Let \( G \) be a group and \( \mathcal F \) a family of subgroups such that:

• For all \( H \in \mathcal F, g \in G \) we have also \( H^g := gHg^{-1} \in \mathcal F _);
• For all \( H, H’ \in \mathcal F \) we have \( H \cap H’ \in \mathcal F \).

For example \( \mathcal F \) can be :

• The trivial subgroup;
• Finite subgroups;
• Virtually cyclic subgroups;
• Free abelian, nilpotent, … subgroups

A model for the classifying space \e E_{\mathcal F}G \) is then a CW-complex \( X \) with a \( G \)-action such that:

• If \( H \in \mathcal F \) then the subset \( X^H \) of points fixed by \( H \) is a contractible subcomplex of \( X \);
• Otherwise \( X^H \) is empty.

For example, if \( \mathcal F = \{\{e\}\} \) then \( E_{\mathcal F}G \) is the classifying space \( EG \) for \( G \). The classifying space can also be defined by the following universal property: it is the only G-complex \( Y \) such that for every \( G \)-action whose point stabilisers are in \( F \), there exists a \( G \)-map \( E_{\mathcal F}G \to Y \) (which is unique up to \( G \)-homotopy).

If \( E_{\mathcal F}G \) has a cocompact model the we say that \( G \) is of type \( F^{\mathcal F} \). In case \( \mathcal F \) is the family of finite subgroups we denote \( E_{\mathcal F}G = \underline E G, F^{\mathcal F} = \underline F \) and if \( F \) is the family of virtually cyclic subgroups \( E_{\mathcal F}G = \underline{\underline E} G,\)\( F^{\mathcal F} = \underline{\underline F} \).

We also define:

• The geometric dimension \( \mathrm{gd}_{\mathcal F}(G) \in {\mathbb Z}_{\ge 0} \cup\{\infty\} \) is the minimal dimension for a model of \( E_{\mathcal F}G \);
• If there exists a model for \( E_{\mathcal F}G \) whose \( k \)-skeleton is cocompact then we say that \( G \) is of type \( F_k^{\mathcal F} \), and if this holds for all \( k \) of type \( F_{\mathcal F}^\infty \).

Examples for \( \mathcal F \) = finite subgroups

• If \( G = D_\infty \) (the infinite dihedral group) the the real line is a model for \( \underline E G \).
• More generally, if \( G \) is a crystallographic group then the corresponding tesselation of Euclidean space is a model for \( \underline E G \).
• If \( G \) is hyperbolic then its Rips complex is a model for \( \underline E G \) (Meintrupp–Schick).
• If \( G = \mathrm{Out}(F_n) \) (resp. \( \mathrm{MCG}(S) \) then the spine of outer space (resp. of Teichmüller space) is a model for \( \underline E G \) (Vogtmann, Mislin).
• More generally, any group acting cocompactly on a CAT(0)-space has it as a model for \( \underline E G \).

Note that for crystallographic groups the geometric dimension \( \underline{\mathrm{gd}}(G) \) is equal to the Hirsch length of \( G \) (the sum \( \sum_i (G_i/G_{i+1})\otimes{\mathbb Q} \) is \( G_i \) is the derived series of \( G \)). This is true more generally for solvable groups, and even for elementary amenable groups (with the definition of Hirsch length suitably extended) by a result of Kropholler–Martínez-Pérez–Nucinkis. This is also equal to the virtual cohomological dimension \( \mathrm{vcd}(G) \).

For the mapping class group it also holds that \( \underline{\mathrm{gd}}(G) = \mathrm{vcd}(G) \) (Aramayona–Martínez-Pérez).

On the other hand, in general both \( \underline{\mathrm{gd}}(G) \) and the class \( \underline F_\infty \) do not behave well under finite extensions. Note that if \( N \to G \to Q \) is an extension with \( Q \) finite and \( E \) torsion-free, and \( X \) is a model for \( EN \) then \( E^Q \) is a model for \( \underline EG \). However it is not cocompact.

Leary–Nucinkis have examples where \( N \) is of type \( F \) but \( G \) is not of type \( \underline F \), and where \( \underline{\mathrm{gd}}(G) > \mathrm{gd}(N) \) with \( Q = \mathfrak A_5 \) (Martínez-Pérez has examples with \( Q = {\mathbb Z} / p \).

There are also examples where \( G \) is of type \( \underline F \) but still \( \underline{\mathrm{gd}}(G) > \mathrm{gd}(N) \) (Petrosyan–Leary, Degrisje–Souto).

The case where \( \mathcal F = \) virtually cyclic subgroups

In case \( G = {\mathbb Z}^2 \) a model for \( \underline{\underline E}G \) is given by the join of an infinite set of edges indexed by the cyclic subgroups (an edge is stabilised by its indexing subgroup). This is a non-cocompact model, but it shows that \( \underline{\underline{\mathrm{gd}}} (G) = 3 \).

Juan-Pineole and Leary construct a 2-dimensional model when \( G \) has a unique maximal virtually cyclic subgroup. They conjecture the following.

Conjecture: G has a cocompact \( \underline{\underline E}G \) if and only if it is virtually cyclic.

An easy result is the following.

Lemma: If \( G \) is \( \underline{\underline F}{}_\infty \) then all normalisers of virtually cyclic subgroups in \( G \) are as well.

Kochloukova–Martínez-Pérez–Nucinkis prove that for solvable \( G \) the conclusion of the lemma implies that \( G \) is polycyclic.

Groves–Wilson prove that an elementary amenable group which is \( \underline{\underline F}{}_0 \) has to be virtually cyclic.

The Lück–Weiermann construction

This is a construction used for almost all known classifying spaces.