A note about splittings of groups and commensurability under a cohomological point of view
Let G be a group, let S be a subgroup with infinite index in G and let FSG be a certain Z2G-module. In this paper, using the cohomological invariant E(G,S,FSG) or simply E~(G,S) (defined in [2]), we analyze some results about splittings of group G over a commensurable with S subgroup which are rela...
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irk-123456789-1545002019-06-16T01:32:42Z A note about splittings of groups and commensurability under a cohomological point of view Maria Gorete Carreira Andrade Ermınia de Lourdes Campelloi Fanti Let G be a group, let S be a subgroup with infinite index in G and let FSG be a certain Z2G-module. In this paper, using the cohomological invariant E(G,S,FSG) or simply E~(G,S) (defined in [2]), we analyze some results about splittings of group G over a commensurable with S subgroup which are related with the algebraic obstruction ``singG(S)" defined by Kropholler and Roller ([8]. We conclude that E~(G,S) can substitute the obstruction ``singG(S)" in more general way. We also analyze splittings of groups in the case, when G and S satisfy certain duality conditions. 2010 Article A note about splittings of groups and commensurability under a cohomological point of view / Maria Gorete Carreira Andrade, Ermınia de Lourdes Campelloi Fanti // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 2. — С. 1–10. — Бібліогр.: 13 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:20J05, 20J06; 20E06 http://dspace.nbuv.gov.ua/handle/123456789/154500 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Let G be a group, let S be a subgroup with infinite index in G and let FSG be a certain Z2G-module. In this paper, using the cohomological invariant E(G,S,FSG) or simply E~(G,S) (defined in [2]), we analyze some results about splittings of group G over a commensurable with S subgroup which are related with the algebraic obstruction ``singG(S)" defined by Kropholler and Roller ([8]. We conclude that E~(G,S) can substitute the obstruction ``singG(S)" in more general way. We also analyze splittings of groups in the case, when G and S satisfy certain duality conditions. |
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Maria Gorete Carreira Andrade Ermınia de Lourdes Campelloi Fanti |
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Maria Gorete Carreira Andrade Ermınia de Lourdes Campelloi Fanti A note about splittings of groups and commensurability under a cohomological point of view Algebra and Discrete Mathematics |
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Maria Gorete Carreira Andrade Ermınia de Lourdes Campelloi Fanti |
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Maria Gorete Carreira Andrade |
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A note about splittings of groups and commensurability under a cohomological point of view |
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A note about splittings of groups and commensurability under a cohomological point of view |
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A note about splittings of groups and commensurability under a cohomological point of view |
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A note about splittings of groups and commensurability under a cohomological point of view |
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A note about splittings of groups and commensurability under a cohomological point of view |
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note about splittings of groups and commensurability under a cohomological point of view |
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Інститут прикладної математики і механіки НАН України |
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A note about splittings of groups and commensurability under a cohomological point of view / Maria Gorete Carreira Andrade, Ermınia de Lourdes Campelloi Fanti // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 2. — С. 1–10. — Бібліогр.: 13 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT mariagoretecarreiraandrade anoteaboutsplittingsofgroupsandcommensurabilityunderacohomologicalpointofview AT ermıniadelourdescampelloifanti anoteaboutsplittingsofgroupsandcommensurabilityunderacohomologicalpointofview AT mariagoretecarreiraandrade noteaboutsplittingsofgroupsandcommensurabilityunderacohomologicalpointofview AT ermıniadelourdescampelloifanti noteaboutsplittingsofgroupsandcommensurabilityunderacohomologicalpointofview |
| first_indexed |
2025-07-14T06:35:33Z |
| last_indexed |
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1837603156710653952 |
| fulltext |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 9 (2010). Number 2. pp. 1 – 10
c© Journal “Algebra and Discrete Mathematics”
A note about splittings of groups
and commensurability under a cohomological
point of view
Maria Gorete Carreira Andrade
Ermı́nia de Lourdes Campello Fanti
Communicated by Komarnytskyj M.Ya.
