# Survey on Classifying Spaces for Families of Subgroups

@article{Lueck2005SurveyOC, title={Survey on Classifying Spaces for Families of Subgroups}, author={Wolfgang Lueck}, journal={arXiv: Geometric Topology}, year={2005}, pages={269-322} }

We define for a topological group G and a family of subgroups \( \mathcal{F} \) two versions for the classifying space for the family \( \mathcal{F} \) , the G-CW-version \( E_\mathcal{F} \) (G) and the numerable G-space version \( J_\mathcal{F} \) (G). They agree if G is discrete, or if G is a Lie group and each element in \( \mathcal{F} \) compact, or if \( \mathcal{F} \) is the family of compact subgroups. We discuss special geometric models for these spaces for the family of compact open… Expand

#### 209 Citations

Classifying spaces for chains of families of subgroups

- Mathematics
- 2018

This thesis concerns the study of the Bredon cohomological and geometric dimensions of a discrete group $G$ with respect to a family $\mathfrak{F}$ of subgroups of $G$. With that purpose, we focus on… Expand

Some results related to finiteness properties of groups for families of subgroups

- Mathematics
- 2018

For a group $G$ we consider the classifying space $E_{\mathcal{VC}yc}(G)$ for the family of virtually cyclic subgroups. We show that an Artin group admits a finite model for $E_{\mathcal{VC}yc}(G)$… Expand

A cofinite universal space for proper actions for mapping class groups

- Mathematics
- 2008

We prove that the mapping class group $\Gamma_{g,n}$ for surfaces of negative Euler characteristic has a cofinite universal space $\E$ for proper actions (the resulting quotient is a finite… Expand

Invariants of hyperbolic 3-manifolds in relative group homology

- Mathematics
- 2013

Let $M$ be a complete oriented hyperbolic $3$--manifold of finite volume. Using classifying spaces for families of subgroups we construct a class $\beta_P(M)$ in the Adamson relative homology group… Expand

Curve complexes versus Tits buildings: structures and applications

- Mathematics
- 2012

Tits buildings\({\Delta }_{\mathbb{Q}}(\mathbf{G})\)of linear algebraic groupsGdefined over the field of rational numbers\(\mathbb{Q}\) have played an important role in understanding partial… Expand

$\boldsymbol{\mathfrak{F}}$-Structures and Bredon–Galois Cohomology

- Mathematics, Computer Science
- Appl. Categorical Struct.
- 2013

The relative Brauer group Br(L/K) of a finite separable non-normal extension of fields L/K as a second Bredon cohomology group is realized and it is shown that this approach is quite suitable for finding nonzero elements in Br(l/K). Expand

Dimension invariants of outer automorphism groups

- Mathematics
- 2016

The geometric dimension for proper actions $\underline{\mathrm{gd}}(G)$ of a group $G$ is the minimal dimension of a classifying space for proper actions $\underline{E}G$. We construct for every… Expand

Complete Bredon cohomology and its applications to hierarchically defined groups

- Mathematics
- Mathematical Proceedings of the Cambridge Philosophical Society
- 2016

Abstract By considering the Bredon analogue of complete cohomology of a group, we show that every group in the class $\cll\clh^{\mathfrak F}{\mathfrak F}$ of type Bredon-FP∞ admits a finite… Expand

On the dimension of the mapping class groups of a non-orientable surface.

- Mathematics
- 2020

Let $\mathcal{N}_g$ be the mapping class group of a non-orientable closed surface. We prove that the proper cohomological dimension, the proper geometric dimension, and the virtual cohomological… Expand

Groups with many finitary cohomology functors

- Mathematics
- 2016

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… Expand

#### References

SHOWING 1-10 OF 104 REFERENCES

Groups acting on finite dimensional spaces with finite stabilizers

- Mathematics
- 1998

Abstract. It is shown that every H
$ \frak {F} $-group G of type
$ \rm{FP}_\infty $ admits a finite dimensional G-CW-complex X with finite stabilizers and with the additional property that for each… Expand

Polycyclic–by–finite groups admit a bounded-degree polynomial structure

- Mathematics
- 1997

Abstract. For a polycyclic-by-finite group
$\Gamma$, of Hirsch length
$h$, an affine (resp. polynomial) structure is a representation of
$\Gamma$ into
${\rm Aff}({\Bbb R}^{h})$ (resp.
${\rm… Expand

The topology of discrete groups

- Mathematics
- 1980

In this paper we explore and exploit a relation between group theory and topology whose existence is suggested by a recent result of D. Kan and W. Thurston [14]: they show that for any connected… Expand

The type of the classifying space for a family of subgroups

- Mathematics
- 2000

Abstract The classifying space E(Γ, F ) for a family F of subgroups of a group Γ is defined up to Γ -homotopy as a Γ - CW -complex E(Γ, F ) such that E(Γ, F ) H is contractible if H belongs to F and… Expand

The virtual cohomological dimension of the mapping class group of an orientable surface

- Mathematics
- 1986

Let F = F ~ r be the mapping class group of a surface F of genus g with s punctures and r boundary components. The purpose of this paper is to establish cohomology properties of F parallel to those… Expand

Some examples of discrete group actions on aspherical manifolds

- Mathematics
- 2003

We construct two classes of examples of virtually torsion-free groups G acting properly cocompactly on contractible manifolds X. In the first class of examples, the universal space for proper actions… Expand

Manifolds with fundamental group a generalized free product. I

- Mathematics
- 1974

This announcement describes new methods in the study and classification of differentiable, PI or topological manifolds with infinite fundamental group. If, for example, Y + l is a closed manifold… Expand

On the Farrell-Jones conjecture for higher algebraic -theory

- Mathematics
- 2003

Here BΓ is the classifying space of the group Γ, and we denote by K−∞(R) the non-connective algebraic K-theory spectrum of the ring R. The homotopy groups of this spectrum are denoted Kn(R) and… Expand

On Algebraic and Geometric Dimensions for Groups with Torsion

- Mathematics
- 2001

We argue that the geometric dimension of a discrete group G ought to be defined to be the minimal dimension of a model for the universal proper G-space rather than the minimal dimension of a model… Expand

On the Universal Space for Group Actions with Compact Isotropy

- Mathematics
- 2003

LetG be a locally compact topological group and EG its universal space for the family of compact subgroups. We give criteria for this space to be G-homotopy equivalent to a d-dimensional G-CW… Expand