are joined by
an x-edge) of ordinary presentation P̃.
Let r1, r2 be two sets of cyclically reduced elements of H ∗ 〈x〉 \ H
such that R1 6= R2 in H ∗ 〈x〉 for any R1 ∈ r
∗
1
and any R2 ∈ r
∗
2
. Thus
it is easy to see that by Corollary 3 we have the following.
If a relative presentation P = 〈H,x; r〉 is orientable and aspherical,
and r = r1
⊔
r2, then the intersection of the normal closures of r1 and
r2 in H ∗ 〈x〉 is equal to the mutual commutant of these normal closures.
For example, it is sufficient for a relative presentation P = 〈H,x; r〉 to
be orientable and satisfy C(p), T (q) with 1/p+ 1/q = 1/2.
(2) In [6] A.Yu. Ol’shanskii considers a hyperbolic group G given by
〈A|O〉 and a group G1 = 〈A|O∪R〉 where R is some symmetrized system
of additional relations. He investigates properties of G1 depending on
R. He introduces the C(ε, µ, λ, c, ρ)−condition to construct torsion and
many other kinds of quotient groups of hyperbolic groups.
Let us show (using the notation and definitions from [6]) that under
the conditions stated in Lemma 6.6 [6] the symmetrized presentation
G1 = 〈A|O∪R〉 is aspherical (in the sense of the present paper). Consider
a spherical diagram ∆ over G1 containing R-faces. Suppose, contrary to
our claim, that ∆ is reduced. Then similar to the proof (a) of Lemma
6.5 [6], we have that ∆ is tame (this notion of [6] is naturally generalized
to spherical diagrams), since the subdiagrms in (a) of Lemma 6.5 [6]
are circular and hence, Lemma 6.6 [6] is true for them. Lemmas 6.1-
6.3 [6] are true for spherical diagrams, it follows from Lemmas 6.3 [6] for
the spherical tame diagram ∆ that 0 > 1 − 21µ, i.e. µ > 1/21. This
contradicts the assumption that µ < 1/30. Hence ∆ is not reduced.
Therefore the presentation G1 = 〈A|O ∪R〉 is aspherical (in the sense of
the present paper), since pictures are dual objects to diagrams.
So by Corollary 3 we have the following.
Let R1, R2 be two symmetrized sets of cyclically reduced elements of
the free group F (A) such that r1 6= tr2t
−1 in a hyperbolic group H =
〈A|O〉 for any r1 ∈ R1, any r2 ∈ R2 and any t ∈ F . Put R = R1 ∪
R2. For any λ > 0, µ ∈ (0, 1/30), c ≥ 0 there are ε ≥ 0 and ρ > 0
such that if the symmetrized presentation G = 〈A|O ∪ R〉 satisfies the
C(ε, µ, λ, c, ρ)−condition then N1N ∩N2N = [N1, N2] ·N and N̄1 ∩ N̄2 =
[N̄1, N̄2] in H (using the notations of Introduction).
It is mentioned in [6] that the standard C ′(µ)-condition (see [3]) is a
particular case of the C(ε, µ, λ, c, ρ)−condition, where H is a free group,
ε = 0, λ = 1, c = 0, ρ = 1.
40 On intersections of normal subgroups in groups
2. Proof of Theorem 1
′
The proof of Theorem 1′ is almost similar to one of Theorem 1 in [2].
In the beginning we list definitions from [2] needed for the proof and
describe necessary generalizations of them.
2.1. Additional definitions
1) Picture P with equator Equ. Subpictures of P .
The difference of this definition from one of [2] is the following. P is a
picture on the sphere S2 over a presentation G = 〈A | R1 ∪ R2 ∪ O〉.
The equator Equ is a fixed simple closed path on S2 not passing through
any vertex of P and dividing S2 into two parts so that one part does
not contain R1-vertices and the other one does not contain R2-vertices.
0-vertices may lie in both hemispheres. Lab(Equ) is denoted by W or
W−1 (the sign depends on the direction of reading). By the choice of the
equator, it follows from Lemma 1 that W ∈ N1N ∩N2N .
