Homological Finiteness Conditions for Modules Over Group Algebras

Cornick, Jonathan; Kropholler, Peter H.

doi:10.1112/s0024610798005729

Cited by 40 publications

(27 citation statements)

References 8 publications

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“…Note that although ( [10], Theorem C) is stated for H F-groups, the given proof shows that the above inequality holds for arbitrary groups. Lemma 2.6 yields (iv).…”

Section: Proof the Connecting Mapsmentioning

confidence: 91%

Groups with Many Finitary Cohomology Functors

Kropholler

2016

Springer Proceedings in Mathematics &Amp; Statistics

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Abstract. For a group G, we study the question of which cohomology functors commute with all small filtered colimit systems of coefficient modules. We say that the functor H n (G, ) is finitary when this is so and we consider the finitary set for G, that is the set of natural numbers for which this holds. It is shown that for the class of groups LHF there is a dichotomy: the finitary set of such a group is either finite or cofinite. We investigate which sets of natural numbers n can arise as finitary sets for suitably chosen G and what restrictions are imposed by the presence of certain kinds of normal or near-normal subgroups. Although the class LHF is large, containing soluble and linear groups, being closed under extensions, subgroups, amalgamated free products, HNN-extensions, there are known to be many not in LHF such as Richard Thompson's group F . Our theory does not extend beyond the class LHF at present and so it is an open problem whether the main conclusions of this paper hold for arbitrary groups. There is a survey of recent developments and open questions. Organizational StatementThis paper lays the foundation stones for a series of papers by the author's former student Martin Hamilton: [13,11,12]. As sometimes happens, this literature has not been published in the order in which it was intended to be read and for this reason I am taking the opportunity of this conference proceedings to include a survey of Hamilton's papers and a discussion of possible future directions. This survey follows and expands upon the spirit of the talk I gave at the meeting.

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“…Note that although ( [10], Theorem C) is stated for H F-groups, the given proof shows that the above inequality holds for arbitrary groups. Lemma 2.6 yields (iv).…”

Section: Proof the Connecting Mapsmentioning

confidence: 91%

Groups with Many Finitary Cohomology Functors

Kropholler

2016

Springer Proceedings in Mathematics &Amp; Statistics

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“…In [2] it was shown that if G is an HF-group then Proof. It was essentially proved in [2] that for any group G…”

Section: O Talellimentioning

confidence: 99%

“…A crucial ingredient of the proof of Theorem I was the following result of Cornick and Kropholler [2]: if G is an HF-group with proj. dim ZG B(G, Z) < ∞ and M is a ZG-module, which is projective as a ZH-module for all finite subgroups H of G, then proj.…”

Section: Introductionmentioning

confidence: 99%

On groups of type Φ

Talelli

2007

Arch. Math.

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We define a group G to be of type Φ if it has the property that for every ZG-module G, proj. dim ZG G < ∞ iff proj. dim Z H G < ∞ for every finite subgroup H of G. We conjecture that the type Φ is an algebraic characterization of those groups G which admit a finite dimensional model for EG, the classifying space for the family of the finite subgroups of G. We also conjecture that the type Φ is equivalent to spli being finite, where spliZG is the supremum of the projective lengths of the injective ZG-modules. Here we prove certain parts of these conjectures.

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“…This is most conveniently expressed in terms of the G-operator ring B(G, Z) of bounded functions from G to Z. This ring was first applied in [10] where it was shown to be free as a ZH-module for each finite subgroup of G. The following is a result of Cornick and Kropholler [6].…”

Section: Notation and Strategymentioning

confidence: 99%

Groups acting on finite dimensional spaces with finite stabilizers

Kropholler

Mislin

1998

Commentarii Mathematici Helvetici

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Abstract. It is shown that every HF-group G of type FP∞ admits a finite dimensional G-CWcomplex X with finite stabilizers and with the additional property that for each finite subgroup H, the fixed point subspace X H is contractible. This establishes conjecture (5.1.2) of [9]. The construction of X involves joining a family of spaces parametrized by the poset of non-trivial finite subgroups of G and ultimately relies on the theorem of Cornick and Kropholler that if M is a ZG-module which is projective as a ZH-module for all finite H ≤ G then M has finite projective dimension.Mathematics Subject Classification (1991). Primary 20J05, 18G20; Secondary 55J05

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Homological Finiteness Conditions for Modules Over Group Algebras

Cited by 40 publications

References 8 publications

Groups with Many Finitary Cohomology Functors

Groups with Many Finitary Cohomology Functors

On groups of type Φ

Groups acting on finite dimensional spaces with finite stabilizers

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