Abstract. Let G be a group, let S be a subgroup with
infinite index in G and let FSG be a certain Z2G-module. In this
paper, using the cohomological invariant E(G,S,FSG) or simply
Ẽ(G,S) (defined in [2]), we analyze some results about splittings of
group G over a commensurable with S subgroup which are related
with the algebraic obstruction “singG(S)" defined by Kropholler and
Roller ([8]. We conclude that Ẽ(G,S) can substitute the obstruction
“singG(S)" in more general way. We also analyze splittings of groups
in the case, when G and S satisfy certain duality conditions.
Introduction
Let (G,S) be a group pair, where G is a group and S is a subgroup of G.
Consider the power set PG of G and the set FG of the finite subsets of G.
Under boolean addition “+”, PG is an addtive group and has a natural
structure of left Z2G-module. It is easy to see that PG ≃ CoindG{1}Z2
(denoted by Z2G) and FG ≃ IndG{1}Z2 ≃ Z2G. Let FSG := {B ⊂ G | B ⊂
F.S for some finite subset F of G }. Clearly, FSG is a Z2G-submodule
of PG. Consider the Z2G-module IndGSZ2S = Z2G ⊗Z2S Z2S with the
The authors participate of Project Number 04/10229-6 - supported by FAPESP.
The first author is also partially supported by MEC-SESu
2000 Mathematics Subject Classification: 20J05, 20J06; 20E06.
Key words and phrases: Splittings of groups, cohomology of groups, commen-
surability.
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2Note about splitt. of groups and comensurability
natural G-action of the induced module (g.(g1 ⊗m) = gg1 ⊗m). It is true
that the modules Z2G⊗Z2S Z2S and FSG are Z2G-isomorphic.
Let resGS,FSG
: H1(G;FSG) → H1(S;FSG) be the restriction map, we
denote it simply by resGS . When [G : S] = ∞, we can define Ẽ(G,S) :=
1 + dimZ2
Ker(resGS ).
As we have stated in [1], Ẽ(G,S) is an algebraic invariant of the cate-
gory C which objects are the group pairs (G,S) with [G : S] = ∞, and
which morphisms are maps ψ : ((G,S),FSG)) → ((L,R),FSG)) consist-
ing of a homomorphism α : G→ L with α(S) ⊂ R and a homomorphism
φ : FSG→ FSG such that φ(α(g).x) = g.φ(x) for all x ∈ FSG.
Some properties of Ẽ(G,S) and its relation to the invariant end ẽ(G,S)
defined by Kropholler and Roller in [9] were studied in [2] and [3].
Now, suppose that H1(G;FSG) ≃ Z2 with generator u. Then the
“obstruction" singG(S) is defined by singG(S) := resGS (u) (see [8]). When
G and S are finitely generated and H1(G;FSG) ≃ Z2, there is necessary
and (under some additional hypotheses) sufficient condition for G to split
over a commensurable with S subgroup. This condition is that singG(S)
is zero ([8]).
The purpose of this paper is to analyze some results about splittings of
a group G over a commensurable with S subgroup obtained, via singG(S),
by Kropholler and Roller (given in [8]), in terms of the invariant Ẽ(G,S).
We show that Ẽ(G,S) can replace, under less hypotheses, the obstruction
singG(S). Initially we recall some definitions and results.
1. Some results about splittings of groups
Definition 1. (i) Let the groups H and K be given by presentations
H =< ger(H); rel(H) >, K =< ger(K); rel(K) >, where ger denotes a
set generators and rel a set of defining relations for each group. Suppose
that S ⊂ H and T ⊂ K are subgroups with a given isomorphism θ : S
∼
→ T .
Then, the free product H ∗S K, of H and K with amalgamated subgroup
S ≃ T , is given by H ∗S K :=< ger(H), ger(K); rel(H), rel(K), s =
θ(s), ∀s ∈ S >.
(ii) Let H be a group, let S and T be subgroups of H with a given iso-
morphism σ : S → T . The HNN-group (or HNN extension) over base
group H, with respect to σ : S ≃ T and stable letter p, is given by
H∗S,σ =< ger(H), p; rel(H), psp−1 = σ(s), ∀s ∈ S > .