2) Boundaries of vertices. North and south vertices. 0-vertices.
The difference of this definition from one of [2] is the following. There
are vertices of three types: R1−, R2− and 0-vertices. An R-vertex is
called north (respectively, south) if it lies in the north (respectively, south)
hemisphere. As in [2], suppose that R1-vertices are north, R2-vertices are
south.
3) Admissible transformations.
The difference of this definition from one of [2] is that admissible trans-
formations can replace W ≡ Lab(Equ) by a word W ′ equal to W to
within an element of [N1, N2] ·N (for simplicity of notation, we will use
the same letter W for the notation W ′). Admissible transformations also
preserve the subdivision of P by Equ into the north and south vertices.
For example, a 0-transformation is admissible if it is performed inside of
any hemisphere.
4) Maps. This definition is the same as one in [2].
5) Components.
The difference of this definition from one of [2] is the following. The de-
finitions of reduced and non-reduced components here are formally the
same as in [2], but one should note that the definition of a dipole in the
present paper differs slightly from one in [2]. A component is called north
(respectively, south) if the corresponding subpicture contains only north
(respectively, only south) vertices and possibly some 0-vertices. A com-
ponent is called a 0-component if it contains only 0-vertices. South and
north components and 0-components are called uniform. A component
is called mixed if the corresponding subpicture contains both south and
O. V. Kulikova 41
north vertices.
6) Countries. Their territories and boundaries. North and south
countries, 0-countries.
This definition is the same as one in [2]. One should only add that a
country is called a 0-country if the corresponding uniform component is
a 0-component.
7) Pieces of equator Equ. This definition is the same as one in [2].
8) Regions of north and south countries. South, north and 0-regions.
This definition is the same as one in [2]. One should only add that a
region is called a 0-region if it doesn’t contain R-vertices.
9) σ-countries. This definition is the same as one in [2].
10) σ-pictures. This definition is the same as one in [2].
2.2. Proof of Theorem 1′ modulo Propositions 1 and 2
The proof of the statement that weak (R1, R2)O-separability follows from
the equality N1N ∩N2N = [N1, N2] ·N is similar to the proof of Corol-
lary 1.
Therefore it remains to prove the converse statement. Let a presen-
tation G = 〈A | R1 ∪R2 ∪O〉 satisfy weak (R1, R2)O-separability.
Since the inclusion [N1, N2] · N ⊂ N1N ∩ N2N always holds, it is
sufficient to prove the inverse inclusion.
Let W be an arbitrary word of N1N ∩ N2N . To show that W ∈
[N1, N2] · N , let us construct two disk pictures. Since W ∈ N1 · N ,
by Lemma 1, there is a disk picture over 〈A|R1 ∪ O〉 with the bound-
ary label equal to the word W . This picture contains only R1-vertices
and 0-vertices. Since W ∈ N2 · N , similarly there is a disk picture over
〈A|R2 ∪ O〉 with the boundary label equal to the word W−1. This pic-
ture contains only R2-vertices and 0-vertices. Pasting together the disk
pictures by their boundaries gives a picture P on S2 with a fixed equator
Equ. Lab(Equ) is equal to W or W−1 depending on the direction of
moving along Equ.
It is obvious that the proof of Theorem 1′ follows from the following
Propositions 1 and 2, which will be proved in the subsections 2.4 and 2.5
below:
Proposition 1. Let a picture P with a fixed equator Equ be over a pre-
sentation G = 〈A | R1 ∪ R2 ∪ O〉. If the presentation satisfies weak
(R1, R2)O-separability, then P is a σ-picture.
Proposition 2. Let a picture P with a fixed equator Equ be over a pre-
sentation G = 〈A | R1 ∪ R2 ∪ O〉. If P is a σ-picture, then the word W
42 On intersections of normal subgroups in groups
along Equ can be reduced to the identity element in the free group by a
finite number of admissible transformations.
