The proper geometric dimension of the mapping class group

Aramayona, Javier; Martı́nez-Pérez, Conchita

doi:10.2140/agt.2014.14.217

Cited by 16 publications

(42 citation statements)

References 19 publications

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“…), as well as examples of groups with

F {cd}_{double-struckZ} G = {cd}_{frakturF} G

(cf. ). As shown in , the finiteness of the

frakturF

‐cohomological dimension of

G

over

double-struckZ

implies that

{Gcd}_{double-struckZ} G = F {cd}_{double-struckZ} G

.…”

Section: Introductionmentioning

confidence: 97%

Gorenstein dimension and group cohomology with group ring coefficients

Emmanouil

Talelli

2018

J. London Math. Soc.

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The Gorenstein cohomological dimension of a group G generalizes the ordinary cohomological dimension of G, in the sense that the two invariants coincide when the latter one is finite. In this paper, we show that the Gorenstein cohomological dimension GcdkG of G over a commutative ring k shares many properties with the cohomological dimension of G over k. For example, if k has finite global dimension, the finiteness of GcdkG implies that GpdkGM is finite for any kG‐module M. Other properties concern the dependence of the Gorenstein cohomological dimension upon the coefficient ring, its behavior with respect to subgroups, the subadditivity with respect to extensions and the formula for the dimension of an ascending union. We also prove that if k has finite global dimension then the finiteness of GcdkG gives a criterion for the conditions cdkG<∞ and hdkG<∞ to be equivalent. Finally, we show that if k is a countable ring and G is a countable group of finite Gorenstein cohomological dimension over k, then GcdkG=supfalse{n:Hn(G,kG)≠0false}. The latter two results are special cases for group algebras of certain assertions that are valid over more general rings.

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“…), as well as examples of groups with

F {cd}_{double-struckZ} G = {cd}_{frakturF} G

(cf. ). As shown in , the finiteness of the

frakturF

‐cohomological dimension of

G

over

double-struckZ

implies that

{Gcd}_{double-struckZ} G = F {cd}_{double-struckZ} G

.…”

Section: Introductionmentioning

confidence: 97%

Gorenstein dimension and group cohomology with group ring coefficients

Emmanouil

Talelli

2018

J. London Math. Soc.

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“…Proof In , this has been proven for connected

S

. Suppose that

S

decomposes into the disjoint union of diffeomorphic copies of its connected components

S = ⊔^{m_{1}} S_{1} \dots ⊔^{m_{q}} S_{q} .

Subsequently, this implies

0true Mod (S) \subseteq \prod_{i = 1}^{q} Mod (S_{i}) ≀ Σ_{m_{i}}

and is of finite index.…”

Section: Mapping Class Groupsmentioning

confidence: 95%

“…So, it gives a model for

\underset{̲}{E} G

. [, Corollary 1.3] of Aramayona and Martínez‐Pérez states that there is a cocompact model for

\underset{̲}{E} G

of minimal dimension

\underset{̲}{prefixgd} G = vcd G

. We need the following minor generalisation of this result.…”

Section: Mapping Class Groupsmentioning

confidence: 96%

“…Proof of Theorem Set

G = Mod (S)

and let

frakturF

and

frakturVc

be the families of finite and virtually cyclic subgroups of

G

, respectively, equipped with the commensurabilty relation on

Vc ∖ F

. By [, Corollary 1.3], there is a cocompact model for

\underset{̲}{E} G

of minimal dimension

\underset{̲}{prefixgd} G = vcd G

. By Proposition , for each infinite cyclic subgroup

H ⩽ G

, there are cocompact models for both

E_{F \cap N_{G} false[H false]} N_{G} false[H false]

and

E_{F [H]} N_{G} false[H false]

of dimension

vcd G

.…”

Section: Mapping Class Groupsmentioning

confidence: 99%

“…The main result of the paper is the following. Since gd Mod(S) = vcd Mod(S) (see [1]), we note that this answers a question by Wolfgang Lück which asks for which countable groups G the inequality gd G − 1 gd G gd G + 1 holds, see for example [15,Problem 9.51]. By Theorem 1.4, the above inequality is always satisfied for the mapping class group of any compact connected orientable surface S with a finite number of boundary components and punctures, and and χ(S) < 0.…”

Section: Introductionmentioning

confidence: 99%

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Hierarchically cocompact classifying spaces for mapping class groups of surfaces

Nucinkis

Petrosyan

2018

Bull. London Math. Soc.

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We define the notion of a hierarchically cocompact classifying space for a family of subgroups of a group. Our main application is to show that the mapping class group Mod (S) of any connected oriented compact surface S, possibly with punctures and boundary components and with negative Euler characteristic has a hierarchically cocompact model for the family of virtually cyclic subgroups of dimension at most vcd Mod (S)+1. When the surface is closed, we prove that this bound is optimal. In particular, this answers a question of Lück for mapping class groups of surfaces.

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