Definition 2. A group G splits over a subgroup S if either G is a HNN-
group H∗S,σ for some subgroup H containing S and some monomorphism
σ from S to H, or G is an amalgamated free product H ∗S K with H 6=
S ≃ T 6= K.
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Definition 3 ([11]). Let G be a group and let PG be the power set of G.
Consider the following submodules of PG: FG which consists of the finite
subsets of G and QG := {A ∈ PG : ∀g ∈ G,A+ gA ∈ FG}. The number
of ends of G is defined by e(G) := dimZ2
(QG/FG) = dimZ2
(PG/FG)G.
Example 1. (a) We have G = Z ∗ Z = Z ∗{1} Z, and e(Z ∗ Z) = ∞;
(b) Z2 ∗ Z2 = Z2 ∗{1} Z2, and e(Z2 ∗ Z2) = 2.
(c)Z = {1}∗{1}, id =< {1}, p, psp−1 = s, ∀s ∈ {1} >=< p >, and e(Z) =
2.
Remark 1. It is known that e(G) can take only the values 0, 1, 2 or ∞
([11], p.176). So, if e(G) ≥ 2, then e(G) = 2 or ∞.
Many important results about splittings of groups, involving the clas-
sical end e(G), were proved in [12] and [13] by Stallings. In the following
result (see [13]), Stallings gave a complete characterization for finitely
generated groups which split over some finite subgroup.
Theorem 1. If G is a finitely generated group, then e(G) ≥ 2 if and only
if G splits over a finite subgroup.
We note that e(Z⊕ Z) = 1 and so Z⊕ Z does not split over a finite
subgroup, but Z⊕Z splits over a infinite subgroup since Z⊕Z= 〈a〉⊕〈b〉=
〈a, b; a.b = b.a〉 = 〈a, b; b−1.a.b = σ(a)〉 = H∗H, id = “Z∗Z, id" is a HNN-
group, where H = 〈a〉 ≃ Z, b is the stable letter, S = T = H and
σ = id : S → T .
The classical end e(G) was generalized for pairs of groups (G,S) by
Houghton in [7] and Scott (using another terminology) in [10]. Following
the terminology from Scott, the number of ends of the pair (G,S) is given
by e(G,S) := dimZ2
(P(G/S)/F(G/S))G.
Remark 2. Scott in [10] has proved many results about splittings of
groups. He tried to generalize Theorem 1, due to Stallings, for groups
which split over infinite subgroups. He showed that “If G splits over a
subgroup S, then e(G,S) ≥ 2" (see [10], Lemma 1.8). The converse of this
result is false in general. In fact, Scott tried to prove the following result:
“e(G,S) ≥ 2 if and only if G splits over some finite extension of S," but
this is also false in general. The main result obtained by Scott was:
Theorem 2 ([10], Theorem 4.1). If G and S are finitely generated groups
and for any g ∈ G− S there is a subgroup G1 of finite index in G such
that G1 contains S but not g, then e(G,S) ≥ 2 if and only if G has a
subgroup T of finite index in G such that T contains S and T splits over
S.
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4Note about splitt. of groups and comensurability
In [8], Kropholler and Roller studied the splitting of a group G over a
commensurable with S subgroup which we will see in the next section.
Here we recall the definition of commensurability.
Definition 4. Two subgroups S and T of a group G are said to be
commensurable if and only if [S : S ∩ T ] <∞ and [T : S ∩ T ] <∞.
Example 2. It is clear that if S is a subgroup of T with [T : S] < ∞,
then T is commensurable with S.
2. The obstruction singG(S) and Ẽ(G,S)
In this section we analyze some results obtained by Kropholler and Roller
in [8], about the obstruction singG(S), under the point of view of the
invariant Ẽ(G,S).