2.3. Some admissible transformations. Auxiliary lemmas
Below we will use the following lemma.
Lemma 2. Let T be a country in P . Let I be an arbitrary piece of the
equator belonging to the territory of T . Then Lab(I) ∈ N1 · N if T is
north, Lab(I) ∈ N2 ·N if T is south, and Lab(I) ∈ N (i.e. Lab(I) = 1
in H) if T is a 0-country.
Proof. Let T be north (the proof in the case of a south country is similar).
The piece I divides the territory of T into two parts T ′ and T ′′. We
consider one of them denoted by T ′. The part T ′ may contain only north
vertices (i.e., R1-vertices), 0-vertices and edges labelled by letters of the
alphabet A. Hence T ′ contains a disk picture over 〈A | R1 ∪ O〉. By
Lemma 1, the word along the boundary of T ′ belongs to N1 · N . Since
the edges intersecting the boundary of T ′ intersect it only in a part which
coincides with I, Lemma 2 in the case of north T follows. The proof in
the case of a 0-country is similar but we need to use Lemma 1 in the case
of empty R.
Further we list admissible transformations, which will be used in the
proof of Propositions 1 and 2. Some of them are the same as in [2]. It is
obvious that they are still admissible in the sense of the present paper.
Also we describe admissible transformations generalizing some transfor-
mations from [2].
1) Isotopy. (It is the same transformation as in [2].)
2) Bridge moves. (It is the same transformation as in [2].)
3) Removing components and edge-circles of P not intersecting Equ. (It
is the same transformation as in [2].)
4) Removing regular 0-regions:
Assume that Equ intersects the territory of a country T in two succes-
sive points such that the part φ of the boundary of T between these two
points and an appropriate part φ̄ of Equ between these two points bound
a 0-region of T . This 0-region (denote it by P 0) forms a disk picture over
H. Hence Lab(∂P 0) = 1 in H. Since edges in P 0 intersecting ∂P 0 inter-
sect it only in φ̄, Lab(φ̄) ≡ Lab(∂P 0) and Lab(φ) ≡ 1. Change Equ by
(Equ \ φ̄) ∪ φ. Then slightly decrease the territory of T to separate Equ
and the part φ of the boundary of T . This transformation corresponds
to a cancellation of a word equal to 1 in H (i.e. belonging to N) in the
equatorial label W . Hence it is admissible.
O. V. Kulikova 43
5) Joining σ-countries. (It is the same transformation as in [2].)
6) Pasting a map with a commutator subpicture in P :
We will use this transformation in the subsection 2.5. In notation of
this subsection, assume that there are two regular regions: one of them
belongs to a north country T1, the other one belongs to a south coun-
try T2. By Lemma 2, the north region contains a picture with a word
w1 ∈ N1 · N written along a piece I2 of the equator, the south region
contains a picture with a word w2 ∈ N2 · N written along a piece I3 of
the equator. Similar to the admissible transformation 6) of [2] construct
a map M containing a picture with the word along the equator equal to
w2w1w2
−1w1
−1. Also similar to [2] a small map Ms in P containing noth-
ing but a part {y = 0} of Equ is replaced by the constructed mapM . This
transformation is admissible because it corresponds to an insertion of the
commutator w2w1w2
−1w1
−1 of the elements from N1 ·N and N2 ·N in the
equatorial label W (i.e. w2w1w2
−1w1
−1 ∈ [N1 ·N,N2 ·N ] ⊂ [N1, N2] ·N).
Lemma 3. Let T be a country in P . If Equ intersects the boundary of T
exactly two times, then a word along the piece of the equator lying inside
the territory of T is equal to the identity element in H (i.e. it belongs
to N).
Proof. Lemma 3 in the case of a 0-country follows from Lemma 2. As-
sume that T is north (the proof in the case of a south country is similar).
The given piece of the equator divides the territory of T into two regions:
north and south. Moreover all R-vertices lie in the north region. The
south region is a regular 0-region. Removing it (see the admissible trans-
formation 4)) gives rise to the case when no edge of T intersects Equ.
