Determining the action dimension of an Artin group by using its complex of abelian subgroups

Davis, Michael W.; Huang, Jingyin

doi:10.1112/blms.12061

Cited by 9 publications

(23 citation statements)

References 21 publications

(63 reference statements)

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“…It follows from that the inclusion of standard infinite subgroups corresponding to the vertices of

L_{⊘}

defines a homomorphism

Φ : A_{L_{⊘}} \to A_{L}

from the

RAAG

A_{L_{⊘}}

to the Artin group

A_{L}

. This leads us to the following.…”

Section: Obstructorsmentioning

confidence: 95%

“…Since

B A S (L)

is a model for

B A

gdim A ⩽ d + 1

. On the other hand, for any

d

‐simplex

σ \in S (L)

, the spherical Artin group

A_{σ}

contains a free abelian subgroup of rank

d + 1

(for example, see ). So,

cd A_{L} ⩾ d + 1

.…”

Section: Simple Complexes Of Groupsmentioning

confidence: 99%

“…Taking the union over all

σ \in Q

, we get a configuration of subgroups

H : = ⋃ H (σ) \subset G

and a configuration of sheets

E H : = ⋃ E H (σ) \subset E G

. The archetype is the case of a general Artin group

A

considered in . This is explained in the next example.…”

Section: Obstructorsmentioning

confidence: 99%

“…For each

σ \in S (L)

W_{σ}

is the corresponding spherical Coxeter group and

D_{σ}

is its Coxeter diagram. As explained in , there is a subdivision

σ_{⊘}

σ

whose vertices correspond to the connected subdiagrams of

D_{σ}

. In other words the vertices of

σ_{⊘}

are the irreducible special subgroups of

W_{σ}

.…”

Section: Obstructorsmentioning

confidence: 99%

“…A configuration of free abelian subgroups in a group

π

has the same meaning as in . There is a simplicial complex

K

and for each simplex

σ \in S (K)

we are given a free abelian subgroup

{double-struckZ}^{σ}

of rank

dim σ + 1

so that

{double-struckZ}^{σ}

is generated by the rank one subgroups corresponding to the vertices of

σ

.…”

Section: Obstructors For Hyperplane Complementsmentioning

confidence: 99%

See 4 more Smart Citations

Action dimensions of some simple complexes of groups

Davis

Schreve

2019

Journal of Topology

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“…It follows from that the inclusion of standard infinite subgroups corresponding to the vertices of

L_{⊘}

defines a homomorphism

Φ : A_{L_{⊘}} \to A_{L}

from the

RAAG

A_{L_{⊘}}

to the Artin group

A_{L}

. This leads us to the following.…”

Section: Obstructorsmentioning

confidence: 95%

“…Since

B A S (L)

is a model for

B A

gdim A ⩽ d + 1

. On the other hand, for any

d

‐simplex

σ \in S (L)

, the spherical Artin group

A_{σ}

contains a free abelian subgroup of rank

d + 1

(for example, see ). So,

cd A_{L} ⩾ d + 1

.…”

Section: Simple Complexes Of Groupsmentioning

confidence: 99%

“…Taking the union over all

σ \in Q

, we get a configuration of subgroups

H : = ⋃ H (σ) \subset G

and a configuration of sheets

E H : = ⋃ E H (σ) \subset E G

. The archetype is the case of a general Artin group

A

considered in . This is explained in the next example.…”

Section: Obstructorsmentioning

confidence: 99%

“…For each

σ \in S (L)

W_{σ}

is the corresponding spherical Coxeter group and

D_{σ}

is its Coxeter diagram. As explained in , there is a subdivision

σ_{⊘}

σ

whose vertices correspond to the connected subdiagrams of

D_{σ}

. In other words the vertices of

σ_{⊘}

are the irreducible special subgroups of

W_{σ}

.…”

Section: Obstructorsmentioning

confidence: 99%

“…A configuration of free abelian subgroups in a group

π

has the same meaning as in . There is a simplicial complex

K

and for each simplex

σ \in S (K)

we are given a free abelian subgroup

{double-struckZ}^{σ}

of rank

dim σ + 1

so that

{double-struckZ}^{σ}

is generated by the rank one subgroups corresponding to the vertices of

σ

.…”

Section: Obstructors For Hyperplane Complementsmentioning

confidence: 99%

See 3 more Smart Citations

Action dimensions of some simple complexes of groups

Davis

Schreve

2019

Journal of Topology

Self Cite

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show abstract

C*-Algebras of Right LCM One-Relator Monoids and Artin–Tits Monoids of Finite Type

Omland

Spielberg

2020

Commun. Math. Phys.

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show abstract

Morse quasiflats II

Huang¹,

Kleiner²,

Stadler³

2023

Advances in Mathematics

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Determining the action dimension of an Artin group by using its complex of abelian subgroups

Cited by 9 publications

References 21 publications

Action dimensions of some simple complexes of groups

Action dimensions of some simple complexes of groups

C*-Algebras of Right LCM One-Relator Monoids and Artin–Tits Monoids of Finite Type

Morse quasiflats II

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