We recall that singG(S) was defined when H1(G;FSG) ≃ Z2 and we
observe that H1(G;FSG) ≃ Z2 is equivalent to ẽ(G,S) = 2, where ẽ(G,S)
denotes the invariant end defined by Kropholler and Roller in [9], which
is also a generalization for pairs of groups (G,S) of the classical invariant
end e(G). In fact, ẽ(G,S) = 1 + dimZ2
H1(G;FSG) if [G : S] = ∞ ([9],
Lemma 1.2).
Moreover, we can easily verify that singG(S) = 0 if and only if
Ker resGS 6= 0 and we have:
Lemma 1. If (G,S) is a group pair with H1(G;FSG) ≃ Z2, then
(i) singG(S) = 0 ⇔ Ẽ(G,S) = 2,
(ii) singG(S) 6= 0 ⇔ Ẽ(G,S) = 1.
Proof. We have [G : S] = ∞ since H1(G;FSG) ≃ Z2, and Ẽ(G,S) =
1 + dimKer resGS . Then,
(i) singG(S) = 0 ⇔ Ker resGS = H1(G,FSG) ≃ Z2 ⇔ Ẽ(G,S) = 2.
(ii) singG(S) 6= 0 ⇔ Ker resGS = 0 ⇔ Ẽ(G,S) = 1.
The following result presents a necessary condition for G to split over
a commensurable with S subgroup, which was proved in [8], and that can
be adapted to the invariant Ẽ(G,S), by means of the last lemma.
Proposition 1 ([8], Lemma 2.4). Let (G,S) be a group pair with finitely
generated S and G. Suppose that H1(G;FSG) ≃ Z2. If G splits over a
commensurable with S subgroup, then Ẽ(G,S) = 2.
Motivated by this fact and considering the invariant Ẽ(G,S) defined
without the restriction H1(G,FSG) ≃ Z2, we believed that it is possible,
through the invariant Ẽ(G,S), to extend the result of the last proposition,
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removing the assumption H1(G,FSG) ≃ Z2. In fact, this is possible (see
Theorem 3 bellow), and the proof is similar to that given in [8], uses
the following lemmas, which proofs have been adapted to the invariant,
without the use of the hypothesis H1(G,FSG) ≃ Z2.
Lemma 2. Let (G,S) be a group pair with finitely generated S and G.
The following conditions are equivalent:
(i) Ẽ(G,S) ≥ 2
(ii)There exists [B] = B+FSG ∈ (
PG
FSG
)G (i.e., B+gB ∈ FSG, ∀g ∈ G)
such that [B] 6= [∅], [B] 6= [G] and SB = B.
Proof. Let F
ε
→→ Z2 be a Z2G projective resolution of Z2. Then F → Z2
is a Z2S projective resolution of Z2, since Z2G is a free Z2S-module.
Consider the exact sequence
0 //FSG
�
� k
//PG //
PG
FSG
//0.
We have the following commutative diagram of chain complexes with
exact rows:
0 // HomG(F,FSG) //
��
HomG(F, PG) //
��
HomG(F,
PG
FSG
) //
��
0
0 // HomS(F,FSG) // HomS(F, PG) // HomS(F,
PG
FSG
) // 0.
