Cohomological finiteness conditions for elementary amenable groups

Kropholler, Peter H.; Martı́nez-Pérez, Conchita; Nucinkis, Brita E. A.

doi:10.1515/crelle.2009.090

Cited by 31 publications

(42 citation statements)

References 30 publications

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“…A version of this fact was conjectured by Gildenhuys and Strebel [10] and proved by the first author [16]. Subsequently this led to a proof that elementary amenable groups of type FP ∞ over Z are constructible [17] and to the construction of classifying spaces for proper actions for such groups, see [16,19,20]. (Results in [17,20] also apply considerably beyond the elementary amenable case.…”

Section: Background Materials and Historical Remarksmentioning

confidence: 93%

Homological dimension of elementary amenable groups

Kropholler

Martı́nez-Pérez

2019

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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In this paper we prove that the homological dimension of an elementary amenable group over an arbitrary commutative coefficient ring is either infinite or equal to the Hirsch length of the group. Established theory gives simple group theoretical criteria for finiteness of homological dimension and so we can infer complete information about this invariant for elementary amenable groups. Stammbach proved the special case of solvable groups over coefficient fields of characteristic zero in an important paper dating from 1970.

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Section: Background Materials and Historical Remarksmentioning

confidence: 93%

Homological dimension of elementary amenable groups

Kropholler

Martı́nez-Pérez

2019

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…where D * is obtained by applying the induction functor associated to M F A 5 → M F Γ to (21) and is therefore of the form…”

Section: An Examplementioning

confidence: 99%

Stable finiteness properties of infinite discrete groups

Bárcenas

Degrijse

Patchkoria

2017

Journal of Topology

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Abstract. Let G be an infinite discrete group. A classifying space for proper actions of G is a proper G-CW-complex X such that the fixed point sets X H are contractible for all finite subgroups H of G. In this paper we consider the stable analogue of the classifying space for proper actions in the category of proper G-spectra and study its finiteness properties. We investigate when G admits a stable classifying space for proper actions that is finite or of finite type and relate these conditions to the compactness of the sphere spectrum in the homotopy category of proper G-spectra and to classical finiteness properties of the Weyl groups of finite subgroups of G. Finally, if the group G is virtually torsion-free we also show that the smallest possible dimension of a stable classifying space for proper actions coincides with the virtual cohomological dimension of G, thus providing the first geometric interpretation of the virtual cohomological dimension of a group.

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“…This invariant is bounded from below by the virtual cohomological dimension vcd(G) of G, which is the cohomological dimension of any finite index torsion-free subgroup of G, and in many interesting cases these two quantities are in fact equal (e.g. see [15,9,1,19,27,2]). On the other hand, there are by now a number of examples showing that gd(G) can be strictly larger than vcd(G) (see [17,22,10,16]).…”

Section: Introductionmentioning

confidence: 99%

Dimension invariants of outer automorphism groups

Degrijse¹,

Souto

2017

Groups Geom. Dyn.

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Abstract. The geometric dimension for proper actions gd(G) of a group G is the minimal dimension of a classifying space for proper actions EG. We construct for every integer r ≥ 1, an example of a virtually torsion-free Gromov-hyperbolic group G such that for every group Γ which contains G as a finite index normal subgroup, the virtual cohomological dimension vcd(Γ) of Γ equals gd(Γ) but such that the outer automorphism group Out(G) is virtually torsion-free, admits a cocompact model for E Out(G) but nonetheless has vcd(Out(G)) ≤ gd(Out(G))−r.

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Cohomological finiteness conditions for elementary amenable groups

Abstract: It is proved that every elementary amenable group of type FP ∞ admits a cocompact classifying space for proper actions.

Cited by 31 publications

References 30 publications

Homological dimension of elementary amenable groups

Homological dimension of elementary amenable groups

Stable finiteness properties of infinite discrete groups

Dimension invariants of outer automorphism groups

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