Cohomology computations for Artin groups, Bestvina–Brady groups, and graph products

Davis, Michael W.; Okun, Boris

doi:10.4171/ggd/164

Cited by 29 publications

(36 citation statements)

References 30 publications

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“…In , Davis and Okun computed the cohomology with compact supports and the

ℓ^{2}

‐cohomology of

X_{t}

for each finite flag complex

L

and any non‐integer

t

. (Throughout this section we will write

X

instead of

X_{L}

to simplify the notation.)…”

Section: Compactly Supported and ℓ2‐cohomologymentioning

confidence: 99%

“…The case

S = Z

of the following theorem is [, Theorem 4.4]. Theorem The

ℓ^{2}

‐Betti numbers of the space

X_{t}^{false(S false)}

for

t \notin Z

are given in terms of the ordinary reduced Betti numbers of vertex links in

L

as follows:

0true β_{false(2 false)}^{n} (X_{t}^{(S)}; G_{L} false(S false)) = \sum_{v \in L^{0}} β^{n} ({St}_{L} false(v false), {Lk}_{L} false(v false)) = \sum_{v \in L^{0}} {\bar{β}}^{n - 1} ({Lk}_{L} false(v false)) .

In the case when both

L

and

\overset{L}{true}

are

double-struckQ

‐acyclic, these are the

ℓ^{2}

‐Betti numbers of the group

G_{L} false(S false)

, and so

β_{(2)}^{n} false(G_{L} (S) false)

does not depend on

S

.…”

Section: Compactly Supported and ℓ2‐cohomologymentioning

confidence: 99%

“…The spectral sequence in which the horizontal differential is taken first collapses to give

H^{*} false({Hom}_{G} (C_{*} false(X_{t} false), N false(G false)) false)

, which has the same von Neumann dimension (in Lück's extended definition ) as the

ℓ^{2}

‐cohomology of

X_{t}^{false(S false)}

. The spectral sequence in which the vertical differential is taken first is the one appearing in the statement of [, Lemma 2.1]. The

E_{1}

‐page of this spectral sequence is identified with simplicial cochains on

L^{'}

with certain non‐constant coefficients; the graded coefficient module for the simplex

σ_{0} < σ_{1} < \dots < σ_{i}

in degree

j

H^{j} false(Y_{σ_{0}}; scriptN (G) false)

.…”

Section: Compactly Supported and ℓ2‐cohomologymentioning

confidence: 99%

“…Similarly, the case

S = Z

of the following theorem is [, theorem 4.3]. In the case of cohomology with compact supports, the relevant spectral sequence still collapses at the

E_{2}

‐page, but the

E_{2}

‐page contains many non‐zero terms in the same total degree, which makes the statement more complicated.…”

Section: Compactly Supported and ℓ2‐cohomologymentioning

confidence: 99%

See 3 more Smart Citations

Uncountably many groups of type FP

Leary

2018

Proc. London Math. Soc.

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show abstract

“…In , Davis and Okun computed the cohomology with compact supports and the

ℓ^{2}

‐cohomology of

X_{t}

for each finite flag complex

L

and any non‐integer

t

. (Throughout this section we will write

X

instead of

X_{L}

to simplify the notation.)…”

Section: Compactly Supported and ℓ2‐cohomologymentioning

confidence: 99%

“…The case

S = Z

of the following theorem is [, Theorem 4.4]. Theorem The

ℓ^{2}

‐Betti numbers of the space

X_{t}^{false(S false)}

for

t \notin Z

are given in terms of the ordinary reduced Betti numbers of vertex links in

L

as follows:

0true β_{false(2 false)}^{n} (X_{t}^{(S)}; G_{L} false(S false)) = \sum_{v \in L^{0}} β^{n} ({St}_{L} false(v false), {Lk}_{L} false(v false)) = \sum_{v \in L^{0}} {\bar{β}}^{n - 1} ({Lk}_{L} false(v false)) .

In the case when both

L

and

\overset{L}{true}

are

double-struckQ

‐acyclic, these are the

ℓ^{2}

‐Betti numbers of the group

G_{L} false(S false)

, and so

β_{(2)}^{n} false(G_{L} (S) false)

does not depend on

S

.…”

Section: Compactly Supported and ℓ2‐cohomologymentioning

confidence: 99%

“…The spectral sequence in which the horizontal differential is taken first collapses to give

H^{*} false({Hom}_{G} (C_{*} false(X_{t} false), N false(G false)) false)

, which has the same von Neumann dimension (in Lück's extended definition ) as the

ℓ^{2}

‐cohomology of

X_{t}^{false(S false)}

. The spectral sequence in which the vertical differential is taken first is the one appearing in the statement of [, Lemma 2.1]. The

E_{1}

‐page of this spectral sequence is identified with simplicial cochains on

L^{'}

with certain non‐constant coefficients; the graded coefficient module for the simplex

σ_{0} < σ_{1} < \dots < σ_{i}

in degree

j

H^{j} false(Y_{σ_{0}}; scriptN (G) false)

.…”

Section: Compactly Supported and ℓ2‐cohomologymentioning

confidence: 99%

“…Similarly, the case

S = Z

of the following theorem is [, theorem 4.3]. In the case of cohomology with compact supports, the relevant spectral sequence still collapses at the

E_{2}

‐page, but the

E_{2}

‐page contains many non‐zero terms in the same total degree, which makes the statement more complicated.…”

Section: Compactly Supported and ℓ2‐cohomologymentioning

confidence: 99%

See 2 more Smart Citations

Uncountably many groups of type FP

Leary

2018

Proc. London Math. Soc.

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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%

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.

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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Cohomology computations for Artin groups, Bestvina–Brady groups, and graph products

Cited by 29 publications

References 30 publications

Uncountably many groups of type FP

Uncountably many groups of type FP

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

Right-Angled Spaces and Groups

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