Given an integer n ≥ 1, a group ${\displaystyle \Gamma }$  is said to be of type F_n if there exists an aspherical CW-complex whose fundamental group is isomorphic to ${\displaystyle \Gamma }$  (a classifying space for ${\displaystyle \Gamma }$ ) and whose n-skeleton is finite. A group is said to be of type F_∞ if it is of type F_n for every n. It is of type F if there exists a finite aspherical CW-complex of which it is the fundamental group.

For small values of n these conditions have more classical interpretations:

• a group is of type F₁ if and only if it is finitely generated (the rose with petals indexed by a finite generating family is the 1-skeleton of a classifying space, the Cayley graph of the group for this generating family is the 1-skeleton of its universal cover);

• a group is of type F₂ if and only if it is finitely presented (the presentation complex, i.e. the rose with petals indexed by a finite generating set and 2-cells corresponding to each relation, is the 2-skeleton of a classifying space, whose universal cover has the Cayley complex as its 2-skeleton).

It is known that for every n ≥ 1 there are groups of type F_n which are not of type F_n+1. Finite groups are of type F_∞ but not of type F. Thompson's group ${\displaystyle F}$  is an example of a torsion-free group which is of type F_∞ but not of type F.^[1]

A reformulation of the F_n property is that a group has it if and only if it acts properly discontinuously, freely and cocompactly on a CW-complex whose homotopy groups ${\displaystyle \pi _{0},\ldots ,\pi _{n-1}}$  vanish. Another finiteness property can be formulated by replacing homotopy with homology: a group is said to be of type FH_n if it acts as above on a CW-complex whose n first homology groups vanish.

Let ${\displaystyle \Gamma }$  be a group and ${\displaystyle \mathbb {Z} \Gamma }$  its group ring. The group ${\displaystyle \Gamma }$  is said to be of type FP_n if there exists a resolution of the trivial ${\displaystyle \mathbb {Z} \Gamma }$ -module ${\displaystyle \mathbb {Z} }$  such that the n first terms are finitely generated projective ${\displaystyle \mathbb {Z} \Gamma }$ -modules.^[2] The types FP_∞ and FP are defined in the obvious way.

The same statement with projective modules replaced by free modules defines the classes FL_n for n ≥ 1, FL_∞ and FL.

It is also possible to define classes FP_n(R) and FL_n(R) for any commutative ring R, by replacing the group ring ${\displaystyle \mathbb {Z} \Gamma }$  by ${\displaystyle R\Gamma }$  in the definitions above.

Either of the conditions F_n or FH_n imply FP_n and FL_n (over any commutative ring). A group is of type FP₁ if and only if it is finitely generated,^[2] but for any n ≥ 2 there exists groups which are of type FP_n but not F_n.^[3]

If a group is of type FP_n then its cohomology groups ${\displaystyle H^{i}(\Gamma )}$  are finitely generated for ${\displaystyle 0\leq i\leq n}$ . If it is of type FP then it is of finite cohomological dimension. Thus finiteness properties play an important role in the cohomology theory of groups.

Finite groups edit

A finite cyclic group ${\displaystyle G}$  acts freely on the unit sphere in ${\displaystyle \mathbb {R} ^{\mathbb {N} }}$ , preserving a CW-complex structure with finitely many cells in each dimension.^[4] Since this unit sphere is contractible, every finite cyclic group is of type F_∞.

The standard resolution ^[5] for a group ${\displaystyle G}$  gives rise to a contractible CW-complex with a free ${\displaystyle G}$ -action in which the cells of dimension ${\displaystyle n}$  correspond to ${\displaystyle (n+1)}$ -tuples of elements of ${\displaystyle G}$ . This shows that every finite group is of type F_∞.

A non-trivial finite group is never of type F because it has infinite cohomological dimension. This also implies that a group with a non-trivial torsion subgroup is never of type F.

Nilpotent groups edit

If ${\displaystyle \Gamma }$  is a torsion-free, finitely generated nilpotent group then it is of type F.^[6]

Geometric conditions for finiteness properties edit

Negatively curved groups (hyperbolic or CAT(0) groups) are always of type F_∞.^[7] Such a group is of type F if and only if it is torsion-free.

As an example, cocompact S-arithmetic groups in algebraic groups over number fields are of type F_∞. The Borel–Serre compactification shows that this is also the case for non-cocompact arithmetic groups.

Arithmetic groups over function fields have very different finiteness properties: if ${\displaystyle \Gamma }$  is an arithmetic group in a simple algebraic group of rank ${\displaystyle r}$  over a global function field (such as ${\displaystyle \mathbb {F} _{q}(t)}$ ) then it is of type F_r but not of type F_r+1.^[8]