Since this transformation corresponds to the cancellation of a word equal
to 1 in H, the original word along the piece of the equator lying inside
the territory of T was equal to the identity element in H.
2.4. Proof of Proposition 1
The proof of Proposition 1 will be divided into several steps (lemmas).
In Step 1 we will show that the picture P with the fixed equator Equ can
be divided into a finite number of uniform components. In Step 2 the
uniform components should be transformed to countries and edges-circles
not belonging to the countries. In Step 3 we should get rid of the edges-
circles not belonging to the countries. In Step 4 the countries should
be divided into σ-countries. In all steps a finite number of admissible
transformations will be used. Therefore we will get that P is a σ-picture.
Note that Steps 2-4 are similar to Steps 2-4 in [2] if one considers 0-
components as north components.
44 On intersections of normal subgroups in groups
All pictures obtained from P will be denoted by P again for simplicity
of notation.
Step 1. Reducing P to a picture containing only uniform components.
In Step 1 we will use the following two admissible transformations.
Operation A: Transformations of reduced mixed components.
Let the presentation G = 〈A | R1 ∪ R2 ∪ O〉 be weakly (R1, R2)O-
separable. By K denote a reduced mixed component.
Since K is a reduced spherical picture, the condition of the weak
(R1, R2)O-separability leads to the existence of a simple closed path γ
dividing the sphere into two parts so that
1) the both parts contain R-vertices;
2) U ≡ Lab(γ) ∈ [N1, N2] ·N ;
3) if one part does not contain south vertices, then the other one does
not contain north vertices.
By the property 3) of γ, the following three cases are possible.
The first case: the path γ divides K into two parts one of which does
not contain north vertices, the other one does not contain south vertices.
The second case: γ dividesK into two parts one of which does not contain
north vertices, the other one contains both south and north vertices. In
these two cases we can assume that some segment ψ of the path γ lies
on Equ. The complement of ψ to γ will be denoted by ¬ψ. One of the
endpoints of ψ will be denoted by p.
The third case: the path γ divides K into two parts each of which
contains both south and north vertices. Consequently, γ is intersected
by Equ and divided by it into segments among which there are segments
lying in the north hemisphere wholly. Fix one of them. By ψ denote its
connected part not intersecting Equ. By p denote one of the endpoints
of ψ. By ¬ψ denote the complement of ψ to γ.
As in [2], in each of these three cases one can assume that all edges
intersecting the path γ intersect it in the segment ψ, because otherwise all
edges intersecting ¬ψ can be moved by isotopy ( the admissible transfor-
mation 1) ) to ψ along the path γ in the direction of the point p starting
successively at the nearest to p edge.
Further, similar to Operation A of Step 1 in [2], in each of these three
cases we select a map M on S2. A new map M ′ is constructed as follows.
Since [N1, N2] · N ⊂ N1N ∩ N2N , the word U along ψ belongs to the
both groups N1 · N and N2 · N . By Lemma 1, one can construct disk
pictures P1 and P2 with the boundary labels respectively U and U−1.
Moreover in the first two cases P1 is constructed over 〈A | R2∪O〉 (using
O. V. Kulikova 45
south vertices and 0-vertices) and P2 is constructed over 〈A | R1 ∪ O〉
(using north vertices and 0-vertices); in the third case the both pictures
P1 and P2 are constructed over 〈A | R1 ∪ O〉 (using north vertices and
0-vertices). Then these pictures are disposed on the new map M ′ similar
to Operation A of Step 1 in [2].
The old map M is cut out from P and replaced by the new one M ′.
By this replacing of maps, the equatorial label W is changed by the
commutator from [N1, N2] ·N in the first two cases and it is not changed
in the third case. After such replacing all R1-vertices lie in the north
hemisphere and all R2-vertices lie in the south one. Therefore this trans-
formation of P is admissible.
