Finiteness properties of groups

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 ${\displaystyle \pi _{1},\ldots ,\pi _{n}}$ 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.