Hence, mapping the functor H∗(−), and recalling the definition of coho-
mology group, we have the following commutative diagram with exact
rows:
0 // H0(G;FSG) //
��
H0(G;PG) //
i
��
H0(G;
PG
FSG
)
δ
//
j
��
H1(G;FSG) //
resG
S
��
· · ·
0 // H0(S;FSG) // H0(S;PG) // H0(S;
PG
FSG
)
ρ
// H1(S;FSG) // · · ·
We have in (i) and (ii) that [G : S] = ∞, and so H0(G;FSG) =
(IndGSPS)
G = 0. By Shapiro’s lemma (PG)G ≃ H0(G;PG) ≃ Z2 and
H1(G;PG) = 0. So we obtain:
0 // (PG)G
β
//
i
��
(
PG
FSG
)G
δ
//
j
��
H1(G;FSG) //
resG
S
��
0
0 // (FSG)
S �
�
// (PG)S
α
// (
PG
FSG
)S
ρ
// H1(S;FSG) // · · ·
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6Note about splitt. of groups and comensurability
Suppose now that (i) is true. If Ẽ(G,S) = 1 + dimKer resGS ≥ 2, there
exists u ∈ H1(G;FSG), u 6= 0 such that resGS u = 0. Since δ is surjective,
there exists [B0] ∈ (
PG
FSG
)G such that u = δ[B0], with [B0] 6∈ Imβ =
{[∅], [G]} since δ[B0] 6= 0 and Im β = Ker δ. Using the commutativity
of the diagram, we obtain ρ(j[B0]) = (resGS ◦δ)[B0] = resGS u = 0. Hence
[B0] = j[B0] ∈ Ker ρ = Imα, and therefore there exists B ∈ (PG)S
such that [B] = α(B) = [B0]. So we have SB = B, [B] ∈ (
PG
FSG
)G and
[B] 6∈ {[∅], [G]} (since [B] = [B0]), which proves (ii).
Conversely, assuming (ii), consider [B] ∈ (
PG
FSG
)G such that [B] 6=
[∅], [B] 6= [G] and SB = B. Thus [B] 6∈ Imβ = Ker δ and therefore
u := δ([B]) 6= 0, with B ∈ (PG)S . By the commutativity of the diagram
we obtain resGS u = resGS (δ[B]) = (ρ ◦ j)([B]) = ρ([B]) = ρ(α(B)) = 0.
Therefore, Ker resGS 6= 0 and so Ẽ(G,S) ≥ 2.
Lemma 3. Let S and T be subgroups of G. If T is commensurable with
S then Ẽ(G,S) ≥ 2 if and only if Ẽ(G, T ) ≥ 2.
Proof. Initially we prove that, if H and K are subgroups of G, with K ≤
H ≤ G and [H : K] = n <∞, then Ẽ(G,K) ≥ 2 implies Ẽ(G,H) ≥ 2.
In fact, if Ẽ(G,K) ≥ 2, then there exists, by Lemma 2, B ⊂ G such
that B + gB ∈ FKG, ∀g ∈ G, [B] 6= [∅], [B] 6= [G] and KB = B. Since
[H : K] <∞ we have that FKG = FHG. Thus
B + gB ∈ FHG, ∀g ∈ G. (1)
Let H0 = {h1, . . . , hn} be a set of representatives for the left cosets hK,
h ∈ H. We have
B +H0B = B + (h1B ∪ . . . ∪ hnB) ⊂ (B + h1B) ∪ . . . ∪ (B + hnB)
⊂ F1H ∪ . . . ∪ FnH, [with Fi ∈ FG, i = 1, . . . , n by (1)]
= (F1 ∪ . . . ∪ Fn)H
Therefore B +H0B ∈ FHG and so [B] = [H0B]. Consider B0 := H0B.
Hence:
(a) B0 + gB0 ∈ FHG, ∀g ∈ G, because
B0 + gB0 = H0B + gH0B = (H0B +B) + (B + gH0B)
= (H0B +B) +B + g(h1B ∪ . . . ∪ hnB)
= (H0B +B) +B + (gh1B ∪ . . . ∪ ghnB)
⊂ (B +H0B) + (B + gh1B) + . . .+ (B + ghnB) ∈ FHG,
where the last affirmation is consequence of (1).
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(b) [B0] 6= [∅] and [B0] 6= [G] since [B0] = [B] and [B] 6= [∅] and [G].
(c) HB0 = B0 since B0 ⊂ HB0 and, using that HH0 ⊂ H (because
H0 ⊂ H), H = h1K∪̇ . . . ∪̇hnK = H0K, KB = B and B0 = H0B,
we obtain HB0 = H(H0B) ⊂ HB = H0KB = H0B = B0. Hence B0
satisfies Lemma 2(ii) for the group pair (G,H) and so Ẽ(G,H) ≥ 2.
