The action dimension of right‐angled Artin groups

Avramidi, Grigori; Davis, Michael W.; Okun, Boris; Schreve, Kevin

doi:10.1112/blms/bdv083

Cited by 18 publications

(78 citation statements)

References 20 publications

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“…Its octahedralization

O K

is defined to be the inverse image of

K

O (V)

O K : = p^{- 1} false(K false) \leq O false(V false) .

Thus, each simplex of

O K

is of the form

(σ, ε)

, where

σ

is a simplex of

K

and

ε : Vert σ \to {\pm 1}

is a function. (This terminology comes from or [, § 8].) We also say that

O K

is the result of doubling the vertices of

K

.…”

Section: Configurations Of Standard Abelian Subgroupsmentioning

confidence: 99%

“…The main results of concern the van Kampen obstruction

{prefixvk}_{Z / 2}^{m} false(O K false)

of an octahedralization. From a nonzero

Z / 2

‐cycle

M \in Z_{k} false(K; double-struckZ / 2 false)

and a

k

‐simplex

normalΔ

in the support of

M

, one produces a

2 k

‐chain

Ω \in C_{2 k} false(scriptC (O K); double-struckZ / 2 false)

on which one can evaluate the van Kampen obstruction.…”

Section: The Obstructor Dimensionmentioning

confidence: 99%

“…From a nonzero

Z / 2

‐cycle

M \in Z_{k} false(K; double-struckZ / 2 false)

and a

k

‐simplex

normalΔ

in the support of

M

, one produces a

2 k

‐chain

Ω \in C_{2 k} false(scriptC (O K); double-struckZ / 2 false)

on which one can evaluate the van Kampen obstruction. If the pair

(M, Δ)

satisfies a certain technical condition, called the ‘

*

‐condition’ in [, p. 122], then

normalΩ

is a cycle and

{prefixvk}_{Z / 2}^{2 k} false(O K false)

evaluates nontrivially on it. This gives the following.…”

Section: The Obstructor Dimensionmentioning

confidence: 99%

“…It is then remarked in that when

k = dim K

and

K

is a flag complex, the

*

‐condition is automatically satisfied. It is also proved in [, Theorem 5.1] that when there is no nontrivial cycle on

K

in the top degree,

{prefixvk}_{Z / 2}^{2 k} false(O K false) = 0

. So, the following is a corollary of Theorem .…”

Section: The Obstructor Dimensionmentioning

confidence: 99%

“…The next theorem is one of our main results. For

RAAG

s, it was proved in . Main Theorem Suppose that

L

is the nerve of a Coxeter system

(W, S)

and that

A

is the associated Artin group.…”

Section: Introductionmentioning

confidence: 99%

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

Davis

Huang

2017

Bull. London Math. Soc.

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Suppose that (W,S) is a Coxeter system with associated Artin group A and with a simplicial complex L as its nerve. We define the notion of a ‘standard abelian subgroup’ in A. The poset of such subgroups in A is parameterized by the poset of simplices in a certain subdivision L⊘ of L. This complex of standard abelian subgroups is used to generalize an earlier result from the case of right‐angled Artin groups to case of general Artin groups, by calculating, in many instances, the smallest dimension of a manifold model for BA. (This is the ‘action dimension’ of A denoted actdimA.) If Hdfalse(L;double-struckZ/2false)≠0, where d=dimL, then actdimA⩾2d+2. Moreover, when the K(π,1)‐Conjecture holds for A, the inequality is an equality.

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“…Its octahedralization

O K

is defined to be the inverse image of

K

O (V)

O K : = p^{- 1} false(K false) \leq O false(V false) .

Thus, each simplex of

O K

is of the form

(σ, ε)

, where

σ

is a simplex of

K

and

ε : Vert σ \to {\pm 1}

is a function. (This terminology comes from or [, § 8].) We also say that

O K

is the result of doubling the vertices of

K

.…”

Section: Configurations Of Standard Abelian Subgroupsmentioning

confidence: 99%

“…The main results of concern the van Kampen obstruction

{prefixvk}_{Z / 2}^{m} false(O K false)

of an octahedralization. From a nonzero

Z / 2

‐cycle

M \in Z_{k} false(K; double-struckZ / 2 false)

and a

k

‐simplex

normalΔ

in the support of

M

, one produces a

2 k

‐chain

Ω \in C_{2 k} false(scriptC (O K); double-struckZ / 2 false)

on which one can evaluate the van Kampen obstruction.…”

Section: The Obstructor Dimensionmentioning

confidence: 99%

“…From a nonzero

Z / 2

‐cycle

M \in Z_{k} false(K; double-struckZ / 2 false)

and a

k

‐simplex

normalΔ

in the support of

M

, one produces a

2 k

‐chain

Ω \in C_{2 k} false(scriptC (O K); double-struckZ / 2 false)

on which one can evaluate the van Kampen obstruction. If the pair

(M, Δ)

satisfies a certain technical condition, called the ‘

*

‐condition’ in [, p. 122], then

normalΩ

is a cycle and

{prefixvk}_{Z / 2}^{2 k} false(O K false)

evaluates nontrivially on it. This gives the following.…”

Section: The Obstructor Dimensionmentioning

confidence: 99%

“…It is then remarked in that when

k = dim K

and

K

is a flag complex, the

*

‐condition is automatically satisfied. It is also proved in [, Theorem 5.1] that when there is no nontrivial cycle on

K

in the top degree,

{prefixvk}_{Z / 2}^{2 k} false(O K false) = 0

. So, the following is a corollary of Theorem .…”

Section: The Obstructor Dimensionmentioning

confidence: 99%

“…The next theorem is one of our main results. For

RAAG

s, it was proved in . Main Theorem Suppose that

L

is the nerve of a Coxeter system

(W, S)

and that

A

is the associated Artin group.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Davis

Huang

2017

Bull. London Math. Soc.

Self Cite

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show abstract

Right-Angled Spaces and Groups

Davis

2024

Ergebnisse Der Mathematik Und Ihrer Grenzgebiete. 3. Folge / a Series of Modern Surveys in Mathematics

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Rational manifold models for duality groups

Avramidi¹

2018

Geom. Funct. Anal.

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We show that a finite type duality group of dimension d > 2 is the fundamental group of a (d + 3)-manifold with rationally acyclic universal cover. We use this to find closed manifolds with rationally acyclic universal cover and some nonvanishing L 2 -Betti numbers outside the middle dimension, which contradicts a rational analogue of a conjecture of Singer.14 Such a manifold would be homotopy equivalent but not properly homotopy equivalent to (Ṫ 2 ) d . 15 Which, in particular, produces a 2d-manifold rationally equivalent to (Ṫ 2 ) d but with very different peripheral properties: its interior can be homotoped to the end, in sharp contrast to the classical (Ṫ 2 ) d manifold.

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The action dimension of right‐angled Artin groups

Cited by 18 publications

References 20 publications

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

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

Right-Angled Spaces and Groups

Rational manifold models for duality groups

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