With the help of such replacing of maps the component K falls into
two components K1 and K2 separated from each other by the path γ. In
the first case the components K1 and K2 are uniform; in the second case
one of the components (denote it by K1) is mixed, the other one (K2) is
south uniform; in the third case each of the components Ki is mixed.
Remark 1. The components K1 and K2 can be non-reduced. The num-
ber of south vertices in each mixed component Ki (K1 in the second case;
K1 and K2 in the third case) is strictly less than the number of south
vertices in the original component K, since during Operation A we added
only north vertices and 0-vertices to obtain the mixed components.
Operation B: Transformations of non-reduced mixed components.
Let K be a non-reduced component. Then there is a dipole in K, i.e.,
there are two R-vertices V ′ and V ′′ such that there is a simple path ψ
joining some points p1 and p2, which lie on the boundaries C1 and C2 of
V ′ and V ′′, so that Lab−1(ψ)Labp1
+(C1)Lab(ψ)Labp2
+(C2) = 1 in H.
Remark 2. The both vertices of a dipole can be either north or south,
since r1 6= tr2t
−1 in H for any ri ∈ Ri(i = 1, 2), t ∈ F .
Evidently, it is possible to surround the vertices of the dipole by a
simple closed path γ passing along ψ and around V ′ and V ′′ such that
Lab(γ) = 1 in H.
Similar to Operation A, one can assume that all edges intersecting
the path γ intersect it in a segment γ̄ not intersecting Equ. By T 0(γ̄)
the component K falls into two components K1 and K2 not connected
with each other. The component K1 contains only V ′ and V ′′ and some
0-vertices and edges. Hence K1 is uniform. The component K2 may be
either uniform or mixed, either reduced or non-reduced.
Remark 3. Operation B does not increase the number of R-vertices.
Therefore the number of R-vertices in each Ki is strictly less than the
46 On intersections of normal subgroups in groups
number of R-vertices in the original component K. In particular, the
number of south vertices in K2 is not more than the corresponding num-
ber in the original component K.
Lemma 4. Let a presentation G = 〈A | R1∪R2∪O〉 be weakly (R1, R2)O-
separable. Then a picture P with a fixed equator Equ falls into a finite
number of uniform components with the help of a finite number of Oper-
ations A and B (being admissible transformations).
Proof. Proof of Lemma 4 is similar to the proof of Lemma 4 in [2].
The rest of the proof of Proposition 1 is similar to the proof of Propo-
sition 1 of [2]. The only difference is the following. Firstly 0-components
and 0-countries should be considered as north and called north (if a 0-
component (a 0-country) does not have a part in the north hemisphere,
it can be removed by the admissible transformation 3)). Secondly one
should use Lemma 1-2 of the present paper instead of Lemma 1-2 of [2]
for construction of disk pictures over 〈A|R1 ∪O〉 in Step 4.
2.5. Proof of Proposition 2
By admissible transformations, P is reduced to a picture containing only
σ-countries.
One can suppose that every regular region of every σ-country contains
R-vertices otherwise it will be so after a finite number of the admissible
transformations 4). In addition if there is a σ-country containing only
0-vertices it can be removed by the admissible transformation 3). Hence
all σ-countries are north or south.
The proof of Proposition 2 repeats the proof of Proposition 2 of [2].
The only difference is that one should use Lemmas 2-3 and corresponding
admissible transformations of the present paper instead of Lemmas 2-3
and corresponding admissible transformations of [2] and take into account
that 1 in [2] denotes the identity in the free group F , Ni in [2] corresponds
to Ni ·N here and [N1, N2] in [2] corresponds to [N1, N2] ·N here.
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[3] R.S.Lindon and P.E.Schupp. Combinatorial group theory, Springer-Verlag, Berlin
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O. V. Kulikova 47
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Contact information
O. V. Kulikova Department of Mechanics and Mathematics
Moscow State University, Vorobievy Gory 1
119992 Moscow, Russia
E-Mail: olga.kulikova@mail.ru
Received by the editors: 02.11.2004
and final form in 13.12.2004.
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