Now, if T is a commensurable with S subgroup of G, then Ẽ(G,S) ≤
Ẽ(G,S ∩ T ) and Ẽ(G, T ) ≤ Ẽ(G,S ∩ T ) ([2], Proposition 7). Hence
Ẽ(G,S) ≥ 2 implies Ẽ(G,S ∩ T ) ≥ 2 and so, by the initially proved
statement, we have Ẽ(G, T ) ≥ 2. Similarly, Ẽ(G, T ) ≥ 2 implies Ẽ(G,S) ≥
2.
Theorem 3. Let (G,S) be a group pair with finitely generated S and G
and [G,S] = ∞. If G splits over a commensurable with S subgroup, then
Ẽ(G,S) ≥ 2. Or equivalently, if Ẽ(G,S) = 1, then G does not split over
any commensurable with S subgroup.
Proof. Suppose that G splits over a commensurable with S subgroup T .
Then, similarly to the proof of Lemma 2.4 in [8], we obtain a set B ⊂ G
satisfying the condition (ii) of Lemma 2, and so Ẽ(G, T ) ≥ 2.
As a consequence of the Theorem, we have the following result in the
duality theory. For concepts and results of duality theory see [4], [5] and
[6].
Corollary 1. If either (G,S) is a duality pair of dimension n over Z2 (or
simply a Dn-pair) with [G : S] = ∞, or G is a duality group of dimension
n (Dn-group) and the homological dimension hdS ≤ n− 2, then G does
not split over any commensurable with S subgroup.
Proof. This follows from the former theorem and the fact that, under the
above hypotheses, Ẽ(G,S) = 1 (see [2], Proposition 8).
Example 3. Consider G =< a > ∗ < b >≃ Z ∗Z and S =< aba−1b−1 >.
We know that (G,S) is a PD2-pair. So, by the previous corollary, G does
not split over any commensurable with S subgroup. In particular, G does
not split over any finite extension of S.
Remark 3. Theorem 3 can be considered as an extension of the Krophol-
ler-Roller’s result since, in the former example, H1(G;FSG) has infinite
dimension (or equivalently, ẽ(G,S) = ∞) and therefore the obstruction
singG(S) is not defined. Moreover, if G and S are as in Example 3, then G
does not split over any commensurable with S subgroup and the invariant
end e(G,S) = ∞ > 2. This example confirms that the Scott’s initial idea
(see Remark 2) is not really true.
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8Note about splitt. of groups and comensurability
Now, consider a group G with subgroups S and K satisfying the
following conditions:
(a) G is a finitely generated group of cohomological dimension cdG ≤ n;
(b) S is a PDn−1-subgroup of G;
(c) H1(G;FSG) ≃ Z2;
(d) cdK ≤ (n− 1) for any subgroup K of G such that (G : K) = ∞.
In [8], §3, the authors have proved the following result considering these
hypotheses:
Proposition 2 ([8], Lemma 3.2). Suppose that [NG(S) : S] = ∞, where
NG(S) denotes the normaliser of S in G. Then G splits over a commen-
surable with S subgroup if and only if, the obstruction singG(S) = 0.
We hoped to generalize the last proposition, removing the hypothesis
(c) and replacing the condition singG(S) = 0 by Ẽ(G,S) ≥ 2. However,
we prove ( see next theorem) that if [NG(S) : S] = ∞, then the hypothesis
(c) is a consequence of the others and so can not be removed. We also
observe that hypothesis (b) can be replaced by (b’): S is a Dn−1-subgroup
of G. So we need the following lemma which proof is similar to the one of
Lemma 3.1 in [8].
Lemma 4. Let (G,S) be a group pair satisfying the conditions (a), (b’)
and (d). If [NG(S) : S] = ∞ then
(i) [G : NG(S)] <∞ , and
(ii) NG(S)/S has an infinite cyclic subgroup of finite index.
Now, we can prove the mentioned result.
Theorem 4. Let (G,S) be a group pair satisfying the conditions (a), (b’)
and (d). Let C ′ be the dualizing module of S. If [NG(S) : S] = ∞ then
G is a Dn-group with dualizing module C such that ResGSC ≃ C ′ and
H1(G;FSG) ≃ Z2.
