Weak commutativity between two isomorphic polycyclic groups

Lima, Bruno César Rodrigues; Oliveira, Ricardo N.

doi:10.1515/jgth-2015-0041

Cited by 18 publications

(20 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…We can apply this lemma in the setting of because Lima and Oliveira proved that

H_{1} false(L, double-struckZ false)

is finitely generated whenever

G

is finitely generated. We recall their construction.…”

Section: On the Structure Of The Group X(g)mentioning

confidence: 99%

“…Thus the abelianisation of

L

is a subgroup of

M = Z G / I_{2} false(G false)

. Lima and Oliveira show in that if

G

is generated by

a_{1}, \dots, a_{m}

, then

M

is generated as an abelian group by 1 and the finitely many monomials

a_{i_{1}} \dots a_{i_{s}}

with

1 ⩽ i_{1} < \dots < i_{j} ⩽ m

.…”

Section: On the Structure Of The Group X(g)mentioning

confidence: 99%

“…For

ρ (L)

, this is covered by Lemma . Lima and Oliveira show in that

M = Z F / I_{2} false(F false)

is generated as an abelian group by

X = {1, a, b, a b}

. The augmentation map

Z F \to Z

(defined by

w \mapsto 1

for all

w \in F

) descends to an epimorphism

μ : M \to Z

defined by

μ (x) = 1

(the generator of

double-struckZ

) for each

x \in X

.…”

Section: On the Structure Of The Group X(g)mentioning

confidence: 99%

“…Proof Consider the LHS spectral sequence in homology for the central extension

1 \to W \to L \to ρ (L) \to 1

and note that

ρ (L) = P

, so the spectral sequence

E_{p, q}^{2} = H_{p} false(P, H_{q} (W, Z) false)

converges to

H_{p + q} false(L, double-struckZ false)

. Then

E_{1, 0}^{\infty} = E_{1, 0}^{2} = H_{1} false(P, double-struckZ false)

E_{0, 1}^{\infty} = E_{0, 1}^{3} = Coker false(d_{2, 0}^{2} false)

, where

d_{2, 0}^{2} : E_{2, 0}^{2} = H_{2} false(P, double-struckZ false) \to E_{0, 1}^{2}

, and there is an exact sequence

E_{0, 1}^{\infty} \to H_{1} false(L, double-struckZ false) \to E_{1, 0}^{\infty} .

By a result of Lima and Oliveira (or Proposition )

H_{1} false(L, double-struckZ false)

is finitely generated. Hence

E_{0, 1}^{\infty} = Coker false(d_{2, 0}^{2} false)

is finitely generated, and since

H_{2} false(P, double-struckZ false)

is finitely generated this impl...…”

Section: X(f) Is Not Of Type Fpmentioning

confidence: 99%

“…Our original proof of Theorem A relied on the basic structural results for

X (G)

established by Sidki in his seminal paper , as well as the VSP criterion for finite presentability established by Bridson, Howie, Miller and Short [, Therorem A], and a finiteness result concerning the abelianisation of a special subgroup

L (G) < X (G)

that was established by Lima and Oliveira . A shortcoming of this original proof was that it did not give an estimate on the number of relations needed to present

X (G)

; it was in seeking to address this that we discovered the proof presented in Section 1.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Weak commutativity and finiteness properties of groups

Bridson

Kochloukova

2018

Bull. London Math. Soc.

View full text Add to dashboard Cite

We consider the group X(G) obtained from G*G by forcing each element g in the first free factor to commute with the copy of g in the second free factor. Deceptively complicated finitely presented groups arise from this construction: X(G) is finitely presented if and only if G is finitely presented, but if F is a non‐abelian free group of finite rank then X(F) has a subgroup of finite index whose third homology is not finitely generated.

show abstract

“…We can apply this lemma in the setting of because Lima and Oliveira proved that

H_{1} false(L, double-struckZ false)

is finitely generated whenever

G

is finitely generated. We recall their construction.…”

Section: On the Structure Of The Group X(g)mentioning

confidence: 99%

“…Thus the abelianisation of

L

is a subgroup of

M = Z G / I_{2} false(G false)

. Lima and Oliveira show in that if

G

is generated by

a_{1}, \dots, a_{m}

, then

M

is generated as an abelian group by 1 and the finitely many monomials

a_{i_{1}} \dots a_{i_{s}}

with

1 ⩽ i_{1} < \dots < i_{j} ⩽ m

.…”

Section: On the Structure Of The Group X(g)mentioning

confidence: 99%

“…For

ρ (L)

, this is covered by Lemma . Lima and Oliveira show in that

M = Z F / I_{2} false(F false)

is generated as an abelian group by

X = {1, a, b, a b}

. The augmentation map

Z F \to Z

(defined by

w \mapsto 1

for all

w \in F

) descends to an epimorphism

μ : M \to Z

defined by

μ (x) = 1

(the generator of

double-struckZ

) for each

x \in X

.…”

Section: On the Structure Of The Group X(g)mentioning

confidence: 99%

“…Proof Consider the LHS spectral sequence in homology for the central extension

1 \to W \to L \to ρ (L) \to 1

and note that

ρ (L) = P

, so the spectral sequence

E_{p, q}^{2} = H_{p} false(P, H_{q} (W, Z) false)

converges to

H_{p + q} false(L, double-struckZ false)

. Then

E_{1, 0}^{\infty} = E_{1, 0}^{2} = H_{1} false(P, double-struckZ false)

E_{0, 1}^{\infty} = E_{0, 1}^{3} = Coker false(d_{2, 0}^{2} false)

, where

d_{2, 0}^{2} : E_{2, 0}^{2} = H_{2} false(P, double-struckZ false) \to E_{0, 1}^{2}

, and there is an exact sequence

E_{0, 1}^{\infty} \to H_{1} false(L, double-struckZ false) \to E_{1, 0}^{\infty} .

By a result of Lima and Oliveira (or Proposition )

H_{1} false(L, double-struckZ false)

is finitely generated. Hence

E_{0, 1}^{\infty} = Coker false(d_{2, 0}^{2} false)

is finitely generated, and since

H_{2} false(P, double-struckZ false)

is finitely generated this impl...…”

Section: X(f) Is Not Of Type Fpmentioning

confidence: 99%

“…Our original proof of Theorem A relied on the basic structural results for

X (G)

L (G) < X (G)

that was established by Lima and Oliveira . A shortcoming of this original proof was that it did not give an estimate on the number of relations needed to present