On the universal space for group actions with compact isotropy

Meintrup, David

doi:10.1090/conm/258/1778113

Cited by 41 publications

(38 citation statements)

References 17 publications

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“…Throughout this section, let G be a discrete group and let F be the family of finite subgroups of G. Recall that the geometric dimension for proper actions gd(G) of G is by definition the smallest possible dimension that a model for EG can have and note that if X is a model for EG, then the cellular chain complexes [30]). This implies that cd(G) gd(G).…”

Section: Geometric Versus Cohomological Dimensionmentioning

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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Section: Geometric Versus Cohomological Dimensionmentioning

confidence: 99%

Stable finiteness properties of infinite discrete groups

Bárcenas

Degrijse

Patchkoria

2017

Journal of Topology

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“…If G is a discrete subgroup of a Lie group L with finitely many path components, then for any maximal compact subgroup K ⊆ L, the space L/K with its left G-action is a model for EG. More information about EG can be found in [14,27,51,58,62].…”

Section: Novikovmentioning

confidence: 99%

K- and L-theory of Group Rings

Lück

2011

Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)

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Abstract. This article will explore the K-and L-theory of group rings and their applications to algebra, geometry and topology. The Farrell-Jones Conjecture characterizes K-and L-theory groups. It has many implications, including the Borel and Novikov Conjectures for topological rigidity. Its current status, and many of its consequences are surveyed. Mathematics Subject Classification (2000). Primary 18F25; Secondary 57XX.Keywords. K-and L-theory, group rings, Farrell-Jones Conjecture, topological rigidity. IntroductionThe algebraic K-and L-theory of group rings -K n (RG) and L n (RG) for a ring R and a group G -are highly significant, but are very hard to compute when G is infinite. The main ingredient for their analysis is the Farrell-Jones Conjecture. It identifies them with certain equivariant homology theories evaluated on the classifying space for the family of virtually cyclic subgroups of G. Roughly speaking, the Farrell-Jones Conjecture predicts that one can compute the values of these K-and L-groups for RG if one understands all of the values for RH, where H runs through the virtually cyclic subgroups of G.Why is the Farrell-Jones Conjecture so important? One reason is that it plays an important role in the classification and geometry of manifolds. A second reason is that it implies a variety of well-known conjectures, such as the ones due to Bass, Borel, Kaplansky and Novikov. (These conjectures are explained in Section 1.) There are many groups for which these conjectures were previously unknown but are now consequences of the proof that they satisfy the Farrell-Jones Conjecture. A third reason is that most of the explicit computations of K-and L-theory of group rings for infinite groups are based on the Farrell-Jones Conjecture, since it identifies them with equivariant homology groups which are more accessible via standard tools from algebraic topology and geometry (see Section 5).The rather complicated general formulation of the Farrell-Jones Conjecture is given in Section 3. The much easier, but already very interesting, special case of a torsionfree group is discussed in Section 2. In this situation the K-and L-groups are identified with certain homology theories applied to the classifying space BG. * The work was financially supported by the Leibniz-Preis of the author. The author wishes to thank several members and guests of the topology group in Münster for helpful comments.

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“…There is an algebraic dimension cdG that bears a close relationship to gdG, analogous to the relationship between cohomological dimension and the minimal dimension of an Eilenberg-Mac Lane space. It can be shown that cdG = gdG except that there may exist G for which cdG = 2 and gdG = 3, and cdG is an upper bound for the cohomological dimension of any torsion-free subgroup of G [23]. In view of this we may split the strong form of Brown's question into two parts, one geometric and one algebraic.…”

Section: Introductionmentioning

confidence: 99%

On dimensions of groups with cocompact classifying spaces for proper actions

Leary

Petrosyan

2017

Advances in Mathematics

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We construct groups G that are virtually torsion-free and have virtual cohomological dimension strictly less than the minimal dimension for any model for EG, the classifying space for proper actions of G. They are the first examples that have these properties and also admit cocompact models for EG. We exhibit groups G whose virtual cohomological dimension and Bredon cohomological dimension are two that do not admit any 2dimensional contractible proper G-CW-complex.

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On the universal space for group actions with compact isotropy

Cited by 41 publications

References 17 publications

Stable finiteness properties of infinite discrete groups

Stable finiteness properties of infinite discrete groups

K- and L-theory of Group Rings

On dimensions of groups with cocompact classifying spaces for proper actions

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