Proof. Under the above hypotheses we have, by the previous lemma, that
NG(S)/S has a subgroup L/S ≃ Z with finite index such that [G : L] <∞.
Consider the short exact sequence 0 → S → L→→ L/S → 0. Since S is a
Dn−1-group with dualizing module C ′ and L/S is a PD1-group, then L is
a Dn-group with dualizing module Hn(L;Z2L) ≃ Z2 ⊗ C ′ ≃ C ′ (as Z2L-
modules) ([4], Theorem 9.10). Hence, using thatG does not have Z2-torsion
(since cdG ≤ n) and [G : L] < ∞, we conclude that (see [4], Theorem
9.9) G is a Dn-group with dualizing module C = Hn(G;Z2G) with
ResGLC ≃ C ′ (as Z2L-modules). Thus S is a Dn−1-group with dualizing
module C ′ ≃ ResGSC, where C is the dualizing module of G. Finally, using
duality and Shapiro’s lemma, we have H1(G;FSG) ≃ Z2.
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M. G. C. Andrade, E. L. C. Fanti 9
In [8], §5, under the hypothesis that G is a PDn-group and S is a
PDn−1-subgroup, the authors proved the following fact:
Theorem 5 ([8], Theorem A). Let G be a PDn-group and S a PDn−1-
subgroup. Then G splits over a commensurable with S subgroup if and
only if singG(S) = 0.
Adapting this result to the invariant Ẽ(G,S) we have:
Theorem 6. Let G be a PDn-group and S a PDn−1-subgroup. Then G
splits over a commensurable with S subgroup if and only if Ẽ(G,S) = 2.
Proof. This follows from the previous theorem and Lemma 1.
Example 4. In the two following cases G and S satisfy the hypotheses
of the former theorem and Ẽ(G,S) = 2 ([2] Example 6 (iii) and (vi),
respectively). So G splits over a commensurable with S subgroup:
(1) G = Z
k and S = Z
k−1, k ≥ 2;
(2) G = (Z⊕Z)⋊Z, where θ : Z → Aut(Z⊕Z) is defined by θ(c)(a, b) =
[
1 0
2 1
]c [
a
b
]
=
[
1 0
2c 1
] [
a
b
]
= (a, 2ca+ b), with the operation in
G defined by ((a, b), c) + ((a1, b1), c1) = ((a, b) + θ(c)(a1, b1), c + c1) =
(a+ a1, b+ b1 + 2ca1, c+ c1) e S = {((a, b), 0); a, b ∈ Z}.
Using the last result and Theorem 3 we have:
Proposition 3. Let G be a finitely generated group, T and S subgroups
of G with S ≤ T ≤ G, [G : T ] < ∞ and [T : S] = ∞. If S and T are
finitely generated and Ẽ(T, S) = 1, in particular, if T is a PDn-group, S
a PDn−1-subgroup, and T does not split over a commensurable with S
subgroup, then also G does not split over a commensurable with S subgroup.
Proof. We have Ẽ(G,S) ≤ Ẽ(T, S) = 1 ([2], Proposition 7). So the result
follows from Theorem 3.
References
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Contact information
M. G. C. Andrade UNESP - Universidade Estadual Paulista
Departamento de Matemática
Rua Cristovão Colombo, 2265
15054-000, São José do Rio Preto - SP
Brazil.
E-Mail: gorete@ibilce.unesp.br
URL: www.mat.ibilce.unesp.br/
personal/gorete.html
E. L. C. Fanti UNESP - Universidade Estadual Paulista
Departamento de Matemática
Rua Cristovão Colombo, 2265
15054-000, São José do Rio Preto - SP
Brazil.
E-Mail: fanti@ibilce.unesp.br
URL: www.mat.ibilce.unesp.br/
personal/erminia.html
Received by the editors: 16.09.2009
and in final form ????.
Maria Gorete Carreira Andrade Ermínia de Lourdes Campello Fanti
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